Free Monoid Research Articles

The universal zero-sum invariant and weighted zero-sum for infinite abelian groups

Let G be an abelian group, and let F ( G ) be the free commutative monoid with basis G. For Ω ⊂ F ( G ) , define the universal zero-sum invariant d Ω ( G ) to be the smallest integer l such that every sequence T over G of length l has a subsequence in Ω . The invariant d Ω ( G ) unifies many classical zero-sum invariants. Let B ( G ) be the submonoid of F ( G ) consisting of all zero-sum sequences over G, and let A ( G ) be the set consisting of all minimal zero-sum sequences over G. The empty sequence, which is the identity of B ( G ) , is denoted by ε . The well-known Davenport constant D ( G ) of the group G can be also represented as d B ( G ) ∖ { ε } ( G ) or d A ( G ) ( G ) in terms of the universal zero-sum invariant. Notice that A ( G ) is the unique minimal generating set of the monoid B ( G ) from the point of view of Algebra. Hence, it would be interesting to determine whether A ( G ) is minimal to represent the Davenport constant or not for a general finite abelian group G. In this paper, we show that except for a few special classes of groups, there always exists a proper subset Ω of A ( G ) such that d Ω ( G ) = D ( G ) . Furthermore, in the setting of finite cyclic groups, we discuss the distributions of all minimal sets by determining their intersections. By connecting the universal zero-sum invariant with weights, we make a study of zero-sum problems in the setting of infinite abelian groups. The universal zero-sum invariant d Ω ; Ψ ( G ) with weights set Ψ of homomorphisms of groups is introduced for all abelian groups. The weighted Davenport constant D Ψ ( G ) (being an special form of the universal invariant with weights) is also investigated for infinite abelian groups. Among other results, we obtain the necessary and sufficient conditions such that D Ψ ( G ) < ∞ in terms of the weights set Ψ when | Ψ | is finite. In doing this, by using the Neumann Theorem on Cover Theory for groups we establish a connection between the existence of a finite cover of an abelian group G by cosets of some given subgroups of G, and the finiteness of weighted Davenport constant.

Commuting Upper Triangular Binary Morphisms

A morphism g from the free monoid X* into itself is called upper triangular if the matrix of g is upper triangular. We characterize all upper triangular binary morphisms g1 and g2 such that g1g2 = g2g1.

Finitely and non-finitely related words

An algebra is finitely related (or has finite degree) if its term functions are determined by some finite set of finitary relations. Nilpotent monoids built from words, via Rees quotients of free monoids, have been used to exhibit many interesting properties with respect to the finite basis problem. We show that much of this intriguing behaviour extends to the world of finite relatedness by using interlocking patterns called chain, crown, and maelstrom words. In particular, we show that there are large classes of non-finitely related nilpotent monoids that can be used to construct examples of: ascending chains of varieties alternating between finitely and non-finitely related; non-finitely related semigroups whose direct product are finitely related; the addition of an identity element to a non-finitely related semigroup to produce a finitely related semigroup.

Discrete Event Systems Theory for Fast Stochastic Simulation via Tree Expansion

Discrete Event Systems Theory for Fast Stochastic Simulation via Tree Expansion

Simplicity of twisted C*-algebras of Deaconu–Renault groupoids

We consider Deaconu–Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2 -cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions.

Solution for 3rd order Cauchy Difference Equation in Free Monoid

— Let k: C ? D be a function, where (C,.) is a Group and (D,+) is an Abelian Group. In this article, the Third Order Cauchy difference of k:E(3) k (c1,c2,c3,c4)= k(c1c2c3c4) – k(c1c2c3) – k(c1c3c4) – k (c1c2c3c4) –k (c2c3c4) + k(c1c2) + k (c1c3) +k(c1c4)+k(c2c3)+ k(c2c4)+k(c3c4) –k(c1) – k(c2) –k(c3) –k(c4) ? c1,c2,c3,c4 C is studied. We give solutions of E(3)k=0 on Free Monoid.

Topologies for error-detecting variable-length codes

Given a finite alphabet A and a quasi-metric d over A⁎, we introduce the relation τd,k⊆A⁎×A⁎ such that (x,y)∈τd,k holds whenever d(x,y)≤k. The error detection capability of variable-length codes is expressed in term of conditions over τd,k. With respect to the prefix metric, the factor one, and any quasi-metric associated with some free monoid (anti-)automorphism, we prove that one can decide whether a given regular variable-length code satisfies any of those error detection constraints.

Post's Correspondence Problem for hyperbolic and virtually nilpotent groups

AbstractPost's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms there exists any non‐trivial such that . PCP for a group takes pairs of group homomorphisms instead, and similarly asks whether there exists an such that holds for non‐elementary reasons. The restrictions imposed on in order to get non‐elementary solutions lead to several interpretations of the problem; we mainly consider the natural restriction asking that and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic , but decidable when is virtually nilpotent, en route also studying this problem for finite extensions. We also consider a different interpretation of the PCP due to Myasnikov, Nikolaev and Ushakov, proving decidability for torsion‐free nilpotent groups.

Strongly Clean Matrix Rings over a Skew Monoid Ring

Let [Formula: see text] be a ring with an endomorphism [Formula: see text], [Formula: see text] the free monoid generated by [Formula: see text] with 0 added, and [Formula: see text] a factor of [Formula: see text] obtained by setting certain monomials in [Formula: see text] to 0 such that [Formula: see text] for some [Formula: see text]. Then we can form the non-semiprime skew monoid ring [Formula: see text]. A local ring [Formula: see text] is called bleached if for any [Formula: see text] and any [Formula: see text], the abelian group endomorphisms [Formula: see text] and [Formula: see text] of [Formula: see text] are surjective. Using [Formula: see text], we provide various classes of both bleached and non-bleached local rings. One of the main problems concerning strongly clean rings is to characterize the rings [Formula: see text] for which the matrix ring [Formula: see text] is strongly clean. We investigate the strong cleanness of the full matrix rings over the skew monoid ring [Formula: see text].

A profinite approach to complete bifix decodings of recurrent languages

Abstract We approach the study of complete bifix decodings of (uniformly) recurrent languages with the help of free profinite monoids. We show that the complete bifix decoding of a uniformly recurrent language F by an F-charged rational complete bifix code is uniformly recurrent. An analogous result is obtained for recurrent languages. As an application of the machinery developed within this approach, we show that the maximal pronilpotent quotient of the Schützenberger group of an irreducible symbolic dynamical system is an invariant of eventual conjugacy.

Free Commutative Monoids in Homotopy Type Theory

We develop a constructive theory of finite multisets in Homotopy Type Theory, defining them as free commutative monoids. After recalling basic structural properties of the free commutative-monoid construction, we formalise and establish the categorical universal property of two, necessarily equivalent, algebraic presentations of free commutative monoids using 1-HITs. These presentations correspond to two different equational theories invariably including commutation axioms. In this setting, we prove important structural combinatorial properties of finite multisets. These properties are established in full generality without assuming decidable equality on the carrier set. As an application, we present a constructive formalisation of the relational model of classical linear logic and its differential structure. This leads to constructively establishing that free commutative monoids are conical refinement monoids. Thereon we obtain a characterisation of the equality type of finite multisets and a new presentation of the free commutative-monoid construction as a set-quotient of the list construction. These developments crucially rely on the commutation relation of creation/annihilation operators associated with the free commutative-monoid construction seen as a combinatorial Fock space.

Self-similar abelian groups and their centralizers

We extend results on transitive self-similar abelian subgroups of the group of automorphisms \mathcal{A}_m of an m -ary tree \mathcal{T}_m by Brunner and Sidki to the general case where the permutation group induced on the first level of the tree, has s\geq 1 orbits. We prove that such a group A embeds in a self-similar abelian group A^* which is also a maximal abelian subgroup of \mathcal{A}_m . The construction of A^{*} is based on the definition of a free monoid \Delta of rank s of partial diagonal monomorphisms of \mathcal{A}_m . Precisely, A^{*} = \overline{\Delta(B(A))} , where B(A) denotes the product of the projections of A in its action on the different s orbits of maximal subtrees of \mathcal{T}_m , and bar denotes the topological closure. Furthermore, we prove that if A is non-trivial, then A^{*} = C_{\mathcal{A}_m} (\Delta(A)) , the centralizer of \Delta(A) in \mathcal{A}_m . When A is a torsion self-similar abelian group, it is shown that it is necessarily of finite exponent. Moreover, we extend recent constructions of self-similar free abelian groups of infinite enumerable rank to examples of such groups which are also \Delta -invariant for s=2 . In the final section, we introduce for m=ns \geq 2 , a generalized adding machine a , an automorphism of \mathcal{T}_{m} , and show that its centralizer in \mathcal{A}_{m} to be a split extension of \langle a \rangle^{*} by \mathcal{A}_s . We also describe important \mathbb{Z}_n [\mathcal{A}_s] submodules of \langle a\rangle^{*} .

Free quasitopological groups

In this paper, we study the topological structure of a universal construction related to quasitopological groups: the free quasitopological group Fq(X) on a space X. We show that free quasitopological groups may be constructed directly as quotient spaces of free semitopological monoids, which are themselves constructed by iterating product spaces equipped with the “cross topology.” Using this explicit description of Fq(X), we show that for any T1 space X, Fq(X) is the direct limit of closed subspaces Fq(X)n of words of length at most n. We also prove that the natural map in:∐i=0n(X⊔X−1)⊗i→Fq(X)n is quotient for all n⩾0. Equipped with this convenient characterization of the topology of free quasitopological groups, we show, among other things, that a subspace Y⊆X is closed if and only if the inclusion Y→X induces a closed embedding Fq(Y)→Fq(X) of free quasitopological groups.

The structure of matrix polynomial algebras

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix-variables over a ground polynomial ring in not necessary commuting variables. These matrix subalgebras of full matrix rings over polynomial rings show up in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, which are interesting noncommutative algebras arising from the study of $\mathrm{Gl}_{n}$-varieties.

Efficient computations with counting functions on free groups and free monoids

We present efficient algorithms to decide whether two given counting functions on nonabelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on nonabelian free groups are cohomologous. We work in the multi-tape Turing machine model with nonconstant-time arithmetic operations. In the case of integer coefficients we construct an algorithm of linear time complexity (assuming that the rank is at least $3$ in the monoid case). In the case of rational coefficients we prove that the time complexity is $O(N\log N)$, where $N$ denotes the size of the input, that is, it is the same as in addition of rational numbers (implemented using the Harvey-van der Hoeven algorithm for integer multiplication). These algorithms are based on our previous work which characterizes bounded counting functions. Bibliography: 20 titles.

Об уравнениях в свободных моноидах и полугруппах с ограничениями на решения

We study algorithmic problems for equations in free monoids and semigroups (equations in words and lengths) with additional restrictions on the solutions. It is proved that it is impossible to construct an algorithm that solves an arbitrary system of equations in words and lengths in a free monoid (free semigroup) of rank 2 with an additional constraint on the solution in the form that one of its components belongs to the language of balanced words or the equality of the projections of two components of the solution into a distinguished free generator to determine whether it has a solution. A similar result is obtained for systems of inequalities in words.

Efficient computations with counting functions on free groups and free monoids

В работе строятся эффективные алгоритмы для проверки, находятся ли две данные считающие функции на неабелевых свободных группах (или моноидах) на ограниченном расстоянии друг от друга, и для проверки, являются ли два данных считающих квазиморфизма на свободных неабелевых группах когомологичными. В качестве модели вычисления нами рассматривается многоленточная машина Тьюринга, для которой арифметические операции не считаются выполнимыми за постоянное время. В случае целочисленных коэффициентов мы строим линейный по времени алгоритм (предполагая, что в случае свободного моноида его ранг не меньше $3$). Для случая рациональных коэффициентов мы доказываем, что временная сложность равна $O(N\log N)$, где $N$ - размер входа, т.е. совпадает со сложностью сложения рациональных чисел (реализованного с помощью алгоритма Харви-ван дер Хувена для умножения целых чисел). Построенные алгоритмы основаны на нашей предыдущей работе, которая дает описание пространства ограниченных считающих функций. Библиография: 20 названий.

Prefix monoids of groups and right units of special inverse monoids

AbstractA prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for certain one-relator monoids, and inverse monoids, can be reduced to solving the membership problem in prefix monoids of certain one-relator groups. Motivated by this, in this paper, we study the class of prefix monoids of finitely presented groups. We obtain a complete description of this class of monoids. All monoids in this family are finitely generated, recursively presented and group-embeddable. Our results show that not every finitely generated recursively presented group-embeddable monoid is a prefix monoid, but for every such monoid, if we take a free product with a suitably chosen free monoid of finite rank, then we do obtain a prefix monoid. Conversely, we prove that every prefix monoid arises in this way. Also, we show that the groups that arise as groups of units of prefix monoids are precisely the finitely generated recursively presented groups, whereas the groups that arise as Schützenberger groups of prefix monoids are exactly the recursively enumerable subgroups of finitely presented groups. We obtain an analogous result classifying the Schützenberger groups of monoids of right units of special inverse monoids. We also give some examples of right cancellative monoids arising as monoids of right units of finitely presented special inverse monoids, and we show that not all right cancellative recursively presented monoids belong to this class.

On the rational subsets of the monogenic free inverse monoid

We prove that the equality problem is decidable for rational subsets of the monogenic free inverse monoid F. It is also decidable whether or not a rational subset of F is recognizable. We prove that a submonoid of F is rational if and only if it is finitely generated. We also prove that the membership problem for rational subsets of a finite J-above monoid is decidable, covering the case of free inverse monoids.

Remark on the identities of the grammic monoid with three generators

Grammic monoids have recently been introduced by Christian Choffrut in terms of the action of the free monoid over a fixed ordered alphabet X on the set of rows of Young tableaux filled with elements from X via Schensted’s insertion. For \(X=\{a,b,c\}\) with \(a<b<c\), Choffrut has identified the grammic monoid on X with the quotient of the plactic monoid on X over the congruence generated by the pair (bacb, cbab). Since \(cbab\ne bcab\) in the latter monoid, the quotient is proper. We show that, nevertheless, the plactic and the grammic monoids with three generators satisfy the same identities.