Class Of Monoids Research Articles

Consistency of Relations over Monoids

The interplay between local consistency and global consistency has been the object of study in several different areas, including probability theory, relational databases, and quantum information. For relational databases, Beeri, Fagin, Maier, and Yannakakis showed that a database schema is acyclic if and only if it has the local-to-global consistency property for relations, which means that every collection of pairwise consistent relations over the schema is globally consistent. More recently, the same result has been shown under bag semantics. In this paper, we carry out a systematic study of local vs. global consistency for relations over positive commutative monoids, which is a common generalization of ordinary relations and bags. Let K be an arbitrary positive commutative monoid. We begin by showing that acyclicity of the schema is a necessary condition for the local-to-global consistency property for K-relations to hold. Unlike the case of ordinary relations and bags, however, we show that acyclicity is not always sufficient. After this, we characterize the positive commutative monoids for which acyclicity is both necessary and sufficient for the local-to-global consistency property to hold; this characterization involves a combinatorial property of monoids, which we call the transportation property. We then identify several different classes of monoids that possess the transportation property. As our final contribution, we introduce a modified notion of local consistency of K-relations, which we call pairwise consistency up to the free cover. We prove that, for all positive commutative monoids K, even those without the transportation property, acyclicity is both necessary and sufficient for every family of K-relations that is pairwise consistent up to the free cover to be globally consistent.

On the word problem for weakly compressible monoids

We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if C is a reversal-closed super-, as defined by Greibach, then M has word problem in C if and only if its compressed left monoid L(M) has word problem in C. As a special case, we may take C to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finally, we show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. This answers a generalization of a question first asked by Zhang in 1992.

Approximating length-based invariants in atomic Puiseux monoids

A numerical monoid is a cofinite additive submonoid of the nonnegative integers, while a Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. Using that a Puiseux monoid is an increasing union of copies of numerical monoids, we prove that some of the factorization invariants of these two classes of monoids are related through a limiting process. This allows us to extend results from numerical to Puiseux monoids. We illustrate the versatility of this technique by recovering various known results about Puiseux monoids.

Matrix homomorphism equations on a class of monoids and non Abelian groupoids

Let α,β∈R. The aim of this paper is to give an explicit description of the solutions f:R×R→M2(C) of the following parametric functional equationsf(x1x2+αy1y2,x1y2+βx2y1)=f(x1,y1)f(x2,y2),that arise from number theory. Depending on the value of β, we present different methods for solving these equations. The solutions of a matrix multiplicative Cauchy functional equation on abelian regular semigroups are given. These results apply to more general equations.

On covers of acts over monoids with condition $$(PF'')$$ and envelopes of S-acts

In 2001, Enochs’ celebrated flat cover conjecture was finally proven, and the proofs have since generated a great deal of interest among researchers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of acts over monoids which concerned with the coessential epimorphisms. In this paper, we restrict our attention to $$(PF'')$$ -covers (coessential-covers that satisfy Condition $$(PF'')$$ ). We give a necessary and sufficient condition for cyclic act to have a $$(PF'')$$ -cover and give some classes of monoids that all cyclic right S-acts have a Condition $$(PF'')$$ -cover, we show that every weakly pullback flat cover is also $$(PF'')$$ -cover and every $$(PF'')$$ -cover is $$(P')$$ -cover, but the converses are not true. Besides we consider the $$\mathcal {PF''}$$ -cover (Enochs’ notion of a cover) and obtain some properties. Finally, we define and investigate the envelopes of S-acts.

S-полигоны над вполне упорядоченным моноидом с модулярной решеткой конгруэнций

This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$–act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$–acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$–acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.

Classes of monoids with applications: formations of languages and multiply local formations of finite groups

The Frattini subgroup $$\varPhi (G)$$ of a group G is the intersection of all maximal subgroups of G. A formation $${\mathfrak {F}}$$ of groups is said to be saturated if $$G/\varPhi (G) \in {\mathfrak {F}}$$ always implies $$G \in {\mathfrak {F}}$$ . A formation of finite groups is saturated iff it is local. A local satellite of $${\mathfrak {F}}$$ is a function with domain the set of all primes whose images are formations of finite groups. If the values of this function are themselves local formations, then this circumstance leads to the definition of multiply local formation. The languages corresponding to n-multiply local and totally local formations of finite groups are described.

Обратная задача для моноида с экспоненциальной последовательностью простых

В работе для произвольного моноида ${M(PE)}$ с экспоненциальной последовательностью простых чисел $PE$ типа $q$ решается обратная задача, то есть нахождение асимптотики для функции распределения элементов моноида ${M(PE)}$, исходя из асимптотики распределения простых чисел последовательности простых чисел $PE$ типа $q$.Для решения этой задачи вводится понятие произвольной экспоненциальной последовательности натуральных чисел типа $q$ и рассматривается моноид, порожденный этой последовательностью. С помощью двух гомоморфизмов таких моноидов задача о распределении плотности сводится к аддитивной задаче Ингама.Показано, что для этого класса моноидов понятие степенной плотности не работает. Введено новое понятие $C$ логарифмической $\theta$-степенной плотности.Показано, что любой моноид ${M(PE)}$ для произвольной экспоненциальной последовательности простых $PE$ типа $q$ имеет $C$ логарифмическую $\theta$-степенную плотность с $C=\pi\sqrt{\frac{2}{3\ln q}}$ и $\theta=\frac{1}{2}$.

Classification of monoids by Condition $(PWP_{ssc})$

Condition $(PWP)$ which was introduced in (Laan, V., {it Pullbacks and flatness properties of acts I}, Commun. Algebra, 29(2) (2001), 829-850), is related to flatness concept of acts over monoids. Golchin and Mohammadzadeh in ({it On Condition $(PWP_E)$}, Southeast Asian Bull. Math., 33 (2009), 245-256) introduced Condition $(PWP_E)$, such that Condition $(PWP)$ implies it, that is, Condition $(PWP_E)$ is a generalization of Condition $(PWP)$. In this paper we introduce Condition $(PWP_{ssc})$, which is much easier to check than Conditions $(PWP)$ and $(PWP_E)$ and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition $(PWP_{ssc})$ will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition $(PWP_{ssc})$ and also a classification of monoids $S$ will be given for which all right $S$-acts satisfying some other flatness properties have Condition $(PWP_{ssc})$.

A note on identities in two variables for a class of monoids

A note on identities in two variables for a class of monoids

On the local k-elasticities of Puiseux monoids

If [Formula: see text] is an atomic monoid and [Formula: see text] is a nonzero non-unit element of [Formula: see text], then the set of lengths [Formula: see text] of [Formula: see text] is the set of all possible lengths of factorizations of [Formula: see text], where the length of a factorization is the number of irreducible factors (counting repetitions). In a recent paper, F. Gotti and C. O’Neil studied the sets of elasticities [Formula: see text] of Puiseux monoids [Formula: see text]. Here, we take this study a step further and explore the local [Formula: see text]-elasticities of the same class of monoids. We find conditions under which Puiseux monoids have all their local elasticities finite as well as conditions under which they have infinite local [Formula: see text]-elasticities for sufficiently large [Formula: see text]. Finally, we focus our study of the [Formula: see text]-elasticities on the class of primary Puiseux monoids, proving that they have finite local [Formula: see text]-elasticities if either they are boundedly generated and do not have any stable atoms or if they do not contain [Formula: see text] as a limit point.

Decidability and independence of conjugacy problems in finitely presented monoids

There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by ∼p, ∼o, and ∼c) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, p-conjugacy is “almost” transitive, ∼c is strictly included in ∼p, and the p- and c-conjugacy problems are decidable with linear complexity on a two-tape Turing Machine. For other classes of monoids, the situation is more complicated. We show that there exists a monoid M defined by a finite complete presentation such that the c-conjugacy problem for M is undecidable, and that for finitely presented monoids, the c-conjugacy problem and the word problem are independent, as are the c-conjugacy and p-conjugacy problems. On other hand, we show that for finitely presented monoids, the o-conjugacy problem is reducible to the c-conjugacy problem.

A correspondence between a class of monoids and self-similar group actions II

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa–Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych. In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be significant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

Markov chains, ${\mathscr R}$-trivial monoids and representation theory

We develop a general theory of Markov chains realizable as random walks on [Formula: see text]-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom–Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

The Ranks of Ideals in Various Transformation Monoids

In this paper we consider various classes of monoids of transformations of a finite chain, including those of transformations that preserve or reverse either the order or the orientation. In line with Howie and McFadden [24], we complete the study of the ranks (and of idempotent ranks, when applicable) of all their ideals.

Boolean topological graphs of semigroups: the lack of first-order axiomatization

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class 𝒞 of algebras let G(𝒞)={G(A)∣A∈𝒞}. Assume that 𝒞 is a class of semigroups possessing a nontrivial member with a neutral element and let ℋ be the universal Horn class generated by G(𝒞). We prove that the Boolean core of ℋ, i.e., the topological prevariety generated by finite members of ℋ equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ℋ. We derive analogous results when 𝒞 is a class of monoids or groups with a nontrivial member.

On the automaticity of singular Artin monoids of finite type

Automaticity is an important concept in group theory as it yields an efficient solution to the word problem and provides other possibilities for effective computation. The concept of automaticity generalizes naturally from groups to monoids and semigroups and the efficiency of the solution of the word problem is preserved when we do this. Whilst this subject has been studied extensively (in both the group and the monoid/semigroup case), there are still some deep and major open problems, including questions concerning the automaticity of certain naturally occurring classes of groups, monoids and semigroups. In this paper, we consider two such classes of monoids, namely the positive singular Artin monoids of finite type and the singular Artin monoids of the finite type. The main purpose here is to show that these monoids are all automatic. When establishing the automaticity of monoids, one obstacle is that we often have asynchronous finite automata recognizing multiplication and we need to establish the existence of synchronous machines accomplishing the same task. Building on the work of Frougny and Sakarovitch, we establish a new criterion for achieving such a transition; this is fundamental in the establishment of the automaticity of the monoids we consider here and may well apply to other naturally occurring classes of monoids and semigroups as well.

Dimension polynomials in the generalized difference case

The aim of this paper is to generalize results on dimension polynomials of difference modules over difference rings for a wider class of rings of difference operators. We introduce the notion of quasi-commutativity, which generalizes the notion of commutativity and enables one to consider wider classes of monoids and groups of endomorphisms. Some properties of quasi-commutative monoids and groups are established; these properties allow us to apply some methods that are almost similar to the ones used in working with free commutative monoids and groups. Also we prove the theorem of existence of the dimension polynomial of generalized difference modules in the cases where the submonoid of endomorphisms is free quasi-commutative. Also the existence of its analog for the case of a direct product of a free quasi-commutative monoid and a finite cyclic group is established.

On a generalization of principal weak flatness property

In this article a generalization of principal weak flatness of acts is defined. Using this new property the characterization of some new classes of monoids are investigated. Furthermore, many known results are generalized.

Classes of Tree Languages and DR Tree Languages Given by Classes of Semigroups

In the first section of the paper we give general conditions under which a class of recognizable tree languages with a given property can be defined by a class of monoids or semigroups defining the class of string languages having the same property. In the second part similar questions are studied for classes of (DR) tree languages recognized by deterministic root-to-frontier tree recognizers.