Commutative 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.

On the ascent of atomicity to monoid algebras

A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid M and an integral domain R are both atomic implies that the monoid algebra R[M] of M over R is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, we prove here that the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors.

On graded-nonnil-Noetherian commutative rings

Our present work is motivated by several papers which have introduced and studied the notion of nonnil-Noetherian rings. Let A = ⊕ α ∈G A α be a commutative ring with unity graded by an arbitrary commutative monoid G. We define A to be graded-nonnil-Noetherian ring if every nonnil graded ideal of A, that is, a graded ideal which is not contained in Nil(A), is finitely generated. The purpose of this paper is to give the graded version of several characterizations, properties and basic results proved in [3], [7] and [4]. We also show the non-triviality of our generalization by giving examples of graded-nonnil-Noetherian rings which are not nonnil-Noetherian and of graded-nonnil-Noetherian rings which are not graded-Noetherian.

On graded going-down domains

Let [Formula: see text] be a torsionless commutative cancellative monoid and [Formula: see text] be a [Formula: see text]-graded integral domain. In this paper, we introduce the notion of graded going-down domains. Among other things, we provide an equivalent condition for graded-Prüfer domains in terms of graded going-down and graded finite-conductor domains. We also characterize graded going-down domains by means of graded divided domains. As an application, we show that the graded going-down property is stable under factor domains.

Associativity and commutativity of partially ordered rings

Consider a commutative monoid (M,+,0) and a biadditive binary operation μ:M×M→M. We will show that under some additional general assumptions, the operation μ is automatically both associative and commutative. The main additional assumption is localizability of μ, which essentially means that a certain canonical order on M is compatible with adjoining some multiplicative inverses of elements of M. As an application we show that a division ring F is commutative provided that for all a∈F there exists a natural number k such that a−k is not a sum of products of squares. This generalizes the classical theorem that every archimedean ordered division ring is commutative to a more general class of formally real division rings that do not necessarily allow for an archimedean (total) order. Similar results about automatic associativity and commutativity are well-known for special types of partially ordered extended rings (“extended” in the sense that neither associativity nor commutativity of the multiplication is required by definition), namely in the uniformly bounded and the lattice-ordered cases, i.e. for (extended) f-rings. In these cases the commutative monoid in question is the positive cone of the partially ordered extended ring. We also discuss how these classical results can be obtained from our main theorem.

Lattices of varieties of plactic-like monoids

We study the equational theories and bases of meets and joins of several varieties of plactic-like monoids. Using those results, we construct sublattices of the lattice of varieties of monoids, generated by said varieties. We calculate the axiomatic ranks of their elements, obtain plactic-like congruences whose corresponding factor monoids generate varieties in the lattice, and determine which varieties are joins of the variety of commutative monoids and a finitely generated variety. We also show that the hyposylvester and metasylvester monoids generate the same variety as the sylvester monoid.

Configuration spaces as commutative monoids

Configuration spaces as commutative monoids

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

Alienation of the quadratic, exponential and d’Alembert functional equations

AbstractLet $$(S,+,0)$$ ( S , + , 0 ) be a commutative monoid, $$\sigma :S\rightarrow S$$ σ : S → S be an endomorphism with $$\sigma ^2=id$$ σ 2 = i d and let K be a field of characteristic different from 2. We study the solutions $$f,g,h:S\rightarrow K$$ f , g , h : S → K of the Pexider type functional equation $$\begin{aligned} f(x+y)+f(x+\sigma y)+g(x+y)=2f(x)+2f(y)+g(x)g(y) \end{aligned}$$ f ( x + y ) + f ( x + σ y ) + g ( x + y ) = 2 f ( x ) + 2 f ( y ) + g ( x ) g ( y ) resulting from summing up the well known quadratic and exponential functional equations side by side. We show that under some additional assumptions the above equation forces f and g to split back into the system of two equations $$\begin{aligned} \left\{ \begin{array}{ll}f(x+y)+f(x+\sigma y)=2f(x)+2f(y)\\ g(x+y)=g(x)g(y)\end{array}\right. \end{aligned}$$ f ( x + y ) + f ( x + σ y ) = 2 f ( x ) + 2 f ( y ) g ( x + y ) = g ( x ) g ( y ) for all $$x,y\in S$$ x , y ∈ S (alienation phenomenon). We also consider an analogous problem for the quadratic and d’Alembert functional equations as well as for the quadratic, exponential and d’Alembert functional equations.

On graded pseudo 2-prime ideals

In this paper, we study graded pseudo 2-prime ideals of graded commutative rings with nonzero identities. Let G be a commutative additive monoid with an identity element 0, and R=⊕g∈G Rg be a commutative graded ring with a nonzero identity element. A proper graded ideal I of R is said to be a graded pseudo 2-prime ideal if whenever ab ∈ I for some homogeneous elements a,b ∈ R, then a²ⁿ ∈ Iⁿ or b²ⁿ ∈ Iⁿ for some n ∈ ℕ. Besides giving many properties of graded pseudo 2-prime ideals, we characterize graded almost valuation domains in terms of our new concept.

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.

Double Johnson filtrations for mapping class groups

In this paper, we first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group [Formula: see text] acting on another group [Formula: see text] equipped with a filtration indexed by a “good” ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid [Formula: see text] of pairs on non-negative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group [Formula: see text] of a surface [Formula: see text] with one boundary component, equipped with the normal subgroups [Formula: see text], [Formula: see text] of [Formula: see text] associated to a standard Heegaard splitting of the [Formula: see text]-sphere. We also consider the case where the group [Formula: see text] is the automorphism group of a free group.

On Certain Properties of Square-free and Radical Elements and Factorizations in Commutative Cancellative Monoids

On Certain Properties of Square-free and Radical Elements and Factorizations in Commutative Cancellative Monoids

NORMAL SUBMONOIDS AND CONGRUENCES ON A MONOID

AbstractA notion of normal submonoid of a monoid M is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf {NorSub}(M)$ of normal submonoids of M is a complete lattice. Joins are explicitly described and the lattice is computed for the finite full transformation monoids $T_n$ , $n\geq ~1$ . It is also shown that $\mathsf {NorSub}(M)$ is modular for a specific family of commutative monoids, including all Krull monoids, and that it, as a join semilattice, embeds isomorphically onto a join subsemilattice of the lattice $\mathsf {Cong}(M)$ of congruences on M. This leads to a new strategy for computing $\mathsf {Cong}(M)$ consisting of computing $\mathsf {NorSub}(M)$ and the so-called unital congruences on the quotients of M modulo its normal submonoids. This provides a new perspective on Malcev’s computation of the congruences on $T_n$ .

On Almost Everywhere K-Additive Set-Valued Maps

Abstract Let X be an Abelian group, Y be a commutative monoid, K ⊂Y be a submonoid and F : X → 2 Y \ {∅} be a set-valued map. Under some additional assumptions on ideals ℐ 1 in X and ℐ 2 in X 2, we prove that if F is ℐ 2-almost everywhere K-additive, then there exists a unique up to K K-additive set-valued map G : X → 2 Y \{∅} such that F = G ℐ 1-almost everywhere in X. Our considerations refers to the well known de Bruijn’s result [1].

On the atomic structure of torsion-free monoids

AbstractLet M be a cancellative and commutative (additive) monoid. The monoid M is atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also, M satisfies the ascending chain condition on principal ideals (ACCP) if every increasing sequence of principal ideals (under inclusion) becomes constant from one point on. In the first part of this paper, we characterize torsion-free monoids that satisfy the ACCP as those torsion-free monoids whose submonoids are all atomic. A submonoid of the nonnegative cone of a totally ordered abelian group is often called a positive monoid. Every positive monoid is clearly torsion-free. In the second part of this paper, we study the atomic structure of certain classes of positive monoids.

Levi-Civita functional equations on commutative monoids with tractable prime ideals

Levi-Civita functional equations on commutative monoids with tractable prime ideals

Fantastic and associative filters in pseudo quasi-ordered residuated systems

An interesting generalization of hoop-algebras and commutative residuated lattices is the concept of quasi-ordered residuated systems (shortly QRS) introduced in 2018 by Bonzio and Chajda. Quasi-ordered residuated system is an integral commutative monoid with two internal binary operations interconnected by a residuation connection. This specificity is the reason for the complexity of this algebraic structure and the existence of a significant number of substructures in it, such as various types of filters. The notion of pseudo quasi-ordered residuated systems was introduced and developed in 2022 by this author, omitting the commutativity requirement in QRSs, discussing, additionally, filters in it. Concept of pseudo QRSs is a generalization of the notion of QRSs. In this report, as a continuation of previous research, in addition to the introduction of concepts of fantastic and associative filters in a pseudo quasi-ordered residuated system, their mutual connection between them is discussed, and some examples are presented.

Eigenvalues of zero-divisor graphs, Catalan numbers, and a decomposition of eigenspaces

ABSTRACT A combinatorial approach is given to compute bases for eigenspaces of zero-divisor graphs of finite Boolean rings. A commutative monoid S of graphs is shown to contain a cyclic submonoid U that determines values of the entries of basis elements, while the members of its complement S ∖ U encode the supports of these elements. Furthermore, every member of S ∖ U is associated with a Catalan-triangle number, which counts the number of basis elements whose supports are determined by the given member. This is established by using a combinatorial interpretation of Catalan-triangle numbers to produce linearly independent sets of eigenvectors.

Cosimplicial monoids and deformation theory of tensor categories

We introduce the notion of n -commutativity ( 0\le n\le\infty ) for cosimplicial monoids in a symmetric monoidal category \mathbf{V} , where n=0 corresponds to just cosimplicial monoids in \mathbf{V} , while n=\infty corresponds to commutative cosimplicial monoids. When \mathbf{V} has a monoidal model structure, we endow (under some mild technical conditions) the total object of an n -cosimplicial monoid with a natural and very explicit E_{n+1} -algebra structure. Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a 1 -commutative cosimplicial monoid and, hence, has an E_2 -algebra structure similar to the E_2 -structure on Hochschild complex of an associative algebra provided by Deligne's conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a 2 -commutative cosimplicial monoid and, therefore, is naturally an E_3 -algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.