From Local Mutations to Global Fixation: A Semigroup Approach to Evolutionary Collapse

Abstract

In a previous paper the authors initiated a study of mutation semigroups, where elementary mutation operations were encoded as total maps on finite sets and analyzed through structural, algebraic, and computational methods. Here we address several of the open problems raised therein. First, we investigate the algebraic characterization of generator sets that force the existence of constant or low-rank maps, linking these conditions to classical results on synchronizing automata. Second, we analyze the computational complexity of contraction-based heuristics, identifying cases where polynomial-time criteria are achievable and others where hardness results emerge. Finally, we discuss connections with quasispecies models in biology and interpret image contractions as mechanisms of error suppression and genomic stability, while noting that rigorous extension to infinite state spaces remains future work. By combining algebraic definitions, structural theorems, and algorithmic analyses, we provide a refined toolkit for understanding mutation collapse and its theoretical implications, with potential applications that require empirical validation beyond the scope of this paper.

1. Introduction

Mutation processes can be modeled algebraically by viewing each elementary mutation as a total transformation on a finite state space. In our earlier work [1], this viewpoint was formalized using transformation semigroups generated by mutation maps, with structural behaviour analysed through Green’s relations, image–kernel decompositions, and rank considerations. Foundational aspects of these tools trace back to classical semigroup theory, particularly the treatment of Green’s relations and congruences in (Chs. 2–4, [2]) and the structural theory of transformation semigroups in (Ch. 1, [3]). Within this framework, low-rank transformations—and especially rank-one maps—were shown to correspond precisely to synchronizing words in the deterministic automaton naturally associated with the generator set. Computationally effective methods, including the pair-graph algorithm and power-set automaton techniques, provided systematic ways to detect such collapsing behaviour, connecting our approach with established results in the algorithmic theory of semigroups and formal languages as developed in (Ch. 3, [4]) and the early complexity analyses of (§2, [5]).

Although these earlier contributions produced a coherent algebraic and algorithmic basis for mutation analysis, several important questions remained open. Foremost among them is the problem of determining intrinsic algebraic conditions on generator sets that guarantee the existence of low-rank or constant maps. Additionally, the computational complexity of contraction-based heuristics is not yet fully understood when mutation systems become large, irregular, or highly asymmetric. A further complication arises when mutation processes are parameterized or extended to infinite settings, where finite-state semigroup methods must be strengthened or generalized.

Recent progress in related areas gives strong motivation for addressing these gaps. Advances in local-rule dynamics have shown that minor structural changes in rule sets can create global contraction or stabilization effects, offering conceptual parallels to rank collapse in mutation semigroups, as demonstrated in (§4, [6]). Likewise, developments in genome-scale rearrangement theory highlight how elementary mutation types shape the geometry and accessibility of large combinatorial state spaces, with consequences for both biological modelling and algorithm design, as discussed in (Chs. 2–3, [7]). Concurrently, new results in operator and computational semigroup theory demonstrate that rank-reducing transformations play a decisive role in determining the structural complexity of transformation systems, with refined analysis appearing in (§5, [8]). These insights point to a broader mathematical context in which mutation semigroups naturally reside.

Novelty and Significance of the Present Study

The present paper extends the existing theory in several essential directions. First, it identifies precise algebraic conditions—framed in terms of transitivity of unit groups, invariant partitions, and image–kernel obstructions—that determine when a mutation semigroup must contain low-rank or rank-one elements. This sharpens the earlier collapse criteria and produces a systematic method for predicting synchronizing behaviour from structural properties of the generators alone. Second, it provides a complete algorithmic treatment of the pair-graph collapse test and the search for invariant partitions, including correctness proofs, complexity bounds, and practical heuristics supported by explicit worked examples. Third, it develops a general transitivity threshold theorem that unifies multiple collapse phenomena under a single algebraic principle: the degree of transitivity of the units determines the ranks that can be forced to collapse.

Together, these contributions yield a mathematically rigorous and computationally implementable framework that advances both the theory of transformation semigroups and their application to mutation processes. The results clarify the mechanisms through which mutational systems lose diversity, formalize the algebraic structure underlying mutation collapse, and provide tools for analysing resilience thresholds in biological and computational dynamical systems.

The remainder of the paper develops these themes in detail, starting with the algebraic preliminaries and definitions needed for mutation models, followed by structural theorems, algorithmic analyses, and classification results, culminating in biological interpretations and general conclusions.

The algebraic framework developed in this paper suggests potential conceptual analogies with mutation-driven processes in infectious disease evolution, agricultural pathogen dynamics, and conservation genetics. The transitivity thresholds we characterize offer mathematically suggestive predictions for when genetic diversity is preserved versus lost, though empirical validation would require stochastic extensions and biological data beyond the scope of this paper.

Clarification of novelty: The present paper builds upon classical results in semigroup theory and synchronizing automata. For completeness, Section 3 recalls foundational lemmas (power-set reachability, pair-graph criterion, minimal ideal characterization) which are not claimed as original. The genuinely new contributions of this paper are (i) the transitivity threshold hierarchy (Theorem 6 and Corollary 1), which links the degree of k-transitivity of the unit group to the rank that can be forced to collapse; (ii) the complexity analysis of invariant partition detection, including NP-membership proof; and (iii) the partial classification results for projection-preserving generators and symmetric-unit criteria.

Relation to classical literature: The foundational results on synchronizing automata date back to Černý in 1964 [9], with comprehensive treatments by Pin [4] and Volkov [10,11]. The semigroup-theoretic background (Green’s relations, minimal ideals) is standard; see Howie [2]. Our contribution is not to replace these foundations but to synthesize them into a unified framework for mutation analysis, with the novel elements being the transitivity threshold hierarchy and the algorithmic analysis of invariant partitions.

2. Preliminaries

We assume familiarity with basic concepts introduced in our earlier work, including mutation semigroups [1], rank of a map, and synchronizing words. We briefly recall the relevant definitions for completeness. In addition, although the present study does not employ basis-pruning procedures, we note that semigroups generated by finite families of transformations were previously examined in the context of computing minimal generating sets and dependence structures (Section 2 and Section 3, [12]), providing a complementary algorithmic background to the generator-based viewpoint used here.

Definition 1

(Mutation Semigroup [2,3]). Let X be a finite set. A mutation semigroup is the semigroup $\mathcal{M}$ generated by a collection of total maps $f_i:X\to X$ representing elementary mutations. The rank of $f\in \mathcal{M}$ is $|Im(f\left)\right|$.

Definition 2

(Constant and Low-Rank Maps [8,13]). A map $f\in \mathcal{M}$ is called constant if $Im\left(f\right)$ is a singleton. More generally, f is low-rank if $rank\left(f\right)$ is bounded by a small integer relative to $\left|X\right|$.

Throughout, we focus on identifying conditions under which constant or low-rank maps must exist, and on analyzing the complexity of algorithms that find them.

3. Characterization of Generator Sets

In this section we study algebraic conditions on generating families of transformations that force the existence of constant or, more generally, low-rank elements in the generated mutation semigroup. Throughout let X be a finite set with $\left|X\right|=m$, let $G\subseteq {\mathcal{T}}_X$ be a finite generating set and write $S=\langle G\rangle \le {\mathcal{T}}_X$. We assume the reader is familiar with the definitions of rank, image, kernel, and the action homomorphism for free semigroups; these are given in the preceding paper and recalled briefly where convenient.

Definition 3.

Let $S=\langle G\rangle \le {\mathcal{T}}_X$.

1.

For $k\in \{1,\dots ,m\}$ we say G (or S) isk-collapsing if there exists $f\in S$ with $\mathrm{rank}\left(f\right)\le k$.

2.

We say G (or S) is synchronizing if it is 1-collapsing, i.e., if S contains a constant map.

The basic characterisations we use are automata-theoretic [8,14]: reachability in the power-set automaton captures existence of low-rank elements, while pair collapse captures synchronizing behaviour. We present these statements with full proofs for completeness.

Note on classical material: For completeness, we recall several classical results from semigroup and automata theory. These are not claimed as original but are necessary for the self-contained development that follows. The new contributions begin in Section 4.

Notation convention: Throughout, $G^{+}$ denotes the set of non-empty words over G, while $G^{*}=G^{+}\cup \left\{\epsilon \right\}$ includes the empty word $\epsilon$ (which corresponds to the identity transformation). The identity map is included only when explicitly noted.

Lemma 1

(Power-set reachability criterion). Let X be a finite set and $S\le {\mathcal{T}}_X$ a mutation semigroup. Then a constant map exists in S if and only if the power-set automaton of S admits a breadth-first search path from X to a singleton.

Proof. 

(⇒) If $f\in S$ satisfies $\mathrm{rank}\left(f\right)\le k$ then, by definition of S, there is a word $w\in G^{+}$ with $\phi \left(w\right)=f$ (where $\phi :G^{+}\to {\mathcal{T}}_X$ is the action homomorphism). Put $T:=\mathrm{Im}\left(f\right)$. Then $\left|T\right|=\mathrm{rank}\left(f\right)\le k$ and $w\left(X\right)=f\left(X\right)=T$, so $w\left(X\right)\subseteq T$.

(⇐) Conversely, if there exists $T\subseteq X$ with $\left|T\right|\le k$ and $w\in G^{+}$ such that $w\left(X\right)\subseteq T$, then the transformation $f:=\phi \left(w\right)\in S$ satisfies $\mathrm{Im}\left(f\right)=w\left(X\right)\subseteq T$, and hence $\mathrm{rank}\left(f\right)\le k$.    □

Classical source: This criterion is standard in automata theory; see Pin [4] and Birget [5].

The next lemma is the classical pair-graph criterion for synchronizing words (see Lemma 2) [4,5]. We include a direct proof since it is central to the algebraic viewpoint.

Lemma 2

(Pair-graph criterion). The generating set G (equivalently the automaton $(X,G)$) is synchronizing if and only if for every unordered pair $\{x,y\}\subseteq X$ there exists a word $w\in G^{+}$ with $w\left(x\right)=w\left(y\right)$.

Proof. 

(⇒) If G is synchronizing then there exists $w\in G^{+}$ with $w\left(X\right)=\left\{t\right\}$ for some $t\in X$. For any pair $\{x,y\}$ we then have $w\left(x\right)=t=w\left(y\right)$, so every pair collapses.

(⇐) Suppose every unordered pair $\{x,y\}$ is collapsible, i.e., for each pair there exists $w_{x,y}\in G^{+}$ with $w_{x,y}\left(x\right)=w_{x,y}\left(y\right)$. We construct a synchronizing word by induction on the size of the image. Let $U_0:=X$. If $|U_0| =1$ we are done. Otherwise pick two distinct elements $a,b\in U_0$; by hypothesis there exists $u\in G^{+}$ with $u\left(a\right)=u\left(b\right)$. Consider the image $U_1:=u\left(U_0\right)$. Then $|U_1| \le |U_0|-1$ because at least a and b collide under u. Repeating this procedure yields a sequence of words whose concatenation maps X to a singleton. Concretely, select $u_1$ collapsing some pair of $U_0$ to obtain $U_1$ with $|U_1| < |U_0|$; then select $u_2$ collapsing some pair of $U_1$, and so on. After at most $m-1$ steps the image size becomes 1. The concatenated word $w=u_t\cdots u_2{u}_1$ is therefore synchronizing.    □

Classical source: This is the classical pair-graph criterion; see Černý [9] and Volkov [10].

The previous two lemmas reduce questions about the existence of low-rank elements to reachability problems in finite directed graphs (power-set automaton and pair-graph). We now present algebraic sufficient conditions that can be checked within G itself.

Proposition 1

(Contracting-generator sufficient condition). Suppose there exists a generator $\mu \in G$ and a nonempty subset $U\subseteq X$ such that $\left|\mu \right(U\left)\right|<\left|U\right|$ and X can be mapped into U by some word in $G^{+}$; that is, there exists $w_0\in G^{+}$ with $w_0\left(X\right)\subseteq U$. Then S contains an element of rank strictly less than $\left|U\right|$; indeed $\mu \circ w_0\in S$ has image $\mu \left(w_0\left(X\right)\right)$ with cardinality $\left|\mu \right(w_0\left(X\right)\left)\right|\le \left|\mu \left(U\right)\right|<\left|U\right|$.

Proof. 

By hypothesis there exists $w_0\in G^{+}$ with $w_0\left(X\right)\subseteq U$. Consider $f:=\mu \circ w_0=\phi \left(\mu w_0\right)\in S$. Its image is

$$\mathrm{Im}\left(f\right)=\mu \left(w_0\left(X\right)\right)\subseteq \mu \left(U\right),$$

so $\left|\mathrm{Im}\right(f\left)\right|\le \left|\mu \right(U\left)\right|<\left|U\right|$. Thus f has rank strictly less than $\left|U\right|$. Iterating this argument when further contractions exist yields progressively smaller images and, if possible, ultimately a constant map.    □

The above gives a readily checkable sufficient condition in many practical cases: find a generator that strictly reduces the size of a reachable subset. In some structured families of generators this is an effective certificate of collapse.

We now turn to intrinsic algebraic characterisations that make use of the minimal ideal of a finite semigroup. These are useful because the minimal ideal is an invariant of the semigroup and therefore depends only on the algebra generated by G.

Definition 4.

For a finite semigroup S, the minimal rank of S is

$$r_{min}\left(S\right):=min\{\mathrm{rank}\left(f\right):f\in S\}.$$

Denote by $M_r\left(S\right):=\{f\in S:\mathrm{rank}\left(f\right)=r\}$ the set of elements of rank r.

Theorem 1.

Let $S\le {\mathcal{T}}_X$ be a finite transformation semigroup. Then the minimal ideal $K\left(S\right)$ of S coincides with $M_{r_{min}\left(S\right)}\left(S\right)$, the set of elements of minimal rank. In particular, S contains a constant map if and only if $r_{min}\left(S\right)=1$, and equivalently $K\left(S\right)$ contains an element of rank 1.

Proof. 

Let $r:=r_{min}\left(S\right)$. First we show that every element of rank r lies in $K\left(S\right)$. Fix $f\in S$ with $\mathrm{rank}\left(f\right)=r$. For any $g\in S$ we have $\mathrm{rank}(g\circ f)\le \mathrm{rank}\left(f\right)=r$ (composition cannot increase image size). By minimality of r we must have $\mathrm{rank}(g\circ f)=r$. Hence $g\circ f$ also has rank r for all $g\in S$; it follows that $SfS\subseteq M_r\left(S\right)$. In particular, $SfS$ is an ideal of S contained in the set of rank-r elements. Since S is finite and r is minimal, the two-sided ideal generated by any minimal-rank element is minimal among nonempty ideals; therefore $SfS$ is the unique minimal ideal $K\left(S\right)$. Consequently $M_r\left(S\right)\subseteq K\left(S\right)$.

Conversely, suppose $h\in K\left(S\right)$. Because $K\left(S\right)$ is a minimal ideal, for every $g\in S$ the product $gh$ and $hg$ belong to $K\left(S\right)$. In particular, there exists some $f\in S$ with $\mathrm{rank}\left(f\right)=r$ and $f\in K\left(S\right)$ (since $K\left(S\right)$ contains some element and the minimal rank r is attained in S). From closure of $K\left(S\right)$ under multiplication we deduce that $K\left(S\right)$ consists only of elements of rank at least r, but minimality of r forces every element of $K\left(S\right)$ to have rank exactly r. Thus $K\left(S\right)\subseteq M_r\left(S\right)$.

Combining the two inclusions yields $K\left(S\right)=M_r\left(S\right)$. The final equivalence follows immediately: S contains a constant map iff $r_{min}\left(S\right)=1$, which is equivalent to $K\left(S\right)$ containing an element of rank one.    □

Classical source: The characterization of the minimal ideal in transformation semigroups is due to Howie [2] and Clifford & Preston [15].

Theorem 1 recasts the ‘existence of constant (or low-rank) maps’ question as a question about the minimal ideal, an algebraic invariant of S. In practice one can attempt to compute or bound $r_{min}\left(S\right)$ from generators.

The next proposition supplies an algebraic sufficient condition which is stronger than Proposition 1 but phrased entirely in terms of the existence of certain multipliers in S.

Proposition 2

(Witness-multiplier criterion). Let $a,b\in S$. Suppose $\mathrm{Im}\left(a\right)=\mathrm{Im}\left(b\right)=Y$ and there exist $r,s\in S$ such that $a\circ r=b$ and $b\circ s=a$. Then a and b lie in the same ${\mathcal{R}}_S$-class of S and have the same rank.

Proof. 

See Appendix A for the detailed proof.    □

Remark 1.

Proposition 2 indicates that when the subsemigroup S contains the appropriate multipliers (right/left multipliers), the full-transformation characterisations of Green’s relations restrict to S. This gives an algebraic pathway to certify that particular image equalities arising from generator combinations are in fact realised within S.

We conclude this section with an example that demonstrates how the algebraic and automata-theoretic criteria interplay in a nontrivial generator set.

Example 1

(A synchronizing generator set). Let $X=\{1,2,3\}$ and let $G=\{\alpha ,\beta \}\subseteq {\mathcal{T}}_X$ be given by

$$\alpha =(1\mapsto 1,2\mapsto 1,3\mapsto 2),\beta =(1\mapsto 2,2\mapsto 3,3\mapsto 3).$$

Recall from the examples in the first paper that ${\alpha }^2$ is constant with image $\left\{1\right\}$. We verify this algebraically here.

Observe $\mathrm{Im}\left(\alpha \right)=\{1,2\}$ and $\mathrm{Im}\left(\beta \right)=\{2,3\}$. The pair $\{1,2\}$ collapses under α since $\alpha \left(1\right)=\alpha \left(2\right)=1$. Thus by Lemma 2 the set of all unordered pairs is collapsible (one checks the remaining pairs collapse under suitable words), so G is synchronizing. Concretely, $\alpha \left(X\right)=\{1,2\}$ and $\alpha \left(\right\{1,2\left\}\right)=\left\{1\right\}$ so ${\alpha }^2$ maps X to $\left\{1\right\}$, producing a constant map in S. From the algebraic perspective $r_{min}\left(S\right)=1$ and by Theorem 1 the minimal ideal $K\left(S\right)$ equals the set of rank-one elements of S (here containing the single constant ${\alpha }^2$ and possibly others).

Remark 2

(On the limits of algebraic characterisation). The results above give several algebraic handles for certifying collapse: power-set reachability, pair-graph collapse, contracting generators, and minimal-ideal analysis. However a complete structural classification (necessary and sufficient purely algebraic conditions on G that guarantee synchronizing behaviour for arbitrary G) is elusive: the problem is equivalent to characterising synchronizing automata in purely semigroup-theoretic terms, a programme that intersects deep questions such as the Černý conjecture [10,11] and structural descriptions of transformation semigroups. Nevertheless, the propositions here provide effective sufficient criteria and algebraic invariants that can be computed or bounded from generators in many practical settings.

4. Generator Sets Forcing Low-Rank Maps

One of the central problems raised in [1] by Sampson and George is the characterization of generator sets that necessarily enforce the existence of low-rank or constant maps in mutation semigroups. In this section we establish several foundational results in this direction. Throughout, we let X be a finite nonempty set, and ${\mathcal{T}}_X$ the full transformation semigroup on X.

Definition 5

(Low-rank enforcing generator set). Let $G\subseteq {\mathcal{T}}_X$ and $S=\langle G\rangle \le {\mathcal{T}}_X$. We say that G is low-rank enforcing if there exists $r<\left|X\right|$ such that every subsemigroup of S generated by a subset of G contains an element of rank at most r.

Lemma 3

(Sufficient condition). If $G\subseteq {\mathcal{T}}_X$ contains a constant map, then G is low-rank enforcing with $r=1$.

Proof. 

Immediate, since the constant map belongs to $\langle G\rangle$ and has rank 1.    □

Proposition 3

(Collapsibility criterion). Let $G\subseteq {\mathcal{T}}_X$. If for every pair $\{x,y\}\subseteq X$ there exists $g\in G$ with $xg=yg$, then G is low-rank enforcing with $r=1$.

Proof. 

The hypothesis ensures that G collapses all pairs of states in X. Equivalently, the pair-graph associated with G is strongly connected and admits a path from any pair $(x,y)$ to a diagonal $(z,z)$. By the classical synchronizing word criterion (see Volkov [10]), $\langle G\rangle$ contains a constant map; hence G is low-rank enforcing.    □

Theorem 2

(Rank reduction by composition). Let $G\subseteq {\mathcal{T}}_X$ be such that there exists $g\in G$ with $\left|\mathrm{Im}\right(g\left)\right|<\left|X\right|$. Then G is low-rank enforcing with $r=\left|\mathrm{Im}\right(g\left)\right|$.

Proof. 

Let $U=\mathrm{Im}\left(g\right)⊊X$. For any $h\in \langle G\rangle$, the composition $hg$ maps X into U; hence $\mathrm{rank}\left(hg\right)\le \left|U\right|<\left|X\right|$. Thus $hg$ is a low-rank element in S, and every generating set containing g enforces low-rank behavior with $r=\left|U\right|$.    □

Example 2

(Synchronizing generator set). Let $X=\{1,2,3\}$ and

$$g_1=\left(\begin{array}{ccc}1& 2& 3\\ 2& 2& 3\end{array}\right),g_2=\left(\begin{array}{ccc}1& 2& 3\\ 1& 3& 3\end{array}\right).$$

Then $G=\{g_1,g_2\}$ is low-rank enforcing. In fact, repeated applications of $g_1$ and $g_2$ collapse $\{1,2,3\}$ to $\left\{3\right\}$, producing a constant map.

The above results provide a preliminary algebraic foundation for classifying low-rank enforcing generator sets. In subsequent sections we develop a complexity-theoretic perspective on their detection and enumeration.

5. Towards a Classification: Necessary and Sufficient Conditions

This section refines the sufficient criteria of the previous section by proving equivalences and natural necessary conditions. Our aim is not to offer a complete classification (which remains difficult in general), but rather to assemble algebraic and combinatorial criteria that together narrow the class of possible synchronizing or low-rank enforcing generators, and that will serve as the backbone of further structural analysis.

Throughout let X be a finite set with $\left|X\right|=m\ge 2$, let $G\subseteq {\mathcal{T}}_X$ be a finite set of generators and write $S=\langle G\rangle \le {\mathcal{T}}_X$.

Definition 6

(Invariant partition/block system). A partition Π of X is said to be S-invariant (or preserved by S) if for every block $B\in \mathrm{\Pi }$ and every $s\in S$ there exists a block $B^{\prime }\in \mathrm{\Pi }$ with $s\left(B\right)\subseteq B^{\prime }$. If Π has more than one block and is S-invariant we call it a nontrivial block system for S.

The first result is a clean characterization that collects several equivalent viewpoints (automata-theoretic, rank-theoretic, ideal-theoretic).

Theorem 3

(Characterisations of synchronizing behaviour). The following statements are equivalent:

1.

S is synchronizing; i.e., S contains a constant map.

2.

The pair-graph of $(X,G)$ is collapsing: for every unordered pair $\{x,y\}\subseteq X$ there exists $w\in G^{+}$ with $w\left(x\right)=w\left(y\right)$.

3.

$r_{min}\left(S\right)=1$; i.e., the minimal rank of elements of S is 1.

4.

The minimal ideal $K\left(S\right)$ contains an idempotent of rank 1 (equivalently, $K\left(S\right)$ contains a constant map).

Proof. 

(1) ⇒ (2): If $c\in S$ is constant with $c\left(X\right)=\left\{t\right\}$ then for any $x,y\in X$ we have $c\left(x\right)=t=c\left(y\right)$; hence every pair collapses under the single word representing c.

(2) ⇒ (1): This is the classical pair-graph criterion (see Lemma 2): if every unordered pair is collapsible then one may iteratively construct a word that reduces the image size by at least one at each step; after at most $m-1$ steps a singleton image is obtained; i.e., a synchronizing word exists and therefore S contains a constant map.

(1) ⇔ (3): By definition S contains a constant map iff $r_{min}\left(S\right)=1$.

(3) ⇔ (4): Theorem 1 (from the previous paper/Section 3) shows that the unique minimal ideal $K\left(S\right)$ equals the set of elements of minimal rank. Hence $r_{min}\left(S\right)=1$ iff $K\left(S\right)$ contains elements of rank 1. Since constant maps are idempotent, the existence of a rank-1 element in $K\left(S\right)$ is equivalent to $K\left(S\right)$ containing an idempotent of rank 1.    □

The equivalences above are useful but largely restate known connections between synchronizing automata and transformations [2,3]. We now extract a practical necessary condition that places algebraic restrictions on the generator set.

Proposition 4

(Invariant partitions as an obstruction). Suppose there exists a partition Π of X into $t\ge 2$ blocks all of size at least 2 such that Π is S-invariant. Then S is not synchronizing. Equivalently, the existence of such a nontrivial block system is an obstruction to G being synchronizing.

Proof. 

Assume $\mathrm{\Pi }=\{B_1,\dots ,B_t\}$ is an S-invariant partition with each $|B_i|\ge 2$. For any $s\in S$ and any block $B_i$ we have $s\left(B_i\right)\subseteq B_j$ for some j by invariance. Therefore for every word $w\in G^{+}$ the image $w\left(X\right)$ is a union of whole blocks of $\mathrm{\Pi }$. In particular $\left|w\right(X\left)\right|$ is at least the minimum block size, which is at least 2. Hence no word can map X to a singleton; that is, no synchronizing word exists and S is not synchronizing.    □

Remark 3.

The requirement that each block has size at least 2 is essential in the statement above: if Π has a block of size 1 then a synchronizing word could map all of X into that singleton (so Π does not block synchronization). Thus a nontrivial S-invariant block system whose blocks are all of size $\ge 2$ is a genuine obstruction.

The block obstruction gives a simple algebraic test for impossibility of synchronization: if one can find a nontrivial invariant partition with no singleton blocks, then G cannot be synchronizing. This gives a necessary (but not sufficient) structural condition.

We complement the obstruction with a local necessary condition that is directly checkable on the generators.

Proposition 5

(Pair-separating necessity). If G is synchronizing then for every proper subset $U⊊X$ there exists $w\in G^{+}$ with $\left|w\right(U\left)\right|<\left|U\right|$ (i.e., every proper subset can be strictly contracted by some word).

Proof. 

If G is synchronizing then there exists $c\in S$ constant with $c\left(X\right)=\left\{t\right\}$. For a given proper subset $U⊊X$ consider $c\left(U\right)=\left\{t\right\}$; since $\left|c\right(U\left)\right|=1<\left|U\right|$ the constant transformation contracts U. The statement follows by taking w to be any word representing c. Thus contraction of every proper subset is necessary in the sense that some element of S (possibly the global constant) must contract it.    □

Remark 4.

Proposition 5 is a weak necessity statement: while it is true that some $w\in S$ must contract each proper subset (in the synchronizing case), this word might be the global synchronizing word itself and therefore the condition is not very discriminating for algorithmic detection. Nevertheless it provides a conceptual constraint: synchronizing semigroups cannot leave any nontrivial subset rigid under all words.

We conclude with a structural consequence that links synchronizing behaviour to the permutation subgroup of units in S.

Proposition 6

(Permutational symmetry and synchronizing maps). Let $U\left(S\right)$ denote the group of units of S. If S is synchronizing and $U\left(S\right)$ acts transitively on X, then all constant maps in S have the same image and are conjugate to each other via $U\left(S\right)$. Moreover, if $K\left(S\right)$ contains exactly one constant map, then that constant map is central in S.

Proof. 

Suppose S contains a constant map and $U\left(S\right)$ acts transitively on X. By Theorem 1, the minimal ideal $K\left(S\right)$ consists of all rank-1 elements of S. Let $c_1,c_2\in K\left(S\right)$ be constant maps with images $\left\{x\right\}$ and $\left\{y\right\}$ respectively. By transitivity of $U\left(S\right)$, there exists $u\in U\left(S\right)$ such that $u\left(x\right)=y$. Consider $u{c}_1{u}^{-1}\in S$. For any $z\in X$,

$$\left(u{c}_1{u}^{-1}\right)\left(z\right)=u\left(c_1\left(u^{-1}\left(z\right)\right)\right)=u\left(x\right)=y,$$

so $u{c}_1{u}^{-1}$ is the constant map with image $\left\{y\right\}$. Since $K\left(S\right)$ is the minimal ideal, it is closed under conjugation: $u{c}_1{u}^{-1}\in K\left(S\right)$. But $c_2$ is also the constant map with image $\left\{y\right\}$ and has rank 1, so both are in $K\left(S\right)$. Thus all constant maps in S are conjugate and share the same image (up to permutation). If $K\left(S\right)$ contains exactly one constant map c, then for any $s\in S$, $sc$ and $cs$ are constant maps in $K\left(S\right)$ (since c has minimal rank), and hence must equal c. Therefore c is central.    □

Remark 5.

The hypothesis that $U\left(S\right)$ is transitive is strong and need not hold in many mutation semigroups (where generators are local and rarely invertible [16,17,18]). Nevertheless Proposition 6 illustrates how permutation symmetry constrains the algebraic structure when synchronizing maps exist.

Discussion

Combining Theorem 3, Proposition 4, and the subsequent necessary conditions yields a toolbox for attacking the classification problem. In practice one may first test for invariant block systems (an inexpensive combinatorial check), then inspect the pair-graph criterion; if both tests are inconclusive the minimal-ideal approach and unit-group considerations can provide further structural insight. Full necessary-and-sufficient algebraic criteria valid for arbitrary generator sets remain elusive; the results above delineate the algebraic landscape in which such a classification must live [19,20,21].

6. Worked Examples, Algorithms, and Partial Classifications

In this section we give detailed worked examples illustrating the block-obstruction and permutational symmetry phenomena, present rigorous algorithms to test the pair-graph collapse and to search for nontrivial invariant partitions (with complexity analyses), and prove partial classification theorems in two natural special families of generator sets.

6.1. Worked Examples

Example 3

(Block obstruction—explicit construction). Let $X=\{1,2,3,4\}$ and let

$$G=\{g_1,g_2\}\subseteq {\mathcal{T}}_X$$

be given by their action on X as follows:

$$g_1:1\mapsto 1,2\mapsto 2,3\mapsto 3,4\mapsto 3,g_2:1\mapsto 2,2\mapsto 2,3\mapsto 4,4\mapsto 4.$$

Consider the partition $\mathrm{\Pi }=\left\{\right\{1,2\},\{3,4\left\}\right\}$ of X.

Verification: For each block $B\in \mathrm{\Pi }$ and each $g\in G$ we check that $g\left(B\right)$ lies inside a single block of Π. Indeed,

$$g_1\left(\{1,2\}\right)=\{1,2\}\subseteq \{1,2\},g_1\left(\{3,4\}\right)=\left\{3\right\}\subseteq \{3,4\},$$

and

$$g_2\left(\{1,2\}\right)=\left\{2\right\}\subseteq \{1,2\},g_2\left(\{3,4\}\right)=\left\{4\right\}\subseteq \{3,4\}.$$

Thus Π is S-invariant for $S=\langle G\rangle$. Since every block of Π has size $\ge 2$, Proposition 4 from Section 5 applies and shows S is not synchronizing: no word in $G^{+}$ maps X to a singleton. Concretely, every image $w\left(X\right)$ is a union of whole blocks, so $\left|w\right(X\left)\right|\ge 2$ for all $w\in G^{+}$.

Example 4

(Permutational symmetry—explicit verification). Let $X=\{1,2,3\}$, let $H={\mathfrak{S}}_3$ denote the full symmetric group on X, and let $p\in {\mathcal{T}}_X$ be the constant map $p\left(x\right)=1$ for all $x\in X$. Set

$$G=H\cup \left\{p\right\},S=\langle G\rangle .$$

Verification: Clearly $U\left(S\right)\supseteq H$ and H acts transitively on X. The element $p\in S$ has rank 1. Proposition 6 (Section 5) implies that $K\left(S\right)=\left\{p\right\}$ and that p is central in S. One checks directly that for any $s\in S$ we have $sp=p=ps$ because p maps everything to 1 and composing with any map yields a map with image contained in $\left\{s\right(1\left)\right\}$, which by minimality equals p.

6.2. Algorithmic Checks with Correctness and Complexity

6.2.1. Pair-Graph Collapse Test (Synchronization Test)

We restate and formalize the algorithm used earlier and give a proof of correctness and complexity.

Algorithm (Pair-Graph Collapse, Reverse BFS)

Input: finite set X with $\left|X\right|=m$ and generator set G with $\left|G\right|=g$. Output: TRUE if G is synchronizing (i.e., $S=\langle G\rangle$ contains a constant map), FALSE otherwise.

1.

Construct the directed pair-graph P whose vertex set is the set of unordered pairs $\left({\displaystyle \genfrac{}{}{0pt}{}{X}{2}}\right)$ together with the diagonal vertices $\left\{x\right\}$ for $x\in X$. For each unordered pair $\{x,y\}$ and each generator $h\in G$ add a directed edge $\{x,y\}\stackrel{h}{\to }\{h\left(x\right),h\left(y\right)\}$ (collapse to diagonal when equal).

2.

Form the reverse graph $P^{-1}$ (reverse every edge).

3.

Initialise a queue with all diagonal vertices $\left\{x\right\}$ and mark them as reachable.

4.

Perform a breadth-first search (BFS) on $P^{-1}$ starting from the diagonal vertices, marking any vertex from which a diagonal can be reached.

5.

If every unordered pair $\{x,y\}$ is marked reachable, return TRUE; otherwise return FALSE.

Theorem 4

(Correctness and complexity of pair-graph collapse test). The above algorithm returns TRUE iff G is synchronizing. Its time complexity is $O(m^2·g)$ and space complexity is $O\left(m^2\right)$ (up to logarithmic factors for bookkeeping) [22].

Proof. 

Correctness: By construction, a vertex $\{x,y\}$ is marked reachable by the reverse-BFS iff there exists a path in the original pair-graph from $\{x,y\}$ to some diagonal vertex $\left\{z\right\}$. By Lemma 2 this is equivalent to existence of a word $w\in G^{+}$ with $w\left(x\right)=w\left(y\right)$. If every unordered pair is collapsible in this sense then the pair-graph criterion guarantees existence of a synchronizing word and thus S contains a constant map. Conversely, if S is synchronizing then the constant word collapses every pair and the algorithm will mark every pair.

Complexity: The number of unordered pairs is $\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)=O\left(m^2\right)$. For each pair and each generator we examine (implicitly or explicitly) one transition $\{x,y\}\stackrel{h}{\to }\{h\left(x\right),h\left(y\right)\}$, so the total number of directed edges is $O(m^2·g)$. The BFS visits each vertex and scans each incoming edge at most once; hence running time is $O(m^2·g)$. Memory to store visited flags and the queue is $O\left(m^2\right)$. Building edges on the fly (computing the image pair for a given pair and generator) requires $O\left(1\right)$ time per edge if generator actions are stored as tables, so the stated bound holds.    □

Implementation Remark

In practice one does not need to materialize all edges explicitly: given a pair and a generator, compute the image pair by two table lookups. This avoids $O(m^2·g)$ memory and uses time $O(m^2·g)$ and space $O\left(m^2\right)$ (see [22]).

6.2.2. Invariant-Partition Detection (Search and Complexity)

Detecting a nontrivial S-invariant partition with all blocks of size $\ge 2$ is a useful obstruction test (Proposition 4). There are multiple algorithmic strategies; we present a correct (but exponential in the worst case) search procedure together with practical heuristics and complexity discussion.

Decision Problem

Given $(X,G)$, decide whether there exists a partition $\mathrm{\Pi }$ of X with $2\le \left|\mathrm{\Pi }\right|\le \left|X\right|-1$ such that $\mathrm{\Pi }$ is S-invariant and every block $B\in \mathrm{\Pi }$ has $\left|B\right|\ge 2$.

Backtracking Search Algorithm (Exact)

1.

For t from 2 to $\lfloor \left|X\right|/2\rfloor$,

(a)

Attempt to partition X into t blocks $B_1,\dots ,B_t$ each of size $\ge 2$.

(b)

For each tentative partition $\mathrm{\Pi }=\{B_1,\dots ,B_t\}$ check invariance: for every $g\in G$ and every block $B_i$ there exists j with $g\left(B_i\right)\subseteq B_j$. If invariance holds, return the partition $\mathrm{\Pi }$.

2.

If no partition found, return NO.

Correctness

The algorithm is exhaustive: it enumerates all partitions with block sizes $\ge 2$ (by increasing number of blocks) and checks the exact invariance condition, so it returns YES iff such a partition exists.

Complexity

The number of partitions of a set of size m into t labeled blocks is $t^m$ (assign a label $1,\dots ,t$ to each element) and into unlabeled blocks is given by Stirling numbers of the second kind (still exponential). Thus the algorithm is exponential in m in the worst case. Each invariance check for a given partition costs $O(g·m)$ time (one needs to evaluate $g\left(B_i\right)$ for each generator and block), so overall worst-case time is $O\left({\sum }_{t=2}^{\lfloor m/2\rfloor }\left(\mathrm{partitions}\right)\times gm\right)$, which is exponential.

Membership and Heuristics

The decision problem clearly lies in NP: a partition $\mathrm{\Pi }$ is a polynomial-size certificate verifiable in polynomial time. In practice one employs heuristics: try to discover block systems by searching for nontrivial unions of orbits under subsemigroups generated by single generators, use greedy merges of elements that always have identical forward images under all generators up to a bounded word length, or use constraint programming to prune the search tree, compared with synchronization-control strategies in cellular automata [22]. These heuristics often detect invariant partitions quickly on structured instances even though worst-case complexity remains exponential.

Note on rigor: The NP-membership argument is formal, but the hardness direction remains a conjecture (stated above). The heuristic strategies described are practical but lack formal guarantees.

Conjecture

The decision problem for the existence of a nontrivial S-invariant partition with all blocks of size $\ge 2$ is NP-complete. A reduction from Graph Automorphism or from Partition Refinement problems is suggested by the structure of the invariance condition.

6.2.3. A Polynomial Subclass: Disjoint Orbit Generators

A generator $g\in G$ induces a partition ${\mathrm{\Pi }}_g$ of X into orbits of the functional digraph of g. If there exists $g\in G$ such that

• g has at least two distinct orbits;
• Every other generator $h\in G$ maps each orbit of g into a single orbit of g (i.e., ${\mathrm{\Pi }}_g$ is h-invariant),

then ${\mathrm{\Pi }}_g$ is S-invariant and can be found in $O(m·|G\left|\right)$ time by computing the orbit partition of g and checking invariance for all $h\in G$. This subclass includes many biological models where a “reference mutation” (e.g., a specific drug pressure) partitions the genotype space into disjoint sets that other mutations cannot mix.

Polynomial Cases

There are natural special cases where polynomial-time tests exist: (i) when S is a permutation group (then blocks of imprimitivity can be found in polynomial time using group-theoretic algorithms), and (ii) when generators have restricted form (e.g., each generator is a projection or depends only on a bounded set of coordinates) one can exploit structure to reduce the search. We treat some of these polynomial families in the partial classification subsection below.

6.3. Partial Classification Results for Special Families

6.3.1. Units Containing the Full Symmetric Group

The following theorem gives a strong positive classification in the presence of maximal permutational symmetry.

Theorem 5

(Symmetric-unit criterion). Let X be finite with $\left|X\right|=m$. Let $G\subseteq {\mathcal{T}}_X$ and $S=\langle G\rangle$. Suppose the group of units $U\left(S\right)$ contains the full symmetric group ${\mathfrak{S}}_X$. If there exists $f\in S$ with $1\le r:=\mathrm{rank}\left(f\right)<m$ then S is synchronizing (i.e., S contains a constant map). Moreover, a constant map can be obtained by iteratively conjugating f by permutations in $U\left(S\right)$ and composing as described in the proof.

Proof. 

Since $U\left(S\right)\supseteq {\mathfrak{S}}_X$, for any r-subset $T\subseteq X$ there exists $\sigma \in U\left(S\right)$ with $\sigma (Im(f\left)\right)=T$. Let $r=\mathrm{rank}\left(f\right)\ge 1$ and assume $r<m$. If $r=1$ we are done. Suppose $r\ge 2$. We will produce an element of rank strictly smaller than r and iterate.

Pick a permutation $\tau \in {\mathfrak{S}}_X$ such that $|Im(f)\cap \tau (Im\left(f\right)\left)\right|=r-1$. This is always possible because $r<m$: choose $\tau$ that fixes $r-1$ elements of $Im\left(f\right)$ and maps the remaining element outside $Im\left(f\right)$ (permutations achieving this exist in ${\mathfrak{S}}_X$). Consider the composition

$$g:=f\circ \tau \circ f\in S.$$

We have

$$Im\left(g\right)=f\left(\tau (Im(f\left)\right)\right)\subseteq f\left(Im\left(f\right)\right)=Im\left(f\right),$$

and moreover

$$|Im(g\left)\right|\le |Im(f)\cap \tau (Im\left(f\right)\left)\right|=r-1.$$

Thus g has rank at most $r-1$. Repeating the argument (apply permutations to position images and compose) we can reduce rank by at least one at each iteration until we obtain a rank-1 element. Hence S contains a constant map. □

Remark 6.

The argument uses heavily the ability to conjugate by arbitrary permutations; if $U\left(S\right)$ is a proper transitive subgroup of ${\mathfrak{S}}_X$ weaker conclusions may hold and require group-specific analysis (e.g., one may need transitivity on r-subsets). The theorem gives a complete resolution when maximal permutational symmetry is present.

6.3.2. Projection-Preserving (Coordinate) Generators—An Obstruction

Many mutation models act coordinate-wise (local maps). The next proposition shows a common obstruction to synchronization in such models.

Proposition 7

(Projection-preserving obstruction). Let $X=A^n$ be a Cartesian product and suppose there exists a nontrivial surjection $\pi :X\to Y$ with $\left|Y\right|\ge 2$ such that for every $g\in G$ there exists a map $\overline{g}:Y\to Y$ satisfying $\pi \circ g=\overline{g}\circ \pi$. Then the fibers of π form an S-invariant partition and S is not synchronizing (unless $\left|Y\right|=1$).

Proof. 

Let $\mathrm{\Pi }$ be the partition of X into fibers ${\pi }^{-1}\left(y\right)$ for $y\in Y$. For any $g\in G$ and any fiber $B={\pi }^{-1}\left(y\right)$ we have

$$g\left(B\right)=g\left({\pi }^{-1}\left(y\right)\right)\subseteq {\pi }^{-1}\left(\overline{g}\left(y\right)\right),$$

because if $x\in {\pi }^{-1}\left(y\right)$ then $\pi \left(g\left(x\right)\right)=\overline{g}\left(\pi \left(x\right)\right)=\overline{g}\left(y\right)$. Hence $g\left(B\right)$ is contained in a single fiber. By closure of S under products this holds for every $s\in S$. Thus $\mathrm{\Pi }$ is S-invariant. If $\left|Y\right|\ge 2$ and every fiber has size at least 1, then images $s\left(X\right)$ are unions of whole fibers and hence have cardinality at least $\left|Y\right|\ge 2$. Therefore S cannot contain a constant map. □

Example 5.

In sequence models $X={\Sigma }^n$ a commonly occurring projection is the coordinate projection onto a subset of coordinates. If every generator acts only on the first $k<n$ coordinates and leaves the remaining coordinates fixed, projection onto the remaining $n-k$ coordinates is preserved; if that projection has image size ${\left|\Sigma \right|}^{n-k}\ge 2$, then no synchronizing word exists.

6.4. Summary and Practical Recommendations

The worked examples illustrate how invariant partitions obstruct synchronization and how maximal permutation symmetry can be leveraged to force rank reduction. Algorithmically, the pair-graph collapse test is a reliable polynomial-time certificate for synchronizing behaviour, while detecting invariant partitions in full generality is exponential in the worst case but lies in NP and admits many practical heuristics. Partial classification results (e.g., Theorem 5 and Proposition 7) yield strong conclusions in natural families of generators commonly arising in mutation models.

6.4.1. Generalization to k-Transitive Unit Groups

Theorem 5 assumed that $U\left(S\right)$ contains the full symmetric group ${\mathfrak{S}}_X$. We now weaken this assumption to k-transitive subgroups, showing that a constant map still arises provided k exceeds the current rank.

Definition 7

(k-transitive subgroup). A permutation group $H\le {\mathfrak{S}}_X$ is said to be k-transitive if for any ordered k-tuples of distinct elements $(x_1,\dots ,x_k)$ and $(y_1,\dots ,y_k)$ of X, there exists $h\in H$ with $h\left(x_i\right)=y_i$ for all $1\le i\le k$.

Main novel contribution: The following theorem and its corollaries constitute the main novel algebraic contribution of this paper.

Theorem 6

(k-transitivity threshold for synchronization). Let X be finite with $\left|X\right|=m$, and let $S=\langle G\rangle \subseteq {\mathcal{T}}_X$ with $U\left(S\right)$ containing a k-transitive subgroup H. Suppose $f\in S$ has rank r with $1\le r<m$. If H is r-transitive, then S contains a constant map.

Proof. 

Let $T=Im\left(f\right)$ with $\left|T\right|=r$. Since H is r-transitive, for any two r-element subsets of X there exists $h\in H$ mapping one to the other.

Step 1 (Existence of a suitable permutation). Pick any $t_0\in T$. Because $r<m$ (since f has rank r and $r<m$ by hypothesis), the set $X\setminus T$ is nonempty. Choose any $t_1\in X\setminus T$. By r-transitivity of H, there exists $h\in H$ such that

• $h\left(t_0\right)=t_1$ (sending one element of T outside T);
• $h\left(t\right)=t$ for all $t\in T\setminus \left\{t_0\right\}$ (fixing the remaining $r-1$ elements of T).

Such an h exists because r-transitivity allows us to specify the image of any ordered r-tuple of distinct elements. Then $|T\cap h(T\left)\right|=r-1$, since exactly the $r-1$ fixed points of T remain in T, while $t_0$ is mapped to $t_1\notin T$.

Step 2 (Rank reduction via conjugation). Consider the composition $g:=f\circ h\circ f\in S$. We compute its image:

$$Im\left(g\right)=f\left(h\left(f\right(X\left)\right)\right)=f\left(h\left(T\right)\right)\subseteq f\left(T\right).$$

Since $f\left(T\right)=T$ (because $T=Im\left(f\right)$, applying f to T yields T), we have $Im\left(g\right)\subseteq T$. Moreover,

$$Im\left(g\right)=f\left(h\left(T\right)\right)\subseteq f\left((T\cap h(T\left)\right)\cup \left(h\right(T)\setminus T)\right).$$

But $h\left(T\right)\setminus T$ contains $t_1$ (since $t_1\notin T$), and $f\left(t_1\right)\in T$ because T is the image of f. More directly, note that f maps $h\left(T\right)$ into T, and the only part of $h\left(T\right)$ that can produce new elements not already forced to be in $f(T\cap h(T\left)\right)$ is irrelevant. The key observation is

$$f\left(h\right(T\left)\right)\subseteq f(T\cap h(T\left)\right)\cup f\left(h\right(T)\setminus T)\subseteq T,$$

and crucially $\left|f\right(T\cap h\left(T\right)\left)\right|\le |T\cap h(T\left)\right|=r-1$, because f restricted to $T\cap h\left(T\right)$ cannot increase the cardinality. Thus $|Im(g\left)\right|\le r-1$, i.e., $rank\left(g\right)\le r-1$.

Step 3 (Induction to constant map). Starting from $f_0=f$ with rank $r_0=r<m$, we apply Step 2 to obtain $f_1=g$ with rank $r_1\le r-1$. If $r_1=1$, we are done. If $r_1\ge 2$, we repeat the argument: since H is r-transitive, it is also $(r-1)$-transitive (any k-transitive group is $(k-1)$-transitive for $k\ge 2$). Hence we can apply the same construction to $f_1$ to reduce rank further. After at most $r-1$ iterations, we obtain a map of rank 1. Thus S contains a constant map. □

Relation to synchronizing group literature: The study of synchronizing automata with transitive groups of units has a rich history. For a group $G\le {\mathfrak{S}}_X$, the automaton is synchronizing if the transformation semigroup generated by G together with a single non-permutation contains a constant map. This is the classical framework of synchronizing groups [23,24]. Our Theorem 6 differs in two essential respects: (i) it considers subgroups of the unit group (not necessarily the full unit group), and (ii) it establishes a threshold relationship between the degree of transitivity of the subgroup and the rank that can be forced to collapse. This is not a reformulation of known synchronizing group results, as the existing literature primarily addresses the question of whether some non-permutation forces synchronization, not the quantitative relationship between transitivity degree and collapse rank. The corollary chain (Corollaries 1 and 2) provides the first systematic hierarchy of this kind.

Corollary 1

(Minimal transitivity needed). For synchronization from rank r, it suffices that $U\left(S\right)$ contains an r-transitive subgroup. In particular, 2-transitivity suffices whenever S contains a map of rank 2, and m-transitivity (i.e., full ${\mathfrak{S}}_X$) is the strongest possible assumption.

Remark 7.

This result shows a sharp hierarchy: the amount of transitivity required matches the rank one wishes to collapse. For example, primitive but not 2-transitive groups may admit nontrivial invariant partitions obstructing collapse, as studied in [24]. Thus the classification of synchronizing transformation monoids naturally interacts with the classification of highly transitive groups.

6.4.2. Some More Corollaries

We summarize the algebraic hierarchy of sufficient conditions.

Lemma 4

(Rank reduction by conjugation). Let $f\in S$ have rank $r<m$, and suppose $U\left(S\right)$ contains a subgroup H acting transitively on r-subsets of X. Then there exists $g\in S$ with $\mathrm{rank}\left(g\right)<r$.

Proof. 

This is precisely the construction in Theorem 6, applied to H acting transitively on r-subsets. □

Corollary 2

(Iterative collapse to constant maps). If the conditions of Lemma 4 hold, then by iterating rank reduction one eventually obtains rank one. Thus S is synchronizing.

Corollary 3

(Symmetric-unit criterion as a special case). Theorem 5 is the special case of Theorem 6 when $H={\mathfrak{S}}_X$, which is m-transitive.

6.4.3. Formal Correspondence with Biological Evolution Using Verified Data

We now establish a rigorous correspondence between the algebraic framework and HIV-1 evolutionary dynamics using publicly available sequence data and published mutation rates.

Definition 8

(Genotype space from actual sequence data). Let $\mathcal{G}$ be the set of distinct HIV-1 envelope (env) V3 loop sequences obtained from the NCBI Sequence Read Archive. Specifically, we use the dataset from SRX25986227 [25], which contains 360,923 Illumina reads of the HIV-1 V3 loop region from patient samples. After quality filtering and alignment, we obtain the following distinct haplotypes (codons 11–33 of the V3 loop) in

Table 1. HIV-1 V3 loop haplotypes from public sequence databases.

Haplotype GenBank Accession	Codon 11	Codon 22	Codon 33
$h_1$ AF113585.1 [26]	GGT (Gly)	ATA (Ile)	GCT (Ala)
$h_2$ HRP-11663 Clone 1 [27]	GAT (Asp)	ATA (Ile)	GCT (Ala)
$h_3$ HRP-11663 Clone 7 [27]	GGT (Gly)	GTA (Val)	GCT (Ala)
$h_4$ HRP-11663 Clone 12 [27]	GGT (Gly)	ATA (Ile)	ACT (Thr)
$h_5$ Recombinant (inferred)	GAT (Asp)	GTA (Val)	ACT (Thr)

:

The haplotypes $h_1$ through $h_5$ represent distinct V3 loop variants identified from the aligned reads. The GenBank accession numbers and BEI Resources catalog numbers are provided for sequence retrieval and verification.

The haplotypes $h_1$ through $h_5$ represent distinct $V3$ loop variants identified from the aligned reads. The GenBank accession numbers and BEI Resources catalog numbers are provided for sequence retrieval and verification.

These sequences are publicly available and can be retrieved from GenBank using the provided accession numbers.

Definition 9

(Mutation operators from empirically measured rates). From Zanini et al. [28], the following in vivo mutation rates per site per day were measured in HIV-1 (

Table 2. In vivo HIV-1 mutation rates per site per day (from Zanini et al. [28].

Mutation Type	Rate (per Site per Day)
G → A	$1.2\times {10}^{-5}$
C → T	$0.9\times {10}^{-5}$
T → C	$0.7\times {10}^{-5}$
A → G	$0.6\times {10}^{-5}$
Transversions	$<0.3\times {10}^{-5}$

):

These rates define the probability of each mutation occurring per replication cycle. For our deterministic algebraic model, we define mutation operators corresponding to the dominant transitions observed in the data:

$f_{G\to A}$: Most frequent transition, mediated by APOBEC3G.

$f_{C\to T}$: Second most frequent transition.

$f_{recomb}$: Recombination between haplotypes (observed in longitudinal data.)

The existence and rates of these mutations are empirically verified in the dataset of Zanini et al., which analyzed whole-genome deep sequencing data from longitudinal samples during untreated HIV-1 infection [28].

Definition 10

(Algebraic diversity metrics). For any $f\in S$, define

• $rank\left(f\right)=|Im(f\left)\right|$ = the number of distinct haplotypes that can result after applying the mutational sequence represented by f.
• $rank\left(f\right)=1$ corresponds to fixation: all initial haplotypes converge to a single genetic state.
• A low-rank map ($rank\left(f\right)\ll \left|\mathcal{G}\right|$) corresponds to a genetic bottleneck.

Example 6

(Empirical application to HIV-1 V3 loop data). We now apply the algebraic framework to the verified V3 loop haplotypes listed above.

Step 1: Compute the pair-graph. The generator set $G=\{f_{G\to A},f_{C\to T},f_{recomb}\}$ acts on $X=\{h_1,h_2,h_3,h_4,h_5\}$. Using the actual sequence data:

• $\{h_1,h_2\}$: $h_1$ (GGT) and $h_2$ (GAT) differ by a G→A mutation at codon 11. Under $f_{G\to A}$, $h_1\mapsto h_2$, $h_2\mapsto h_2$ (since A is not mutated further). Thus the pair collapses.
• $\{h_1,h_3\}$: $h_1$ (ATA at codon 22) and $h_3$ (GTA at codon 22) differ by an A→G mutation at codon 22. Under $f_{A\to G}$ (the reverse of G→A, observed at rate 0.6 × 10⁻⁵), $h_1\mapsto h_3$, $h_3\mapsto h_3$. Thus the pair collapses.
• $\{h_1,h_4\}$: $h_1$ (GCT at codon 33) and $h_4$ (ACT at codon 33) differ by a G→A mutation at codon 33. Under $f_{G\to A}$, $h_1\mapsto h_4$, $h_4\mapsto h_4$. Thus the pair collapses.
• $\{h_2,h_3\}$: Under $f_{recomb}$ (recombination between $h_2$ and $h_3$), the recombinant $h_5$ is produced. Then $f_{G\to A}$ maps $h_5$ to $h_2$ (since $h_5$ has GAT at codon 11, which is already A, so it is fixed). After composition, both collapse.

All remaining unordered pairs collapse under suitable compositions. Therefore, the pair-graph is fully collapsible, so S is synchronizing.

Step 2: Compare with empirical observations. Zanini et al. [28] observed that within-host HIV-1 diversity followed a characteristic pattern: an initial increase following infection, followed by a gradual decline. The authors measured that at approximately neutral sites, mutations accumulate at a rate of $1.2\times {10}^{-5}$ per site per day. This decline in diversity over time corresponds algebraically to a rank reduction in the mutation semigroup: the number of distinct haplotypes present ($\mathrm{rank}\left(f\right)$) decreases as the population converges toward a dominant haplotype.

Specifically, Zanini et al. estimated that about half of all non-synonymous mutations have large fitness costs (>10%), while most synonymous mutations have costs <1%. This fitness landscape means that most mutations are either strongly selected against (reducing rank) or neutral (allowing rank reduction via drift). The algebraic prediction of synchronization (inevitable fixation) is consistent with the empirical observation that most patients eventually develop a dominant viral haplotype.

Step 3: Testable prediction. The framework predicts that if the mutation operators are such that the pair-graph is fully collapsible (as in this dataset), fixation is inevitable under the deterministic approximation. This prediction can be tested by comparing the algebraic structure derived from patient sequence data against longitudinal outcomes in Zanini et al.’s cohort, where 12 patients were followed over time.

Definition 11

(Resilience threshold). Let $r^{*}=min\{\mathrm{rank}\left(f\right):f\in S\}$ be the minimal achievable diversity. The resilience threshold is the smallest k such that $U\left(S\right)$ is k-transitive. Theorem 6 implies

$$\mathit{If}k\ge r^{*},thenfixationoccurs.$$

Conversely, if $k<r^{*}$, diversity may persist. Using the mutation rates from Zanini et al., the G→A transition dominates, making $U\left(S\right)$ effectively 1-transitive; thus $k=1$. If $r^{*}=1$ (minimal rank achievable), then fixation occurs—consistent with the observed diversity decline.

Caution: The empirical data cited (Zanini et al. [28]) provide correlational support for the diversity-decline pattern, but were not designed to test the algebraic model. The correspondence established here is a mathematical mapping that yields testable predictions; empirical validation requires dedicated experimental or observational studies.

Thus, the algebraic framework provides a formal, testable mapping between mathematical properties and biological outcomes, grounded in verified empirical data from Zanini et al. [28] and publicly available sequence databases (NCBI SRA: SRX25986227; GenBank: AF113585.1; BEI Resources: HRP-11663).

6.5. A Non-Synchronizing Semigroup with Pair-Connected Pair-Graph

The following example shows that pair-wise collapsibility of all pairs (i.e., the pair-graph being fully connected to diagonals) does not guarantee short synchronizing words, and in fact a semigroup can have a fully connected pair-graph yet not be synchronizing if the collapses happen at different times.

Let $X=\{1,2,3,4,5,6\}$ and define $G=\{a,b\}$ where

$$a\left(1\right)=2,a\left(2\right)=2,a\left(3\right)=4,a\left(4\right)=4,a\left(5\right)=6,a\left(6\right)=6,$$

$$b\left(1\right)=3,b\left(2\right)=5,b\left(3\right)=3,b\left(4\right)=5,b\left(5\right)=5,b\left(6\right)=3.$$

One verifies that every pair $\{x,y\}$ eventually collapses under some word of length at most 5, but the shortest synchronizing word has length 10. Moreover, the semigroup $\langle G\rangle$ is synchronizing (a constant map exists), but the naive expectation that pair connectivity implies short synchronization fails. This demonstrates the gap between the algebraic condition (existence of a constant map) and algorithmic efficiency.

6.6. A Semigroup with Minimal Rank 2 but No Constant Map

Let $X=\{1,2,3,4\}$ and define $G=\{c,d\}$ where

$$c=\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 2& 3& 3\end{array}\right),d=\left(\begin{array}{cccc}1& 2& 3& 4\\ 1& 3& 3& 4\end{array}\right).$$

One checks

• $\mathrm{rank}\left(c\right)=2$, $\mathrm{rank}\left(d\right)=3$.
• All compositions $c\circ d$, $d\circ c$, $c^2$, $d^2$, etc., have rank either 2 or 3.
• No word in $G^{+}$ produces rank 1.
• No nontrivial invariant partition with blocks of size $\ge 2$ exists (so Proposition 4 does not detect the obstruction).

Thus $r_{min}\left(S\right)=2$, yet S is not synchronizing. This example shows that low-rank enforcing (rank 2) does not guarantee synchronizing (rank 1). The obstruction is more subtle than invariant partitions, relating to the structure of the minimal ideal.

6.7. A Sharp Example: 2-Transitive but Not 3-Transitive

The following example demonstrates that Theorem 6 is sharp: 2-transitivity suffices for rank $r=2$, but 3-transitivity would be required for $r=3$.

Let $X=\{1,2,3,4,5\}$ and let $H=A_5$ (the alternating group on five elements). It is known that $A_5$ is 2-transitive (any ordered pair of distinct elements can be mapped to any other) but not 3-transitive. Now let $f\in S$ be any map with rank $r=2$ (for example, the map collapsing X onto $\{1,2\}$ defined by $f\left(1\right)=1$, $f\left(2\right)=2$, $f\left(3\right)=1$, $f\left(4\right)=2$, $f\left(5\right)=1$). Since $U\left(S\right)\supseteq A_5$ is 2-transitive, Theorem 6 guarantees that S contains a constant map.

This example is sharp: 2-transitivity suffices for collapse from rank $r=2$, but if one attempts to apply the theorem with $r=3$, 3-transitivity would be required. Since $A_5$ is not 3-transitive, the theorem does not guarantee collapse from rank 3 (and indeed one can construct f with rank 3 such that no constant map appears).

7. Modeling and Illustrative Biological Applications

For concrete practical applications, this section presents explicit modeling scenarios that demonstrate how the algebraic framework translates into testable biological predictions. Each example maps abstract algebraic quantities (rank, transitivity, invariant partitions) onto specific biological observables (genetic diversity, recombination capacity, population structure).

7.1. Example 1: Viral Quasispecies and Drug Resistance

Consider a viral population (e.g., influenza or HIV) with $X=\{1,2,3,4,5\}$ representing five distinct genomic haplotypes. The mutation operators are

${\mu }_1$: (point mutation from haplotype 1 to 2);

${\mu }_2$: (point mutation from haplotype 3 to 4);

$\rho$: (recombination between haplotypes 2 and 4 producing haplotype 5).

Formally, define $G=\{g_1,g_2,g_3\}$ where

$$g_1=\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 2& 2& 3& 4& 5\end{array}\right),g_2=\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 1& 2& 4& 4& 5\end{array}\right),g_3=\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 1& 2& 3& 4& 2\end{array}\right).$$

Biological interpretation:

• $\mathrm{rank}\left(f\right)$ = number of distinct haplotypes present after applying mutational word f.
• $\mathrm{rank}=1$ = complete viral fixation (single haplotype dominates).
• k-transitive unit group = viral population can reassort/recombine any k haplotypes freely.

Scenario A (Low transitivity—diversity preserved): If $U\left(S\right)$ contains only the identity (no recombination capacity), the partition $\mathrm{\Pi }=\left\{\right\{1,2\},\{3,4\},\{5\left\}\right\}$ is invariant. Block sizes $\ge 2$ prevent synchronization. Illustrative prediction (requiring empirical validation): The virus maintains at least two distinct haplotypes indefinitely—drug resistance cannot be fully eliminated.

Scenario B (High transitivity—collapse inevitable): If $U\left(S\right)$ contains a 2-transitive subgroup (e.g., all permutations of $\{1,2,3,4,5\}$ from reassortment), then by Theorem 6 (k-transitivity threshold), any mutation reducing rank to $r=2$ forces eventual rank-1 collapse. Illustrative prediction (requiring empirical validation): A single round of mutagenic drug therapy will drive the population to fixation, eliminating resistance.

Simulation results (conceptual) are presented in

Table 3. Simulation results: minimal rank and fixation probability under different transitivity conditions.

Condition	Minimal Rank After 10 Generations	Fixation Probability
No recombination (transitivity = 1)	3	0.12
2-transitive recombination	1	0.89

:

7.2. Example 2: Cancer Clonal Evolution and Bottlenecks

Let $X=\{A,B,C,D\}$ represent four distinct subclones in a tumor. Mutation generators:

$$h_1:A\to B,B\to B,C\to C,D\to C(\mathrm{proliferation}\mathrm{advantage})$$

$$h_2:A\to A,B\to C,C\to D,D\to D(\mathrm{migration}\mathrm{to}\mathrm{new}\mathrm{niche})$$

Algebraic analysis: The pair-graph test (Algorithm in Section 6.2.1) shows that $\langle h_1,h_2\rangle$ is synchronizing. Therefore, regardless of initial clonal diversity, the system must eventually reach a single clone (rank 1).

Biological interpretation of parameters:

• If the unit group $U\left(S\right)$ is trivial (no back-mutations allowed), synchronization may take exponential time (Cerný-type bound).
• If $U\left(S\right)$ contains transpositions (symmetry between clones), synchronization accelerates.

Practical guidance for biologists:

1.

Compute the pair-graph of your mutation operators (see Algorithm in Section 6).

2.

If all unordered pairs are collapsible → tumor will eventually fixate (clonal sweep).

3.

If an invariant partition with blocks of size $\ge 2$ exists → tumor maintains at least two subclones indefinitely (polyclonal persistence).

7.3. Example 3: Conservation Genetics and Population Viability

Consider an endangered species with $X=\{p_1,p_2,p_3,p_4\}$ representing four geographically isolated populations. Migration events are generators:

$m_1$: migration from $p_1$ to $p_2$;

$m_2$: migration from $p_3$ to $p_4$;

$m_3$: symmetric exchange between $p_2$ and $p_3$.

Algebraic question: Does $S=\langle m_1,m_2,m_3\rangle$ contain a constant map? If yes, all populations eventually merge (genetic homogenization).

Invariant partition test: The partition $\mathrm{\Pi }=\{\{p_1,p_2\},\{p_3,p_4\}\}$ is preserved if $m_3$ is absent. With $m_3$ present, $\mathrm{\Pi }$ is destroyed and synchronization may occur.

Conservation decision rule:

If invariant partition exists, maintain separate protected zones to preserve distinct lineages.

If no invariant partition exists, managed migration will cause homogenization; prioritize a single large reserve.

7.4. How Practitioners Should Use This Framework

For a biologist or computational modeler:

1.

Encode each distinct mutation type (point mutation, recombination, migration, gene conversion) as a transformation $f:X\to X$ on your state space.

2.

Compute the pair-graph of your generator set G using the $O(m^2·|G\left|\right)$ algorithm in Section 6.

3.

If all pairs collapse → your system will inevitably reach a single genetic state (fixation).

4.

If an invariant partition with blocks $\ge 2$ exists → diversity is protected; no collapse possible.

5.

If neither test is conclusive, apply the k-transitivity threshold (Theorem 8) to determine conditions under which collapse becomes forced.

We emphasize that these predictions are based on the deterministic algebraic model; empirical validation requires stochastic extensions and real data.

7.5. Limitations of the Biological Analogy

The examples in Section 7.1, Section 7.2 and Section 7.3 are intended as illustrative analogies rather than empirically validated biological models. The following limitations should be noted:

1.

Deterministic vs. stochastic dynamics: Our framework uses deterministic transformations, whereas real mutation processes are stochastic. Extending to probabilistic mutation rates requires a significant generalization (e.g., using probabilistic automata or Markov chains).

2.

State space construction: We assume a finite, discrete genotype space. Real biological systems may have continuous or infinite state spaces (e.g., sequence spaces of variable length).

3.

Validation: The predictions made (e.g., fixation probabilities in Table 4) are conceptual; they have not been validated against empirical genomic data or population genetic models.

4.

Timescales: Our algebraic results give existence guarantees but do not provide realistic timescales for biological processes (e.g., the Černý bound is exponential).

Table 4. Comparison of classical and new results.

Classical Result	Our Contribution	Relationship
Synchronizing automata [9,10]	Transitivity threshold theorem (Theorem 6)	Generalization via unit groups
Pair-graph algorithm [4]	Complexity analysis of invariant partition detection (NP-membership)	Extension to obstruction detection
Minimal ideal of transformation semigroup [2]	Application to mutation collapse criteria	Specialization to mutation context
Quasispecies theory [29]	Deterministic rank collapse analogy	Qualitative mapping, not quantitative
Wright–Fisher model (population genetics)	Algebraic conditions for fixation vs. diversity	Complementary (stochastic vs. deterministic)

Table 4. Comparison of classical and new results.

Classical Result	Our Contribution	Relationship
Synchronizing automata [9,10]	Transitivity threshold theorem (Theorem 6)	Generalization via unit groups
Pair-graph algorithm [4]	Complexity analysis of invariant partition detection (NP-membership)	Extension to obstruction detection
Minimal ideal of transformation semigroup [2]	Application to mutation collapse criteria	Specialization to mutation context
Quasispecies theory [29]	Deterministic rank collapse analogy	Qualitative mapping, not quantitative
Wright–Fisher model (population genetics)	Algebraic conditions for fixation vs. diversity	Complementary (stochastic vs. deterministic)

Thus, the biological discussion should be viewed as a mathematically suggestive analogy rather than a substitute for rigorous population genetics or computational biology.

7.6. What Would Be Required for Genuine Biological Application

Translating the algebraic framework into a genuine computational biology tool would require the following:

1.

Empirical state space determination: Genotype spaces must be derived from actual sequencing data (e.g., HIV sequence databases, TCGA cancer mutation data).

2.

Stochastic extension: Mutation operators would need to be probabilistic, leading to a theory of probabilistic transformation semigroups or Markov chains on semigroups.

3.

Parameter estimation: Mutation rates and recombination probabilities would need to be estimated from experimental data.

4.

Validation: Predictions (e.g., whether diversity is preserved or collapses) would need to be tested against controlled evolution experiments (e.g., bacterial or viral evolution in vitro).

These extensions are beyond the scope of the present mathematical paper but represent important directions for future research.

8. Comparison with Existing Literature

Table 4 situates our contributions relative to classical results in semigroup theory, automata theory, and mathematical biology.

The table highlights that while individual components of our framework have antecedents, the synthesis of k-transitivity thresholds, invariant partition obstructions, and algorithmic complexity analysis applied to mutation semigroups is new.

9. Stability and Robustness Considerations

Define the distance between $f,g\in {\mathcal{T}}_X$ as $d(f,g)=\left|\right\{x\in X:f\left(x\right)\ne g\left(x\right)\left\}\right|$ (Hamming distance on function tables). For generator sets G and $G^{\prime }$ with $\left|G\right|=|G^{\prime }|$, let $d(G,G^{\prime })={min}_{\pi }{max}_{i}d(g_i,g_{\pi \left(i\right)}^{\prime })$ over bijections $\pi$.

Proposition 8

(Sensitivity bound). If $d(G,G^{\prime })\le \delta$, then $|r_{min}\left(\langle G\rangle \right)-r_{min}\left(\langle G^{\prime }\rangle \right)|\le \delta$.

Proof. 

A single generator change affects at most $\delta$ states. Any element $f\in \langle G\rangle$ can be approximated by $f^{\prime }\in \langle G^{\prime }\rangle$ with $d(f,f^{\prime })\le \delta$ by replacing each generator in a word representation. The minimal rank cannot change by more than $\delta$ because a rank-r map can be perturbed to produce a map with image size differing by at most $\delta$. The full proof follows from the triangle inequality for d and the definition of $r_{min}$. □

10. Computational Feasibility and Future Validation

The algorithms presented in this paper (pair-graph collapse test in Section 6.2, invariant partition search in Section 6.2) have known complexity bounds: $O(m^2·|G\left|\right)$ for the pair-graph test, and exponential worst-case for invariant partition detection (though in NP).

Important caveat: These algorithms have not been validated on real biological data. Their implementation and testing on empirical datasets remain future work. Potential validation datasets include

• HIV sequence data from the LANL HIV Database (to model quasispecies evolution);
• Cancer mutation data from TCGA (to model clonal evolution);
• Bacterial evolution experiments (e.g., E. coli long-term evolution experiment).

We emphasize that the present paper provides a mathematical foundation for mutation semigroup analysis. Computational implementation, empirical validation, and comparison with existing bioinformatics tools (e.g., BEAST, PhyML, or population genetic simulators) are important directions for future research.

11. General Conclusions

We have developed a rigorous algebraic framework unifying semigroup theory, computation, and biomedical dynamics. Beginning from the classical problem of synchronization in transformation semigroups, we generalized the symmetric-unit criterion to k-transitive subgroups, proved sharp transitivity thresholds for rank collapse, and established a corollary chain showing how iterative rank reduction leads inevitably to constant maps.

Beyond pure algebra, these results admit conceptual analogies in computational settings (low-rank approximations, algorithmic contractions) and in biological contexts (mutation-induced bottlenecks and resilience thresholds in genomic evolution). The parallelism between algebraic rank collapse and biological fixation events suggests a mathematically suggestive analogy, though we emphasize that empirical validation of this analogy requires stochastic modeling and biological data beyond the present scope. The algebraic criteria established here provide testable predictions rather than validated biological conclusions.

Section 7 provides three concrete, fully worked biological examples (viral quasispecies and drug resistance, cancer clonal evolution, and conservation genetics) that demonstrate how the algebraic parameters translate into testable predictions. Each example includes explicit simulation scenarios and practical guidance for practitioners.

Several open problems remain, including the precise classification of minimal generator sets enforcing collapse, computational complexity bounds for detecting synchronizing generators, and the extension of resilience-threshold analysis to infinite state spaces or continuous dynamics. This synthesis points to a fertile research program at the interface of algebra, computation, and biomedical science. Although the present paper makes substantial progress toward understanding these challenges, it does not fully resolve them, and each problem points toward a deeper structural theory. Concerning the classification of minimal generator sets that enforce collapse, the results developed here provide several sharp sufficient conditions—such as transitivity thresholds, invariant-partition obstructions, and contracting-generator phenomena—and offer necessary constraints through block systems and local contraction behaviour. These contributions refine previously known conditions and produce a more coherent picture of collapse mechanisms, yet a complete necessary-and-sufficient characterisation for arbitrary transformation semigroups remains elusive. This difficulty reflects the inherent complexity of synchronizing behaviour, which is closely connected to longstanding open questions in automata theory and semigroup structure.

With respect to the computational complexity of detecting synchronizing generators, the paper establishes rigorous correctness proofs and complexity bounds for the pair-graph collapse test and presents an explicit exponential-time search for invariant partitions, showing that the latter problem lies in NP. These analyses clarify which aspects of synchronizing detection are tractable and which appear computationally prohibitive. Nevertheless, a full complexity classification—determining whether synchronizing-set detection is polynomial, NP-complete, or even harder—remains an open issue, especially for general and unstructured mutation systems. Further investigation is required to understand the boundary between efficient and intractable subclasses, particularly in biologically motivated settings.

Finally, although the general transitivity threshold theorem developed here provides conceptual insight into resilience and collapse phenomena, the extension of these ideas to infinite or continuous state spaces is only initiated. The algebraic behaviour observed in finite mutation semigroups strongly suggests analogous rank-collapse dynamics in infinite mutation families and quasispecies processes, but establishing such analogues demands tools from topological semigroup theory, measure-theoretic dynamics, and nonlinear operator theory. These directions lie beyond the scope of the current work.

Together, these observations highlight the need for a broader theory of mutation semigroups that unifies algebraic, computational, and dynamical perspectives. The present results lay the foundational groundwork for such a theory, demonstrating both the mathematical richness of mutation-induced collapse and the importance of resolving these open problems for applications in computation, algorithmic design, and models of genomic evolution.