Search Results (72)

In this paper we present a detailed study of bonded knots and their related structures, integrating recent developments into a single framework. Bonded knots are classical knots endowed with embedded bonding arcs modeling physical or chemical bonds. We consider bonded knots in three categories (long, standard, and tight) according to the type of bonds, and in two categories, topological vertex and rigid vertex, according to the allowed isotopy moves, and we define invariants for each category. We then develop the theory of bonded braids, the algebraic counterpart of bonded knots. We define the bonded braid monoid, with its generators and relations, and formulate the analogues of the Alexander and Markov theorems for bonded braids in the form of L-equivalence for bonded braids. Next, we introduce enhanced bonded knots and braids, incorporating two types of bonds (attracting and repelling) corresponding to different interactions. We define the enhanced bonded braid group and show how the bonded braid monoid embeds into this group. These models capture the topology of chains with inter and intra-chain bonds and suggest new invariants for classifying biological macromolecules. Full article

We ask whether there exist non-isomorphic trace monoids over a fixed alphabet that have the same average parallelism. This question is related to the bivariate generating series $F$ which counts traces by their height and length; trace monoids with the same $F$ also possess the same average parallelism. The series $F$ is known to be rational and has been calculated efficiently via the symmetries of the dependence graph, when the latter is connected. We investigate the existence of non-isomorphic dependence graphs (over a common fixed alphabet) with the same series $F$. Using fractional graph isomorphisms and certain equitable partitions of the Cartier-Foata clique automaton, we prove two classification results. First, we show that all$2$-regular independence graphs of the same order share the same generating series $F$ if and only if they have the same number of triangular connected components. Secondly, for any $d\ge 2$, all triangle-free $d$-regular independence graphs of the same order—except for the complete bipartite graph $K_{d,d}$—share this property. The smallest instance of this result for $d=3,$ is the pair consisting of the cube graph $Q_3$ and the Wagner graph $M_8$, both on eight vertices. Full article

A left group is a semigroup that is the direct product of a left zero semigroup and a group. In this paper, we investigate some group-like properties of left groups. In particular, we characterise monogenic left group monoids and prove some fundamental results on left group morphisms. We also characterise Green’s equivalences on left groups and, analogously to groups, establish a correspondence between congruences and normal sub-left groups. Finally, we characterise DSC left groups. Full article

This article proposes a unified concept of topological types of convergence for nets of multifunctions between topological spaces. Any kind of convergence is representable by a (2n + 2)-tuple, n = 0, 1, …, of two special functions u and l, such that their compositions $ul$ and $lu$ create the Choquet supremum and infimum operations, respectively, on the filters considered in terms of the upper Vietoris topology on the range hyperspace of the considered multifunctions. Convergence operators are defined by establishing the order of composition of the functions from such (2n + 2) tuples. An allocation of places for the two distinguished functions in a convergence operator reflects the structure of the used (2n + 2)-tuple. A monoid of special three-parameter functions called products describes the set of all possible structures. The monoid of products is the domain space of the convergence operators. The family of all convergence operators forms a finite monoid whose neutral element determines the pointwise convergence and possesses the structure determined by the neutral element of the monoid of products. We demonstrate the construction process of every convergence operator and show that the notions of the presented concept can characterize many well-known classical types of convergence. Of particular importance are the types of convergence derived from the concept of continuous convergence. We establish some general theorems about the necessary and sufficient conditions for the continuity of the limit multifunctions without any assumptions about the type of continuity of the members of the nets. Full article

The usual inputs for a causal identification task are a graph representing qualitative causal hypotheses and a joint probability distribution for some of the causal model’s variables when they are observed rather than intervened on. Alternatively, the available probabilities sometimes come from a combination of passive observations and controlled experiments. It also makes sense, however, to consider causal identification with data collected via schemes more generic than (perfect) passive observation or perfect controlled experiments. For example, observation procedures may be noisy, may disturb the variables, or may yield only coarse-grained specification of the variables’ values. In this work, we investigate identification of causal quantities when the probabilities available for inference are the probabilities of outcomes of these more generic schemes. Using process theories (aka symmetric monoidal categories), we formulate graphical causal models as second-order processes that respond to such data collection instruments. We pose the causal identification problem relative to arbitrary sets of available instruments. Perfect passive observation instruments—those that produce the usual observational probabilities used in causal inference—satisfy an abstract process-theoretic property called marginal informational completeness. This property also holds for other (sets of) instruments. The main finding is that in the case of Markovian models, as long as the available instruments satisfy this property, the probabilities they produce suffice for identification of interventional quantities, just as those produced by perfect passive observations do. This finding sharpens the distinction between the Markovianity of a causal model and that of a probability distribution, suggesting a more extensive line of investigation of causal inference within a process-theoretic framework. Full article

A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of persistent homology in this context, its interaction with the ordering, and the repercussions of merging multigraphs in the calculation of Betti numbers. For the latter, an extended version of the incremental algorithm is provided. The ideas developed here are mainly oriented to the original example described by the author and others in the context of the formalization of the notion of embodiment in Neuroscience. Full article

Causal discovery involves searching intractably large spaces. Decomposing the search space into classes of observationally equivalent causal models is a well-studied avenue to making discovery tractable. This paper studies the topological structure underlying causal equivalence to develop a categorical formulation of Chickering’s transformational characterization of Bayesian networks. A homotopic generalization of the Meek–Chickering theorem on the connectivity structure within causal equivalence classes and a topological representation of Greedy Equivalence Search (GES) that moves from one equivalence class of models to the next are described. Specifically, this work defines causal models as propable symmetric monoidal categories (cPROPs), which define a functor category ${\mathcal{C}}^P$ from a coalgebraic PROP P to a symmetric monoidal category $\mathcal{C}$. Such functor categories were first studied by Fox, who showed that they define the right adjoint of the inclusion of Cartesian categories in the larger category of all symmetric monoidal categories. cPROPs are an algebraic theory in the sense of Lawvere. cPROPs are related to previous categorical causal models, such as Markov categories and affine CDU categories, which can be viewed as defined by cPROP maps specifying the semantics of comonoidal structures corresponding to the “copy-delete” mechanisms. This work characterizes Pearl’s structural causal models (SCMs) in terms of Cartesian cPROPs, where the morphisms that define the endogenous variables are purely deterministic. A higher algebraic K-theory of causality is developed by studying the classifying spaces of observationally equivalent causal cPROP models by constructing their simplicial realization through the nerve functor. It is shown that Meek–Chickering causal DAG equivalence generalizes to induce a homotopic equivalence across observationally equivalent cPROP functors. A homotopic generalization of the Meek–Chickering theorem is presented, where covered edge reversals connecting equivalent DAGs induce natural transformations between homotopically equivalent cPROP functors and correspond to an equivalence structure on the corresponding string diagrams. The Grothendieck group completion of cPROP causal models is defined using the Grayson–Quillen construction and relate the classifying space of cPROP causal equivalence classes to classifying spaces of an induced groupoid. A real-world domain modeling genetic mutations in cancer is used to illustrate the framework in this paper. Full article

Cyber-physical systems (CPSs) are fundamental components of Industry 4.0 production environments. Their interconnection is crucial for the successful implementation of distributed and autonomous production plans. A particularly relevant challenge is the optimal scheduling of tasks that require the collaboration of multiple CPSs. To ensure the feasibility of optimal schedules, two primary issues must be addressed: (1) The design of global systems emerging from the interconnection of CPSs; (2) The development of a scheduling formalism tailored to interconnected Industry 4.0 settings. Our approach is based on a Category Theory formalization of interconnections as compositions. This framework aims to guarantee that the emergent behaviors align with the intended outcomes. Building upon this foundation, we introduce a formalism that captures the assignment of operations to cyber-physical systems. Full article

The aims of this paper are two-fold. First, we present the result of the decomposition on the iterations of a Collatz transform into arithmetic sequences. With this, we prove that in Furstenberg topology, the set of (odd) integers with an infinite stopping time is closed and nowhere dense. Then, we move our considerations to some monoids $\mathbb{L}$ in $\mathbb{N}$, where we define a suitably modified Collatz transform, and we present some results of numerical investigations on the behaviour of these modified transforms. Full article

A semigroup S is termed a generalized Clifford semigroup (GC-semigroup) if it forms a strong semilattice of $\pi$-groups. This paper explores necessary and sufficient conditions for a GC-monoid to be nearly complete within certain subclasses. These subclasses are distinguished by the nature of their linking homomorphisms, which may be bijective, surjective, injective, or image trivial. The findings provide a deeper understanding of the structural integrity and completeness of GC-monoids, contributing valuable insights to the theoretical framework of semigroup theory. Applications of this study span various fields, including cryptography for secure algorithm design, coding theory and quantum computing for advanced quantum algorithms. The established criteria also support further research in mathematical biology and automorphic theory, demonstrating the broad relevance and utility of nearly complete GC-monoids. Full article

Let M be an Abelian monoid. A necessary and sufficient condition for the class $\mathcal{A}r{m}_M$ of all Armendariz rings relative to M to coincide with the class $\mathcal{A}rm$ of all Armendariz rings is given. As a consequence, we prove that $\mathcal{A}r{m}_M$ has exactly three cases: the empty set, $\mathcal{A}rm$, and the class of all rings. If N is an Abelian monoid, then we prove that $\mathcal{A}r{m}_{M\times N}=\mathcal{A}r{m}_M\bigcap \mathcal{A}r{m}_N$, which gives a partial affirmative answer to the open question of Liu in 2005 (whether R is $M\times N$-Armendariz if R is M-Armendariz and N-Armendariz). We also show that the other Armendariz-like rings relative to an Abelian monoid, such as M-quasi-Armendariz rings, skew M-Armendariz rings, weak M-Armendariz rings, M-$\pi$-Armendariz rings, nil M-Armendariz rings, upper nil M-Armendariz rings and lower nil M-Armendariz rings can be handled similarly. Some conclusions on these classes have, therefore, been generalized using these classifications. Full article

Let $\mathbb{S}$ be a numerical monoid, while a ${\mathcal{P}}_{\mathrm{fin}}\left(\mathbb{S}\right)$-monoid S is a monoid generated by a finite number of finite non-empty subsets of $\mathbb{S}$. That is, S is a non-cancellative commutative monoid obtained from the sumset of finite non-negative integer sets. This work provides an algorithm for computing the ideals associated with some ${\mathcal{P}}_{\mathrm{fin}}\left(\mathbb{S}\right)$-monoids. These are the key to studying some factorization properties of ${\mathcal{P}}_{\mathrm{fin}}\left(\mathbb{S}\right)$-monoids and some additive properties of sumsets. This approach links computational commutative algebra with additive number theory. Full article

The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a notion of “flat quantale morphism” as an abstraction of flat ring morphisms. For this, we start from a characterization of the flat ring morphism in terms of the ideal residuation theory. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat ring morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional coherent quantales are obtained, formulated in terms of Going-down, Going-up, and localization. We also prove two characterization theorems for the coherent quantales of dimension at most one. The results of the paper can be applied both in the theory of commutative rings and to other algebraic structures: F-rings, semirings, bounded distributive lattices, commutative monoids, etc. Full article

A gap a of a numerical semigroup S is fundamental if $\{2a,3a\}\subseteq S.$ In this work, we will study the set $\mathcal{B}\left(a\right)=\left\{S\mid S\mathrm{is}\mathrm{a}\mathrm{numerical}\mathrm{semigroup}\mathrm{and}a\mathrm{is}\mathrm{a}\mathrm{fundamental}\mathrm{gap}\mathrm{of}S\right\}.$ In particular, we will give an algorithm to compute all the elements of $\mathcal{B}\left(a\right)$ with a given genus. The intersection of two elements of $\mathcal{B}\left(a\right)$ is again one element of $\mathcal{B}\left(a\right).$ A $\mathcal{B}\left(a\right)$-irreducible numerical semigroup is an element of $\mathcal{B}\left(a\right)$ that cannot be expressed as an intersection of two elements of $\mathcal{B}\left(a\right)$ containing it properly. In this paper, we will study the $\mathcal{B}\left(a\right)$-irreducible numerical semigroups. In this sense we will give an algorithm to calculate all of them. Finally, we will study the submonoids of $(\mathbb{N},+)$ that can be expressed as an intersection (finite or infinite) of elements belonging to $\mathcal{B}\left(a\right).$ Full article

The skew generalized power series ring $R\left[\right[S,\omega \left]\right]$ is a ring construction involving a ring R, a strictly ordered monoid $(S,\le )$, and a monoid homomorphism $\omega :S\to End\left(R\right)$. The ring $R\left[\right[S,\omega \left]\right]$ is a common generalization of ring extensions such as (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) Mal’cev–Neumann series rings, and (skew) monoid rings. In this paper, we study the nilpotent elements of skew generalized power series rings and the relationships between the properties of the rings R and $R\left[\right[S,\omega \left]\right]$ expressed in terms of annihilators, such as weak symmetry, weak zip, and the nil-Armendariz and McCoy properties. We obtain results on transferring the weak symmetry and weak zip properties between the rings R and $R\left[\right[S,\omega \left]\right]$, as well as sufficient and necessary conditions for a ring R to be $(S,\omega )$-nil-Armendariz or linearly $(S,\omega )$-McCoy. Full article