Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

: As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Mal’cev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal. Conceptualisation, writing—ﬁrst M.B.; methodology, analysis,


Introduction
There are various notions of symmetry in the sciences and in mathematics. Algebraic structures are usually considered symmetric if they have a lot of automorphisms or, more generally, endomorphisms. For a universal algebra (a structure without relations) the automorphism group/endomorphism monoid consists of all those permutations/selfmaps of the carrier set that commute with all fundamental operations of the algebra. Studying commuting operations in more generality leads to the notion of the centraliser clone of an algebra, or simply of a set of operations if the carrier set is understood from the context. The unary part of the centraliser (clone) is then exactly the endomorphism monoid of the algebra, and the fundamental operations of the algebra are said to witness this monoid.
This article is concerned with algebras on a specific carrier set A that are 'maximally symmetric' in the sense that their endomorphism monoid is a co-atom in the lattice of all possible endomorphism monoids of algebras on A. As the monoids in this lattice are the unary parts of (all) centraliser clones on A, they are called centralising monoids, and the co-atoms of this lattice are also referred to as maximal centralising monoids.
The study of centralisers in algebra goes back to Cohn [1] (Chapter III. 3), and notably to Kuznecov [2] who first established logical methods for their investigation, exploiting closure under operations whose graphs are primitive positively definable from given operations. Kuznecov allegedly also discovered all 25 centraliser clones among Post's lattice, a fact later re-proved by Hermann [3]. Danil'čenko [4][5][6][7] continued the work of Kuznecov by determining all 2986 centraliser clones on sets of three elements [8,9]. On carrier sets of size four and beyond, currently no good overview of the lattice of centralisers exists. Between 1974 and 1976 Harnau [10][11][12][13][14][15] worked on centralisers of unary operations, which are dual to centralising monoids in terms of the Galois connection induced by commutation of finitary operations (see Section 2). Centralising monoids of single unary operations, i.e., monounary algebras, were investigated in [16][17][18], showing, for example, which centralising monoids of this type are equal to the centralising monoid they describe as a witness [16] (Theorem 4.1, p. 8, Theorem 5.1, p. 10), and which of them have a unique unary operation as their witness [18] (Theorems 3.1 and 3.3, p. 4659 et seq.). Research on centralising monoids was further pushed forward in a series of papers [19][20][21][22][23][24][25] by Machida and Rosenberg, linking in particular maximal centralising monoids to the five types of functions appearing in Rosenberg's Classification Theorem [26] for minimal clones. As a consequence, all 192 centralising monoids on three-element sets were determined in [21,23], and all 10 maximal centralising monoids among them were identified.
This research is part of the project, begun in [27], to take the results of [23] to the next level, that is, to determine all maximal centralising monoids on four-element carrier sets. According to Proposition 3 it is necessary for this to find all centralising monoids on {0, 1, 2, 3} induced by single functions of each of the five types of Rosenberg's Theorem and to determine the ones among them which are maximal proper transformation monoids under set inclusion. As an initial step centralising groups of majority operations and semiprojections were studied in [28]. Extending the methods of [23], the case of majority operations on {0, 1, 2, 3} was already completed in [27], and further investigated under a different aspect in [29]. In this article we tackle the cases of unary operations (permutations of prime order and non-identical retractive operations), binary idempotent operations and ternary minority operations arising as Mal'cev operations corresponding to Boolean groups.
We determine all centralising monoids on 4 = {0, 1, 2, 3} witnessed by binary idempotent operations, and we classify both the witnessing operations and the monoids up to conjugacy by inner automorphisms from Sym 4. We also exhibit the maximal monoids among them, in general, and again, up to conjugacy. With respect to unary operations, we establish that on every carrier set of size at least four, every single transposition or every product of disjoint transpositions without fixed points and every non-identical retraction witnesses a centralising monoid which is a co-atom in the lattice of those centralising monoids witnessed by sets of unary operations, that is, in the lattice of endomorphism monoids of unary algebras. However, we also show that, given at least four elements, no transposition and no permutation with a single fixed point, as well as no retraction fixing all but one element can ever witness a maximal centralising monoid. For {0, 1, 2, 3} we improve this by showing that no permutation at all will witness a maximal centralising monoid. Finally, based on results of Länger and Pöschel [30] about strongly constantively rigid relational systems, we give a new proof of the known fact [21] (Theorem 5.1) that the centralising monoid of a constant on an at least three-element set is indeed maximal. We use the same technique to prove that-with the exception of the two-element set-the centralising monoid of a Mal'cev operation of a Boolean group is always maximal.

Preliminaries
We start by introducing basic notation with respect to sets, functions and relations, followed by fundamental facts regarding their Galois theory based on preservation and commutation. We also give a brief overview of formal concept analysis in order to be able to switch between different Galois connections. In the second part of this section we present background theory on clones, in particular centraliser clones, and centralising monoids that we shall use extensively to derive our main results.

Notation and Basic Concepts
We write N = {0, 1, 2 . . . } for the set of all natural numbers (finite ordinals) and N + for the positive ones. It will be convenient for us to understand every n ∈ N as the set of its ordinal predecessors n = {0, . . . , n − 1} as in the von Neumann model of natural numbers. The cardinality (size) of a set A is denoted by |A|; if |A| < ℵ 0 , then we often pick a canonical representative for it, e.g., n = {0, . . . , n − 1} where n = |A|.
Given sets A, B, C and functions f : A −→ B and g : B −→ C, we denote their composition by g • f : A −→ C and mean the function mapping each element a ∈ A to (g • f )(a) := g( f (a)) ∈ C. The set of all functions from A to B is symbolised as The full image of f is also denoted as im( f ) := f [A]. The restriction of f to an arbitrary subset U ⊆ A is the function f | U : For n ∈ N we understand tuples x ∈ A n as maps from n = {0, . . . , n − 1} to A that are simply written as x = (x 0 , . . . , x n−1 ). Sometimes we allow ourselves to deviate from this standard notation if some other indexing like x = (a, b, c) or x = (x 1 , . . . , x n ) seems more convenient. As tuples x ∈ A n are maps, we have in particular im(x) = {x 0 , . . . , x n−1 } and we can compose with f ∈ B A or α ∈ n I , giving f • x = ( f (x 0 ), . . . , f (x n−1 )) ∈ B n and re-indexed tuples For a set A and n ∈ N, every f ∈ O  a (x) = a for all x ∈ A n . We collect all constant operations on A in the set C A = c (n) a a ∈ A ∧ n ∈ N + . We also need some more specific operations given by identities. An operation f ∈ O A is idempotent, if it satisfies f (x, . . . , x) = x for all x ∈ A. A ternary operation f ∈ O (3) A is called a majority operation, if f (x, x, y) = f (x, y, x) = f (y, x, x) = x for all x, y ∈ A; it is called a minority operation if f (x, x, y) = f (x, y, x) = f (y, x, x) = y for all x, y ∈ A; finally, it is a Mal'cev operation if f (x, x, y) = f (y, x, x) = y for all x, y ∈ A. Among unary operations we need permutations f ∈ Sym A, i.e., bijective self-maps f ∈ O (1) A , and retractive operations f ∈ O (1) A satisfying f • f = f (sometimes also called idempotent since they are idempotent elements in the semigroup O (1) For all m, n ∈ N and g 1 , . . . , g n ∈ O (m) A , we can form (g 1 , . . . , g n ) : A m −→ A n , called the tupling of g 1 , . . . , g n , sending x ∈ A m to (g 1 (x), . . . , g n (x)) ∈ A n . In this way, we can compose finitary operations g 1 , . . . , A as f • (g 1 , . . . , g n ). A (concrete) clone of non-nullary operations on A is any set F ⊆ O A that is closed under this form of composition and satisfies J A ⊆ F.
For m ∈ N an m-ary relation on a set A is any subset ⊆ A m of m-tuples. Any m-ary operation f ∈ O (m) A can be understood as an (m + 1)-ary relation via its graph f • := { (x 1 , . . . , x m , g(x 1 , . . . , x m )) | x 1 , . . . , x m ∈ A}. We extend this notation elementwise to sets of operations by putting F A binary relation ⊆ A 2 is just a set of pairs; we define its inverse (sometimes also called dual) to be If is reflexive, symmetric and transitive, that is (x, y), (y, z) ∈ implies (x, z) ∈ for all x, y, z ∈ A, then belongs to Eq(A), the set of all equivalence relations on A. By R A = { ⊆ A m | m ∈ N + } we denote the set of all finitary relations on A.
While functions can be composed, new relations can be constructed from given ones using logical expressions. A primitive positive formula ϕ (in prenex normal form) over a given relational signature consists of a prefix of finitely many existentially quantified variables (possibly none) followed by a finite non-empty conjunction of atomic predicates that correspond to the given signature (relations) and have been substituted by some tuple of variables. If the relation symbols are associated with concrete relations, the set of all satisfying value assignments to (a superset of) the free variables of ϕ determines a finitary relation on A, which is called primitive positively definable from the given relations. For a finite carrier set A, a set Q ⊆ R A is said to be a relational clone on A if it contains the diagonal ∆ A := { (x, x) | x ∈ A} and is closed under all relations that are primitive positively definable from members of Q (for infinite carrier sets stronger closure properties are required). As this is an implicational definition, all relational clones on A form a closure system. The least relational clone on A containing a particular set R ⊆ R A will be denoted by [R] R A and is computed by adding to R all relations that are primitive positively definable from R ∪ {∆ A }.
Clones of operations and relational clones are connected in the following way: for m, n ∈ N and f : A n −→ A, ⊆ A m we say that f preserves if and only if is a subuniverse of the algebra A; f m . This means that for every (r 0 , . . . , r n−1 ) ∈ n the m-tuple obtained by the composition f • (r 0 , . . . , r n−1 ) of f with the tupling of r 0 , . . . , r n−1 stays inside , that is, for every (m × n)-matrix R ∈ A m×n the columns of which all belong to , the m-tuple obtained by applying f row-wise to R remains in . We then call an invariant of f or f a polymorphism of . The preservation relation induces a Galois connection between O A and R A , giving rise to the derivation operators Q → Pol A Q and F → Inv A F, collecting all polymorphisms in O A of every relation ∈ Q ⊆ R A and all invariant relations ∈ R A for all given operations f ∈ F ⊆ O A , respectively. Moreover, we declare for n ∈ N and . Now for every Q ⊆ R A the set Pol A Q forms a clone on A, and for every F ⊆ O A the invariants Inv A F are a relational clone containing ∅ (since we omit nullary operations in our clones). Therefore, Inv A Pol A Q ⊇ [Q ∪ {∅}] R A and Pol A Inv A F contains the clone generated by F. On finite carrier sets A, these inclusions are equalities [31,32]; for infinite A local interpolation operators (and a strengthened definition of relational clone) need to be added to close the gap [33,34].
Besides the Galois connection induced by preservation, we shall encounter several other Galois connections that differ by the inducing relation or by restrictions of the domains they link. Switching between them and the associated closure systems can be expressed well in the framework of formal concept analysis [35,36], providing terminology, notation and theory for the manipulation of Galois connections on a general level. The basic object of formal concept analysis is that of a formal context K = (G, M, I) where I ⊆ G × M is any binary relation between sets G (commonly called objects) and M (usually called attributes). A context K induces two derivation operators of a Galois connection between G and M in the natural way, and specifies exactly between which sets this Galois connection is to be understood. The Galois closed sets on the side of objects are called extents and the ones on the side of attributes are referred to as intents. When we keep the relation I, but pass on to a subset H ⊆ G of the objects or N ⊆ M of the attributes (or both), we form a subcontext K = (H, N, I ∩ (H × N)). Though technically incorrect, it is customary to omit the intersection ∩ (H × N) when specifying a subcontext, as the restriction becomes clear from stating the sets of objects and attributes. How the lattices of Galois closed sets (extents and intents) of K and K are related is discussed in Chapter 3 of [35] (p. 97 et seqq.). We will mostly aim for context manipulations where the closure system of intents (and thus the lattice structure) does not change, such as object clarification and object reduction. We shall explain more details at the appropriate place in the text and give pointers to the literature there.
Centrally for this paper will be the Galois connection of commutation given by the . In this respect we say that an n-ary operation f ∈ O A , if g( f (X)) = f g X T holds for any matrix X ∈ A m×n , where f (X) := ( f (X(i, ·))) i∈m denotes the tuple obtained by applying f rowwise to the matrix X, and similarly for g and the transposed matrix X T . This commutation condition will be denoted by f ⊥ g; it is easy to see that it is a symmetric property, i.e., we have f ⊥ g if and only if g ⊥ f . Therefore, the two Galois derivatives induced by K c coincide: they map F ⊆ O A to its centraliser F * = { g ∈ O A | ∀ f ∈ F : g ⊥ f }, and the associated closure operator sends F to F * * , the bicentraliser of F. A routine verification shows that g ∈ F * ⇐⇒ g ∈ Pol A F • ⇐⇒ g • ∈ Inv A F holds for F ∪ {g} ⊆ O A , so F * = Pol A F • , i.e., commutation can be rephrased in terms of preservation of graphs of operations. Hence, up to isomorphism, K c is the subcontext of finitary operations and relations with preservation, where R A is restricted to As mentioned in the introduction, when studying centralising monoids we only look at the unary parts F * (1) of centraliser clones, that is to say, we are dealing with the A , ⊥ , of K c . The intents of K, i.e., Galois closed sets M = F * (1) for some F ⊆ O A , are exactly all centralising monoids on A. Moreover, every set F ⊆ O A describing M in this way, is a witness of M, and we can equivalently express that M is witnessed by F via saying that M = End A; F . A centralising monoid M ⊆ O (1) A is maximal if it is a co-atom in the intent lattice of K, i.e., a proper centralising monoid which is maximal under set inclusion.

Fundamental Results on Clones, Centralisers and Centralising Monoids
The following simple observation exhibits necessary conditions that can be used to describe functions in a particular centraliser clone (or its unary part). The utility of said conditions was already observed by Harnau, cf. Lemma 2.6 and Satz 2.15 of [10], in the context of centralisers of single unary operations f ∈ O where for n ∈ N and f ∈ O (n) A ker( f ) := (x 1 , . . . , x n , y 1 , . . . , y n ) ∈ A 2n f (x 1 , . . . , x n ) = f (y 1 , . . . , y n ) , Second, we recollect the following helpful characterisation of centralising monoids.
A , ⊥ , they correspond via that Galois connection of commutation to an atom F in the lattice of extents of K. Such an atom must be generated (as an extent) by a single non-trivial object (1) by virtue of the Galois connection given by K. So M is singly witnessed by a non-trivial operation f ; however, even more is known about these witnesses: they can always be chosen as a generator of a minimal clone of minimum arity, a so-called minimal function. Minimal functions are separated by Rosenberg's Classification Theorem [26] for minimal clones into five distinct categories ((II) is a special case of (V), but is often listed on its own).
Theorem 4 (See [26]). On a finite set A every minimal function f is of one of the following types: A and f ∈ Sym A is a permutation of prime order or a retractive operation A is a ternary (minority) Mal'cev operation arising as f (x, y, z) := x + y + z for x, y, z ∈ A from a Boolean group A; + ; A is a ternary majority operation; A is a proper semiprojection of arity n where 3 ≤ n ≤ |A|.
In this paper, we address centralising monoids witnessed by a minimal function of types (I)-(III) with a special focus on the set {0, 1, 2, 3}; those relating to type (IV) have already been considered in [27], the ones of type (V) are part of ongoing research.
Before we can tend to this problem in more detail, we need further background information on clones, such as the centraliser of a constant operation.
Lemma 5 (Lemma 1.9 in [10], Lemma 5.2 in [21]). For any n ∈ N and a ∈ A we have a if and only if for all x 11 , . . . , x mn ∈ A the condition

Proof. We have
The following characterisation is also very well known.

Lemma 7.
For a clone F on any set A the following statements are equivalent: F is a clone of idempotent operations, i.e., For every f ∈ F and all a ∈ A we have f (a, . . . , a) = a. (c) Proof. Condition (b) simply spells out the preservation of every singleton set {a} in (a). From this it follows that every f ∈ F (1) satisfies f (a) = a for every a ∈ A, i.e., f = id A . For F contains all projections, (b) implies (c). Now since F is a clone, with every f ∈ F also f • (id A , . . . , id A ) ∈ F (1) . If (c) holds, then f • (id A , . . . , id A ) ∈ {id A }, and this shows statement (b).

Lemma 8. For any set
Proof. For |A| < 2 this is trivially true. For |A| ≥ 2, no constant map is idempotent, so, by Corollary 6, can only consist of projections. The converse inclusion is trivial.
When proving that a certain centralising monoid is maximal, it will be necessary to compute the centralising monoid M * * (1) generated by some set M ⊆ O (1) A and to show that it is the full transformation monoid on A. To do this we shall always demonstrate the seemingly stronger condition that the centraliser of M is trivial. The following lemma shows that this is in fact a necessary step to take whenever |A| ≥ 3. A . By Lemma 1, this means the functions in F preserve the kernel of any unary operation, so they preserve every equivalence relation on A. Therefore, F ⊆ Pol A Eq(A), and for |A| ≥ 3 the latter clone equals J A ∪ C A as is shown in Example 3.3 of [30] (p. 136). Now A , so Lemma 8 shows that (c) must be satisfied.
Note that for |A| = 2, Lemma 9 fails as, e.g., the clone L of all affine linear functions, which is the centraliser of the (ternary minority) Mal'cev function on the two-element set, contains all unary operations. So To establish for a subset M ⊆ O A condition (c) of Lemma 9 or to prove at least M * (1) = {id A }, we shall exploit a theorem of Länger and Pöschel [30] on (strongly) constantively rigid systems of (binary) relations.
A ; moreover, if all relations in Q are reflexive, then F ⊆ J A ∪ C A .
Proof. By our assumption we have F ⊆ Pol A Q and, in particular, F (1) ⊆ Pol (1) A Q. We use Theorem 2.13 of [30] to infer that Q is constantively rigid, i.e., Pol A . If, additionally, Q only contains reflexive relations, then Theorem 2.3 of [30] states that constantive rigidity of Q is equivalent to strong constantive rigidity, that is, Remark 11. There are two more conditions, (A) and (C), in [30], which can also be employed to ensure (strong) constantive rigidity of systems Q of (binary reflexive) relations, see, e.g., Theorems 2.11 and 2.12 of [30]. In particular, via Theorem 2.12, (B) and (C) together may be used to obtain (strong) constantive rigidity also when |A| = 3, and (C) automatically follows from (B) if all the relations in Q are symmetric (see Lemma 2.8 in [30]), for example, equivalence relations. More concrete constantively rigid systems of relations are provided in Section 3 of [30], some of them also being strongly constantively rigid.
, which contradicts the assumption on M.
Corollary 13. Let A be any set with |A| ≥ 3 and M ⊆ O A be such that for each a ∈ A there is s ∈ M satisfying a / ∈ fix(s). Moreover assume that for all pairwise distinct x, y, z, u ∈ A there is s ∈ M (1) such that s(x) = s(y) and s(z) = s(u), then M * = J A .

Corollary 14.
Let A be any set with |A| ≥ 4 and M ⊆ O A be such that for each a ∈ A there is s ∈ M satisfying a / ∈ fix(s). Moreover assume that A M * in Corollary 12.

Monoids Witnessed by Unary Operations
There are two types of unary minimal functions in Rosenberg's Theorem 4. The first are permutations of prime order, that is, their cycle structure consists only of fixed points and cycles of length p for some prime p. The second are idempotent or retractive These are exactly those non-identical unary operations which fix every point of their image.

Computational Results for {0, 1, 2, 3}
We started by computing a commutation table of all 256 unary operations on the A , ⊥ in the language of [35,36]. Being a 256 × 256 Boolean matrix this table is already confusingly big and we therefore do not present it here. For purposes of verification, in Listing 1 we have instead added simple Python code that allows anybody to reproduce such a table if desired. Using standard algorithms presented in Section 2.1 of [35] or Chapter 2 of [36], one may compute from K 1 that there are exactly 1485 centralising monoids on {0, 1, 2, 3} that can be witnessed by sets of unary operations. However, we are only interested in the coatoms of this large lattice. By virtue of the Galois connection represented by K 1 , every co-atom M is the Galois derivative N * (1) of an atom N on the dual side of the Galois connection. Since in the lattice of closed sets of any closure operator the atoms must be singly generated, it follows that (1) . Hence, to obtain the co-atoms, we only need to iterate over the 255 rows of K 1 belonging to each f ∈ O The result is that there are 49 co-atoms in the lattice of closed sets of K 1 , and subsequently we shall explain our computational findings in terms of the types of unary operations from Rosenberg's Theorem, i.e., permutations of prime order and non-identical retractions. In the following two subsections we shall then give theoretical evidence why the 49 co-atoms arise on {0, 1, 2, 3}. It is worth noting that these 49 co-atoms in the lattice belonging to K 1 are merely candidates for maximal centralising monoids, and we shall indeed demonstrate that some co-atoms of K 1 fail to be maximal among all centralising monoids, that is, are no co-atoms with respect to O A , O f o r f [1] i n r a n g e (4): f o r f [2] i n r a n g e (4): f o r f [3] i n r a n g e (4): i n r a n g e (4): f o r g [1] i n r a n g e (4): f o r g [2] i n r a n g e (4): f o r g [3] i n r a n g e (4): fgcommute = True f o r x i n r a n g e (4): row [ j ] = 1 # indicates commutation of f and g e l s e : row [ j ] = 0 # indicates non -commutation of f and g j = j + 1 p r i n t ( separator . join (map( s t r , row ))) # output table row On a four-element set there are three types of permutations of prime order: threecycles, transpositions or products of two disjoint transpositions. Up to conjugacy these can be represented by the following three functions on {0, 1, 2, 3}: (0, 1, 2), (0, 1), (0, 1)(2, 3). There are 2 · ( 4 1 ) = 8 three-cycles (4 choices of the unique fixed point and 2 for the image of the first cycle element), but it will turn out that centralising monoids witnessed by these operations are not maximal (Lemma 16), not even with respect to K 1 . However, the monoids described by the other prime permutations are maximal when considered among the monoids witnessed by only unary operations (i.e., the closure system of K 1 ). Yet again, they will not be maximal in the lattice of all centralising monoids on {0, 1, 2, 3}, see Corollary 22. We have ( 4 2 ) = 6 transpositions (choosing the two transposed elements) and 1 2 · ( 4 2 ) = 3 products of disjoint transpositions since such a function can be obtained by choosing the first transposed pair or the second one. Altogether there are 9 prime permutations (of order 2) that describe maximal monoids of K 1 .
The non-trivial retractive functions on a four-element set can be separated into the following types: being constant (one-element image, there are 4 such functions), having a two-element image, or having a three-element image (there are ( 4 3 ) · ( 3 1 ) = 12 such functions, first choosing the image and then the value of the element outside the image within it). The retractions with a two-element image can moreover be split up into those mapping the two elements outside the image to the same value and those mapping them to both distinct values of the image. In both subcases we have ( 4 2 ) · 2 = 12 functions as we need to choose the two-element image and then one of two ways how the elements outside the image can be mapped into it. Retractions of all four types witness centralising monoids that are maximal among those centralising monoids witnessed by unary functions, see Corollary 27, and there are 3 · 12 + 4 = 40 non-trivial retractions on {0, 1, 2, 3}. Moreover, it is known with a ∈ A is a maximal centralising monoid in general (that is on any finite set A with |A| ≥ 3 in the lattice of all centralising monoids), so the 4 centralising monoids of constants are co-atoms with respect to O A , O A , ⊥ , not just K 1 . In total, we have 40 + 9 = 49 centralising monoids on {0, 1, 2, 3} that are maximal in the 1485-element lattice of those monoids that can be witnessed by sets of unary operations. These can be grouped together into 6 conjugacy types that can be represented as {(0, 1)} * (1) , where the retractions f 24 , f 16 and f 17 are given as f 24 • (0, 1, 2, 3) = (0, 1, 2, 0), f 16 • (0, 1, 2, 3) = (0, 1, 0, 0) and f 17 • (0, 1, 2, 3) = (0, 1, 0, 1). From the explicit calculations (and the characterisations in the following subsections) it follows that As (0, 1)(2, 3) has no fixed points but (0, 1) and f 17 have, the monoid Therefore, the co-atoms of the lattice of intents of K 1 fall into exactly 6 distinct isomorphism classes of monoids. Moreover, we computed that all 1485 monoids in the lattice of intents of K 1 can be separated into 106 classes up to element-wise conjugacy (cf. Section 4 for more explanation).

Monoids Witnessed by Permutations
In Lemma 4.11 of [27] (see also Lemma 4.10 of [37]) we provided a characterisation of when a finitary operation f ∈ O A on a finite set A commutes with a permutation s ∈ Sym A, based on examining orbits of the permutation group generated by s. Under the assumption that f is a function the given condition was sufficient to ensure that f ⊥ s, but blindly fulfilling the orbit condition could also lead to some value assignments for f that would contradict the assumption of f being a function. Thus, some care had to be taken when working with Lemma 4.11 from [27], but this was never much of an issue when studying majority operations f on small sets as in [27]. Here we improve the mentioned characterisation by placing an additional necessary condition on the choice of the function values such that no contradictions can occur.
In particular if for all x ∈ T a value f (x) is chosen such that the size of the orbit of f (x) is a divisor of the size of the orbit of x, then there is a unique extension of this partial definition to an n-ary function f ∈ {s} * by defining f (s Proof. Let m := |S| be the order of s. Lemma 4.11 of [27] shows that f ⊥ s if and only if for The length of the orbit of x is the least common multiple of the lengths of the orbits of the entries of x and each of these lengths divides m, so divides m.
it follows from the definition of and the commutation condition that ) for all 1 ≤ j < and t divides , where t is the size of the orbit of f (x), then s t ( f (x)) = f (x) and we can show by induction on i ∈ N that s i ( f (x)) = f (x). The case i = 0 is trivial and the step from which implies commutation, see, e.g., Lemma 4.11 in [27].
If the divisor condition on f (x) is satisfied for x ∈ T with orbit of size , then certainly f (s j • x) := s j ( f (x)) can (and must to ensure f ∈ {s} * ) be defined for 1 ≤ j < as the tuples s j • x 0≤j< are pairwise distinct by the choice of as the size of the orbit of x.
For T is a transversal of the orbits on A n , this completely defines a function f ∈ O (n) A ; the second part of the proof above shows that such f actually commutes with s due to the divisor condition.
From Lemma 15 it follows again for every n-ary f ∈ {s} * that if s • x = x, that is, x ∈ (fix(s)) n , then f (x) must belong to an orbit of size 1. Hence, f (x) ∈ fix(s) and f ∈ Pol A {fix(s)}. So we obtain once more one of the necessary conditions for functions in the centraliser that were given in Lemma 1.
We can easily adapt an argument of Harnau's [10] (Lemma 2.9, p. 345), given for centraliser clones of unary operations, to centralising monoids. This shows that on a four-element set no three-cycle can witness a maximal centralising monoid (nor a maximal centraliser clone, by Harnau's result).
Lemma 16. Let A be a set with 2 ≤ |A| < ℵ 0 , and s ∈ Sym A be a permutation such that {s} * (1) is a maximal centralising monoid, then the number of fixed points of s is not 1.
Proof. Assume for a contradiction that s ∈ Sym A witnesses a maximal centralising monoid and has a ∈ A as its unique fixed point. Since {s} * (1) is maximal, we have s p = id A = s for some prime p. Applying the above Lemmas 1 and 5 we then conclude that A due to |A| ≥ 2. As s = id A , its cycle representation has at least one p-cycle (a 1 , . . . , a p ), and a / ∈ a 1 , . . . , a p because it is fixed by s. We define f ∈ Pol A and {s} * (1) cannot be a maximal centralising monoid (not even maximal among just those witnessed by unary operations).
The following lemma describes (for the case when is a prime) the centralising monoids witnessed by permutations of prime order. Lemma 17. Let s ∈ Sym A be a permutation on a finite set A that is a product of t ≥ 0 disjoint cycles of length ≥ 2 and has |A| − t · fixed points. That is, s is of the form holds if and only if both of the following conditions are satisfied: Proof. We simplify the condition given in Lemma 15. A transversal of the orbits of the cyclic permutation group generated by s is given by T = a i 0 0 ≤ i < t ∪ fix(s). The condition in Lemma 15 expressed for those x ∈ T that are fixed by s is exactly equivalent to f ∈ Pol A {fix(s)} since the only positive divisor of 1 is 1. For all x ∈ T \ fix(s), i.e., x = a i 0 for some 0 ≤ i < t, the length of the orbit generated by x is , so the divisibility condition in Lemma 15 is fulfilled for any choice of the value f (x) = f (a i 0 ) ∈ A as the only orbit sizes are 1 or . The remaining part of the condition translates to the equality of The following corollary is also evident.
Next we prove that the centraliser of any proper supermonoid of a centralising monoid witnessed by a fixed-point free product of disjoint transpositions or a single transposition with at least two fixed points is idempotent. Hence, by Lemma 7, the centralising monoid of such a transposition is maximal in the lattice of all centralising monoids witnessed by unary operations (the intent lattice of K 1 ).

Lemma 19.
For a permutation s ∈ Sym A on A with 4 ≤ |A| < ℵ 0 of order two that can be written as a fixed-point free product s = (a 1 , In particular, for every a ∈ A we can pick b ∈ A \ {a} and the partial definition g(a) := b = a can be extended to some g ∈ {s} * (1) ⊆ M with a / ∈ fix(g). Hence, the initial assumption of Corollary 14 is satisfied. To show condition (B"), take distinct x, y, z ∈ A and extend the partial definition g(x) := y = z to some g ∈ {s} * (1) ⊆ M. For condition (D") consider distinct x, y, z, u ∈ A and some h ∈ M \ {s} * . Since h does not commute with s there is a ∈ A such that for b := s(a), p := h(a) and q := h(b) the inequality q = h(b) = h(s(a)) = s(h(a)) = s(p) holds. We can define some g ∈ {s} * (1) with g(p) = g(q) = y. Namely, if p = q, this is clear, and if q = p, then q / ∈ {p, s(p)} and thus belongs to a different orbit than p, wherefore the assignment of y on a transversal containing {p, q} can be extended to some g ∈ {s} * (1) . Moreover, we defineg(x) := a and g(z) := b. If z = s(x), then this assignment is compatible with condition (b) of Lemma 17 becauseg(s(x)) =g(z) = b = s(a) = s(g(x)), so it can be extended to someg ∈ {s} * (1) . If z = s(x), then z / ∈ {x, s(x)} and thus belongs to a distinct orbit than x, wherefore the given assignment on a transversal including {x, z} can be extended to someg ∈ {s} * (1) . As a consequence, we have g • h •g ∈ M, and it satisfies g(h(g(x))) = g(h(a)) = g(p) = y and g(h(g(z))) = g(h(b)) = g(q) = y = u. Invoking Corollary 14, M * (1) = {id A }, and by Lemma 7, M * has only idempotent operations.
As a subcase we assume that h(b) = h(0) = a for all b ∈ fix( f ). By Lemma 17 there is g ∈ { f } * (1) with g(h(0)) = g(a) = a and g(x) = t for all x ∈ A \ {a} where t ∈ fix( f ) \ {a}. Such a t exists as |A| ≥ 4; in particular g(h(1)) = t. Since c , and as above these preimages are primitive positively definable from M • as g 1 , g 2 , h, f , c (1) A , defined by g z (x) = x for x ∈ {0, 1} and g z (x) = z for x ∈ fix( f ) and both z ∈ {a, b}, belong to { f } * (1) ⊆ M. Proof. We first note that b is well defined since A \ {0, 1} = fix(s) and thus for any x, y ∈ fix(s) we have s(y) = y = b(x, y). In particular, b(y, y) = y for y ∈ fix(s), and b(y, y) = y for y ∈ {0, 1}, too. Hence, b(y, y) = y for all y ∈ A, so b is idempotent.
The third and last case is that For the opposite inclusion we consider any f ∈ {b} * (1) that is not constant and have to show f ∈ {s} * (1) . First we demonstrate that f ∈ Pol , we obtain f (y) ∈ fix(s).

Monoids Witnessed by Retractive Operations
We now turn our attention from permutations of prime order to unary retractive A (also called idempotent unary operations), that is, such f : In particular, since for every x ∈ A n the tuple f • x ∈ (im( f )) n = (fix( f )) n , for any choice of values h(x) ∈ fix( f ) for x ∈ (fix( f )) n there are (generally not unique) extensions h ∈ { f } * (n) .

Proof. For every individual x ∈ A n the commutation condition
For all x ∈ A n \ (fix( f )) n we use the reformulation, while for all x ∈ (fix( f )) n we keep the original condition, which simplifies to f (h(x)) = h(x) as f • x = x. However, this is just expressing that h(x) ∈ fix( f ) for every For the second part of the lemma, we observe that once values h(x) ∈ fix( f ) are selected for each tuple x ∈ (fix( f )) n , the condition h( adds a requirement to the commutation condition, and (for unary operations) it is useful to rephrase Lemma 23 in terms of these sets.

Corollary 24. For a retraction f ∈ O
(1) Proof. We use Lemma 23 for n = 1 and note that for j ∈ I and a ∈ A j we have f (a) = j and h( f (a)) = h(j) =: A for any retraction f ∈ O (1) A \ { f } * (1) by Corollary 24.

Lemma 25. Let f ∈ O
(1) A be a retraction,   x ∈ A i we have h a,t (i) = i and the condition in Corollary 24 turns into h a,t (x) ∈ {i} ∪ A i , which is satisfied for a as a, t ∈ A j and clearly holds for all other x. So h a,t ∈ { f } * (1) . If t ∈ A with ∈ I \ {j}, we put h a,t (j) := , h a,t (x) := t for x ∈ A j and h a,t (x) := x else. Clearly, h a,t ∈ Pol A {fix( f )} as h a,t (j) = ∈ fix( f ) and h a,t (i) = i for all i ∈ I \ {j}. For such i and x ∈ A i we have h a,t (x) = x ∈ {i} ∪ A i , and for j and x ∈ A j we have Referring to Corollary 24, we distinguish as subcases whether := h(j) ∈ I or not. First suppose that ∈ I, then h(a) which is in accordance with Corollary 24 since g(i) = j. Thus, g ∈ { f } * (1) ⊆ M and im(g) = {a, j}. To define a functiong ∈ { f } * (1) with im(g) = B, we observe that there is which agrees with Corollary 24 asg( ) = t. Finally, for x ∈ A i where i ∈ I \ { }, we putg(x) := x, which also complies with Corollary 24. Hence, g ∈ { f } * (1) ⊆ M and im(g) = B. Therefore, we get The remaining subcase of the second case is that h ∈ Pol A {fix( f )} and ∈ fix( f ) \ I. By the condition in Corollary 24, we have h(a) ∈ A \ {h(j)} = A \ { } for the violating element a ∈ A j with j ∈ I. Thus, we get  Proof. Clearly, for every x ∈ A \ {a} we have g(x, x) = f (x) = x; moreover, g(a, a) = a, so g ∈ C (1) * A is idempotent (Corollary 7) and so C (1) (1) . Consider now any h ∈ { f } * (1) ; we have to show that h ∈ {g} * . For this take (x, y) ∈ A 2 . First we assume that (x, y) = (a, a). There is t ∈ {x, y} with t ∈ A \ {a} = fix( f ). Since by Corollary 24 (a, a). Now the definition of g implies g(h(x), h(y)) = f (h(x)) = h( f (x)) = h(g(x, y)). The remaining possibility is that (x, y) = (a, a), and here h(g(a, a)) = h(a) = g(h(a), h(a)), for g is idempotent. Therefore, h ∈ {g} * . Conversely, let h ∈ {g} * (1) . If h = c (1) t for some t = a, then h ∈ { f } * because t belongs to A \ {a} = fix( f ), cf. Lemma 5. Now consider h ∈ {g} * (1) \ C A and any x ∈ A. As h is not constant, there is y ∈ A such that h(y) = h(x). It certainly follows that {(x, y), (h(x), h(y))} ⊆ A 2 \{(a, a)}, so h( f (x)) = h(g(x, y)) = g(h(x), h(y)) = f (h(x)). We conclude that h ∈ { f } * (1) . A with fix( f ) = A \ {a} for some a ∈ A; however, these do not witness a maximal centralising monoid.
Proof. Maximality with respect to the closure system of the operator X → X * (1) * (1) follows from Lemma 25; for functions with a single non-fixed point non-maximality in general is evident from Lemma 26.
Lemma 25 and Corollary 27 deal with all non-identical retractions on any set and successfully explain our computational results regarding those centralising monoids on A = {0, 1, 2, 3} that can be witnessed by just unary operations. For retractions fixing all but one element, we have that the centralising monoid is never maximal; for constant retractions it is known that the centralising monoid is always maximal, see Corollary 29 below. For the other types of centralising monoids witnessed by non-constant retractions, we currently do not know more than what is stated in Corollary 27. On a four-element set there are two types of such retractions, both having a two-element image. Our computations from [27] and Section 4 show that the corresponding monoids are not contained in any proper centralising monoid witnessed by a unary or (idempotent) binary or majority operation on A = {0, 1, 2, 3}, but at the moment we cannot exclude that there might be a monoid witnessed by a semiprojection, which is properly larger.
We conclude the discussion of monoids witnessed by retractions by giving a short and new proof of the fact that on sets with at least three elements every centralising monoid witnessed by a constant is actually maximal. This has been shown before in [21] (Theorem 5.1, p. 157). Proof. By Lemma 5, c

Lemma 28. Let
for any a ∈ A \ {0}. Hence, the initial precondition of Corollary 13 has been established. We shall use the operations from T  A by a combination of Lemmas 2 and 9.

Results
We summarise the results of the current section in the following theorem. |A| ≥ 4 and f ∈ Sym A is a transposition of just two elements, • |A| ≥ 4 is even and f ∈ Sym A is a product of disjoint transpositions without fixed points, A is a centralising monoid that is maximal among those monoids on A witnessed by unary operations. If |A| ≥ 3 and f is constant, then Corollary 31. On a four-element set no maximal centralising monoid is witnessed by a permutation nor a retraction with a three-element image; however all constants witness maximal centralising monoids. Moreover, every permutation of order 2 and every non-identical retraction witnesses a centralising monoid forming a co-atom in the lattice of all centralising monoids witnessed by unary operations.

Monoids Witnessed by Binary Idempotent Operations
Each binary idempotent operation g ∈ C * (2) A satisfies g(x, x) = x for all x ∈ A. Therefore, on A = {0, 1, 2, 3} there are 4 4 2 −4 = 4 12 = 16,777,216 binary idempotent operations, which are candidates for binary minimal functions. In a very similar way as shown in Listing 1 we used a (slightly more complicated) c++ implementation to brute force enumerate all these binary operations g, and for each of them produced a characteristic vector in {0, 1} 256 of {g} * (1) . Whenever we found someg ∈ C * (2) A with {g} * (1) = {g} * (1) for a previously enumerated g, we did not store {g} * (1) , that is, we dropped duplicate rows from the context K 2 := C * (2) A , ⊥ . We also removed the binary projections that describe the full transformation monoid O A , ⊥ with a 10,263-element subset G ⊆ C * (2) A of binary idempotent non-projections that has the same closure system of monoids (intents) as K 2 .
Proof. Straightforward (time consuming) computation from the given data.
As in [27], we also use conjugation by inner automorphisms to simplify the situation.
We define the conjugate f s ∈ O (n) are conjugate if there is s ∈ Sym A such that g = f s , and we can hence form conjugacy classes of functions from this equivalence relation. It is helpful to extend the conjugation notation to sets F ⊆ O A by an element-wise definition: for s ∈ Sym A we stipulate F s := { f s | f ∈ F}. Clearly, this conjugation notion respects any composition structure that F may have. The following observation is an immediate consequence from the fact Lemma 33 (cf. Observation 5.4 in [27]). For all n ∈ N, F ⊆ O A and permutations s ∈ Sym A we have F s * (n) = F * (n) s .

Proof. We have
from [27]. By the injectivity of f → f s the latter equals Mainly relevant in the context of centralising monoids is the case n = 1 of Lemma 33, where it states that the conjugate by s of a centralising monoids witnessed by F is the centralising monoid witnessed by the conjugate F s of F. As the centralising monoid F * (1) is isomorphic to its conjugate via f → f s , it is a useful initial step to simplify a large list of monoids given by witnesses via separating conjugacy classes of the witnesses. In a second turn one can then classify the remaining monoids according to element-wise conjugacy.
In this respect we obtained the following results: the 10,263 idempotent binary nonprojections constituting the objects G of K 2 can be classified into 2274 conjugacy types by inner automorphisms. Similarly, the 1236 binary operations inG ⊆ G can be grouped together into 142 conjugacy classes (their representatives are marked in Appendix B by over-or underlining), and the 612 idempotent binary operations describing maximal elements with respect to K 2 fall into 77 conjugacy types (representatives are marked in Appendix B by ↑ and over-or underlining, the monoids are listed in Appendix A, the classification of witnesses into conjugacy classes is given in Appendix C). The respective numbers of monoids up to element-wise conjugacy are therefore at most as big as these values, but generally reduce further. For example, the 1236 monoids {g} * (1) for g ∈G can be separated into 83 classes up to element-wise conjugacy. They correspond to those operations in Appendix B that are overlined. The other numbers are contained in the following theorem. A , 563 proper ones that can be witnessed by singletons, and 42 that are inclusion maximal proper monoids (among monoids witnessed by binary idempotent non-projections). These 42 monoids are shown in Appendix A.1 and correspond to those operations in Appendix B that are overlined and marked with ↑.

Monoids Witnessed by Mal'cev Operations of Boolean Groups
In this section, we are going to consider a set A carrying a Boolean group A; +, −, 0 , i.e., fulfilling the law x + x ≈ 0. Such a group is necessarily Abelian, and since every nonunit element has order two, there is a natural Z 2 -vector space structure on A by defining 1 · a := a and 0 · a := 0 for all a ∈ A. It follows from this that if A is finite, then |A| = 2 d for some d ∈ N. As the group is Boolean, it satisfies −x ≈ x and any linear combinations simply reduce to sums. All identities holding in A clearly also transfer to its powers, and notationally we shall not distinguish between the (infix) addition on A or on a power, as it is common in linear algebra.
Following case (III) of Rosenberg's Theorem 4, we consider the Mal'cev (minority) A given as f (x, y, z) := x − y + z = x + y + z for x, y, z ∈ A. Our goal is to prove via Corollary 12 that { f } * (1) always is a maximal centralising monoid on A, given |A| ≥ 4.
From the theory of maximal clones on finite sets, the centraliser of the Mal'cev operation of any Abelian p-group for prime p is very well known, see, e.g., [38] (4.3.13 Lemma, p. 107) and note that the quaternary relation G given in [38]  A be its Mal'cev operation and g ∈ O (n) A with n ∈ N. For L := { f } * we have g ∈ L ⇐⇒ ∀x, y, z ∈ A n : g(x + y + z) = g( f • (x, y, z)) = f (g(x), g(y), g(z)) = g(x) + g(y) + g(z) ⇐⇒ ∀x, y, z, u ∈ A n : u = x + y + z =⇒ g(u) = g(x) + g(y) + g(z) ⇐⇒ ∀x, y, z, u ∈ A n : Since the centraliser L = { f } * of the Mal'cev operation of a p-group is the clone of affine (linear) functions, it will be necessary to review a few basic facts regarding affine functions and affine (in)dependence.
Elements a 1 , . . . , a n in an affine space A are affinely independent if and only if for every linear combination 0 = λ 1 a 1 + · · · + λ n a n with λ 1 + · · · + λ n = 0 necessarily all coefficients vanish: λ 1 = · · · = λ n = 0. Rephrasing this, they are affinely dependent precisely if there is a linear combination 0 = λ 1 a 1 + · · · + λ n a n with λ 1 + · · · + λ n = 0 with some non-zero coefficients in it. Of course, we can remove all the coefficients which are zero from the sum without changing this condition. Thus, {a 1 , . . . , a n } are affinely dependent if and only if there is a subset ∅ = S ⊆ {a 1 , . . . , a n } and non-zero coefficients λ s for s ∈ S such that 0 = ∑ s∈S λ s s and ∑ s∈S λ s = 0. In the context of an affine space over Z 2 , all λ s ∈ Z 2 \ {0} = {1} are equal to 1. Hence, over Z 2 a set {a 1 , . . . , a n } is affinely dependent if there is a non-empty subset S ⊆ {a 1 , . . . , a n } with ∑ s∈S s = 0 and 0 = ∑ s∈S 1 = |S| mod 2. Thus, a 1 , . . . , a n are affinely independent over Z 2 if every non-empty sum of evenly many of these elements is distinct from the zero-vector 0.
The following special cases of affine independence over Z 2 will be relevant for our proof: a 1 , a 2 ∈ A are affinely independent if and only if a 1 = a 2 since a 1 + a 2 = 0 is equivalent to a 1 = a 2 ; three general elements a 1 , a 2 , a 3 ∈ A are affinely independent if and only if |{a 1 , a 2 , a 3 }| = 3, i.e., they are pairwise distinct. Moreover, a 1 , a 2 , a 3 , a 4 ∈ A are affinely independent exactly if |{a 1 , a 2 , a 3 , a 4 }| = 4 and a 1 + a 2 + a 3 + a 4 = 0. Therefore, four pairwise distinct elements a 1 , a 2 , a 3 , a 4 of A are affinely independent precisely if a 1 + a 2 + a 3 + a 4 = 0.
The importance of affine independence in our situation comes from the following extendability property, which is part of a typical introductory course to linear algebra (see, e.g., Satz 6.3.6 in [39] (p. 156)).

Proposition 36.
Let A and A be affine spaces over the same field and B ⊆ A be an affine basis of A. Then for every partial assignment (g(b)) b∈B ∈ A B there is a unique extension to an affine linear map g : A −→ A .
Since every affinely independent set I ⊆ A can be extended to an affine basis B ⊆ A, every partial assignment (g(x)) x∈I ∈ A I is extendable to an affine map g : A −→ A . A with h / ∈ { f } * , i.e., the condition given in Lemma 35 is falsified by h. We proceed through a series of claims.

Claim 1.
There are pairwise distinct affinely dependent x, y, z, u ∈ A for which there is a function g ∈ M with (x, y) ∈ ker(g) and (z, u) / ∈ ker(g).

Proof.
To prove this claim we use the operation h / ∈ { f } * (1) . This means there are (affinely dependent) x + y + z + u = 0 in A for which h(x) + h(y) + h(z) + h(u) = 0. If any two elements from (x, y, z, u) were equal, then because of x + y + z + u = 0 also the other two would have to be equal. It would follow that h(x) + h(y) + h(z) + h(u) = 0 for it would be a sum of two pairs of equal summands. This contradicts the assumption on x, y, z, u. Therefore, we know that x + y + z + u = 0 and h(x) + h(y) + h(z) + h(u) = 0 and x, y, z, u are pairwise distinct. These conditions are completely symmetric under permutation of the four values. If there are any two that are sent by h to the same value, then because of 0 = h(x) + h(y) + h(z) + h(u) the other two are not in the kernel of h, and h and some permutation of (x, y, z, u) will show Claim 1. Now let us assume that all four h-values are pairwise distinct. Then h(x), h(y), h(z), h(u) are affinely independent, and by Proposition 36 we choose some affine linear map g ∈ L (1) ⊆ M satisfying g(h(y)) := h(x) and g(h(w)) := h(w) for w ∈ {x, z, u}.
. So g • h will show Claim 1.

Claim 2.
For all pairwise distinct affinely dependent x, y, z, u ∈ A there is a function g ∈ M with (x, y) ∈ ker(g) and (z, u) / ∈ ker(g).
Proof. Let x, y, z, u ∈ A be pairwise distinct. If they are affinely dependent, the result follows from Claim 2; hence suppose that x, y, z, u are affinely independent. In this case we use Proposition 36 to construct g ∈ L (1) ⊆ M with g(x) = g(y) = g(z) = 0 = g(u), so that (x, y) ∈ ker(g) and (z, u) / ∈ ker(g).
We now take advantage of Corollary 12 to complete the proof of Lemma 37. Condition (D') is provided by Claim 3; condition (B') holds as any pairwise distinct x, y, z ∈ A are affinely independent, and so we may again rely on Proposition 36 to pick some g ∈ L (1) ⊆ M with g(y) = g(x) = g(z). Similarly, for each a ∈ A this single point is affinely independent, and so Proposition 36 can be invoked to define some g ∈ L (1) ⊆ M with g(a) = a, i.e., a / ∈ fix(g). Consequently, Corollary 12 implies M * = J A . A . Namely, take pairwise distinct x, y, z ∈ A and let u := x + y + z. Since |{x, y, z}| = 3, u / ∈ {x, y, z}. We define h(u) = 0 and h(a) := 0 for a ∈ A \ {u}, in particular h(x) = h(y) = h(z) = 0. If h were affine linear, then, by Lemma 35, we would have h(u) = h(x) + h(y) + h(z) = 0 + 0 + 0 = 0, contradicting the definition of h. Hence,

Remark 39.
Four-element sets are the smallest ones where Theorem 38 (non-vacuously) holds. Three-element sets do not support Boolean groups-for this reason monoids witnessed by Mal'cev operations did not appear in the analyses given in [21,23]; and for |A| ≤ 2 we have L (1) = O  Acknowledgments: Both authors gladly acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Programme. Edith Vargas-García gratefully acknowledges support by the Asociación Mexicana de Cultura A.C.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.