Commutative Monoid Research Articles

Commutative semigroups whose endomorphisms are power functions

For any commutative semigroup S and positive integer m the power function f: S rightarrow S defined by f(x) = x^m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

Length-factoriality in commutative monoids and integral domains

An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x∈M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in the setting of integral domains, length-factoriality can be taken as an alternative definition of a unique factorization domain. However, being a length-factorial monoid is, in general, weaker than being a factorial monoid (i.e., a unique factorization monoid). Here we further investigate length-factoriality. First, we offer two characterizations of a length-factorial monoid M, and we use such characterizations to describe the set of Betti elements and obtain a formula for the catenary degree of M. Then we study the connection between length-factoriality and purely long (resp., purely short) irreducibles, which are irreducible elements that appear in the longer (resp., shorter) part of any unbalanced factorization relation. Finally, we prove that an integral domain cannot contain purely long and purely short irreducibles simultaneously, and we construct a Dedekind domain containing purely long (resp., purely short) irreducibles but not purely short (resp., purely long) irreducibles.

On algebraic abstractions for concurrent separation logics

Concurrent separation logic is distinguished by transfer of state ownership upon parallel composition and framing. The algebraic structure that underpins ownership transfer is that of partial commutative monoids (PCMs). Extant research considers ownership transfer primarily from the logical perspective while comparatively less attention is drawn to the algebraic considerations. This paper provides an algebraic formalization of ownership transfer in concurrent separation logic by means of structure-preserving partial functions (i.e., morphisms) between PCMs, and an associated notion of separating relations. Morphisms of structures are a standard concept in algebra and category theory, but haven't seen ubiquitous use in separation logic before. Separating relations. are binary relations that generalize disjointness and characterize the inputs on which morphisms preserve structure. The two abstractions facilitate verification by enabling concise ways of writing specs, by providing abstract views of threads' states that are preserved under ownership transfer, and by enabling user-level construction of new PCMs out of existing ones.

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.

Commutative Monoid Formalism for Weighted Coupled Cell Networks and Invariant Synchrony Patterns

This paper presents a framework based on matrices of monoids for the study of coupled cell networks. We formally prove within the proposed framework, that the set of results about invariant synchrony patterns for unweighted networks also holds for the weighted case. Moreover, the approach described allows us to reason about any multiedge and multiedge-type network as if it was single edge and single-edge-type. Several examples illustrate the concepts described. Additionally, an improvement of the coarsest invariant refinement algorithm to find balanced partitions is presented that exhibits a worst-case complexity of $ \mathbf{O}(\vert\mathcal{C}\vert^3) $, where $\mathcal{C}$ denotes the set of cells.

Enriched categories of correspondences and characteristic classes of singular varieties

For the category $\mathscr V$ of complex algebraic varieties, the Grothendieck group of the commutative monoid of the isomorphism classes of correspondences $X \xleftarrow f M \xrightarrow g Y$ with a proper morphism $f$ and a smooth morphism $g$ (such a

D-ultrafilters and their monads

For a number of locally finitely presentable categories K we describe the codensity monad of the full embedding of all finitely presentable objects into K. We introduce the concept of D-ultrafilter on an object, where D is a “nice” cogenerator of K. We prove that the codensity monad assigns to every object an object representing all D-ultrafilters on it. Our result covers e.g. categories of sets, vector spaces, posets, semilattices, graphs and M-sets for finite commutative monoids M.

On Hollow acts

In this paper, we complete an early result by R. Khosravi and M. Roueentan on the hollow act and obtained new properties and characterizations for this notion continues the account of hollow act for commutative monoids begun on 2019. This notion (hollow acts) represents a dual notion to the uniform acts which were submit by M. Roueentan and M. Sedaghatjoo on 2017. An S-act Ms is referred to as hollow act if every subact Ns of Ms is small where a subact Ns is called small (or superfluous) in Ms if for every subact Hs of MS, NS∪HS=MS implies Hs=Ms. Equally, we reformulate this definition in another words as follows: an S-act Ms is referred to as hollow if whenever N1,N2 are subacts of Ms and N1 ∪N2 = Ms implies either N1=Ms or N2= Ms. Conditions under which subacts are inheriting the property of the Hollow act have been examined. As well as the condition on the quotient subact to be Hollow act was studied. Other properties of the small subact differ from those properties which were early studied by the author are investigated. In the interests of simplicity, the relationship between hollow acts and cyclic act is considered. As a consequence, conditions to coincide those classes are shown. Ultimately, the notion of the Hollow act was used to study when the endomorphism monoid will be local.

Commutative algebraic monoid structures on affine surfaces

Abstract We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n - 1 n-1 or 𝑛 on a normal affine variety of dimension 𝑛.

On strongly primary monoids, with a focus on Puiseux monoids

Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. It is well-known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is Noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux monoids are primary monoids, and we provide conditions ensuring that they are strongly primary. Then we study local and global tameness of strongly primary Puiseux monoids; most notably, we establish an algebraic characterization of when a Puiseux monoid is globally tame. Moreover, we obtain a result on the structure of sets of lengths of all locally tame strongly primary monoids.

When Are Graded Rings Graded S-Noetherian Rings

Let Γ be a commutative monoid, R=⨁α∈ΓRα a Γ-graded ring and S a multiplicative subset of R0. We define R to be a graded S-Noetherian ring if every homogeneous ideal of R is S-finite. In this paper, we characterize when the ring R is a graded S-Noetherian ring. As a special case, we also determine when the semigroup ring is a graded S-Noetherian ring. Finally, we give an example of a graded S-Noetherian ring which is not an S-Noetherian ring.

On the number of possible resonant algebras

We investigate the number of distinct resonant algebras depending on the generator content, which consists of the Lorentz generator, translation, and new additional Lorentz-like and translation-like generators. Such algebra enlargements originate directly from the Maxwell algebra and implementation of the S-expansion framework. Resonant algebras, being sub-class of the S-expanded algebras, should find use in the construction of gravity and supergravity models along some other applications. The undertaken task of establishing all the possible resonant algebras is closely related to the subject of finding commutative monoids (semigroups with the identity element) of a particular order, were we additionally enforce the parity condition.

Commutative Topological Semigroups Embedded into Topological Abelian Groups

In this paper, we give conditions under which a commutative topological semigroup can be embedded algebraically and topologically into a compact topological Abelian group. We prove that every feebly compact regular first countable cancellative commutative topological semigroup with open shifts is a topological group, as well as every connected locally compact Hausdorff cancellative commutative topological monoid with open shifts. Finally, we use these results to give sufficient conditions on a commutative topological semigroup that guarantee it to have countable cellularity.

Factorizations in upper triangular matrices over information semialgebras

An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a bound for the number of atoms in each of its factorizations. These two properties have been systematically studied since they were introduced by Anderson, Anderson, and Zafrullah in 1990. Noetherian domains satisfy the BFP, while Dedekind domains satisfy the FFP. It is well known that, for commutative cancellative monoids (and, in particular, for multiplicative monoids of integral domains), FFP ⇒ BFP ⇒ ACCP ⇒ atomic. For n≥2, we show that each of these four properties transfers back and forth between an information semialgebra S (certain commutative cancellative semiring) and the multiplicative monoid Tn(S)• consisting of n×n upper triangular matrices over S. We also show that a similar transfer behavior takes place if one replaces Tn(S)• by its submonoid Un(S) consisting of upper triangular matrices with units along their main diagonals. As a consequence, we find that the atomic chain FFP ⇒ BFP ⇒ ACCP ⇒ atomic also holds for the two classes comprising the noncommutative monoids Tn(S)• and Un(S). Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible.

Ambidexterity and the universality of finite spans

Pursuing the notions of ambidexterity and higher semiadditivity as developed by Hopkins and Lurie, we prove that the span $\infty$-category of $m$-finite spaces is the free $m$-semiadditive $\infty$-category generated by a single object. Passing to presentable $\infty$-categories we obtain a description of the free presentable $m$-semiadditive $\infty$-category in terms of a new notion of $m$-commutative monoids, which can be described as spaces in which families of points parameterized by $m$-finite spaces can be coherently summed. Such an abstract summation procedure can be used to give a formal $\infty$-categorical definition of the finite path integral described by Freed, Hopkins, Lurie and Teleman in the context of 1-dimensional topological field theories.

Geometric and combinatorial aspects of submonoids of a finite-rank free commutative monoid

If F is an ordered field and M is a finite-rank torsion-free monoid, then one can embed M into a finite-dimensional vector space over F via the inclusion M↪gp(M)↪F⊗Zgp(M), where gp(M) is the Grothendieck group of M. Let C be the class consisting of all monoids (up to isomorphism) that can be embedded into a finite-rank free commutative monoid. Here we investigate how the atomic structure and arithmetic properties of a monoid M in C are connected to the combinatorics and geometry of its conic hull cone(M)⊆F⊗Zgp(M). First, we show that the submonoids of M determined by the faces of cone(M) account for all divisor-closed submonoids of M. Then we appeal to the geometry of cone(M) to characterize whether M is a factorial, half-factorial, and other-half-factorial monoid. Finally, we investigate the cones of finitary, primary, finitely primary, and strongly primary monoids in C. Along the way, we determine the cones that can be realized by monoids in C and by finitary monoids in C.

Strongly Fully Stable Acts Relative to an Ideal

The purpose of this paper is to introduce and investigate strongly fully stable acts relative to an ideal as a concept generalizing strongly fully stable modules relative to an ideal, but is stronger than that of fully stable acts. In this study, we consider some properties and characterizations of the class of strongly fully stable acts relative to an ideal, as well as the relations between this class and other classes. Among these classes are quasi-injective acts, strongly quasi-injective acts, acts which satisfy Baer's criterion, acts satisfying the strongly Baer's criterion, and duo acts. The product of strongly fully stable acts relative to an ideal need not be a strongly fully stable act relative to that ideal. The coproduct of any family of strongly fully stable acts relative to an ideal need not be a strongly fully stable act relative to that ideal. Also, we have that strongly fully stable acts relative to an ideal are equivalent to an $S$-act that satisfies the strongly Baer's criterion relative to an ideal $I$ for cyclic subacts. The strongly fully stable act relative to $I$ is equivalent to the strongly quasi-injective act relative to the ideal $I$ and duo act with a commutative monoid.

On (2; 2)-regular Non-associative Ordered Semigroups via Its Semilattices and Generated (Generalized Fuzzy) Ideals

The main motivation behind this paper is to study some structural properties of a non-associative structure as it hasn't attracted much attention compared to associative structures. In this paper, we introduce the concept of an ordered A*G**-groupoid and provide that this class is more generalized than an ordered AG-groupoid with left identity. We also define the generated left (right) ideals in an ordered A*G**-groupoid and characterize a (2; 2)-regular ordered A*G**-groupoid in terms of these ideals. We then study the structural properties of an ordered A*G**-groupoid in terms of its semilattices, (2; 2)-regular class and generated commutative monoids. Subsequently, compare -fuzzy left/right ideals of an ordered AG-groupoid and respective examples are provided. Relations between an -fuzzy idempotent subsets of an ordered A*G**-groupoid and its -fuzzybi-ideals are discussed. As an application of our results, we get characterizations of (2; 2)-regular ordered A*G**-groupoid in terms of semilattices and -fuzzy left (right) ideals. These concepts will help in verifying the existing characterizations and will help in achieving new and generalized results in future works.

Certain properties of difference operator and stability of Fr\xe9chet functional equation

Let M be a commutative monoid, and let B be a Banach space. We give a new recursive method to obtain a Gǎvruţa-type stability result for the functional equation Δyn+1f(x):=∑k=0n+1(−1)n+1+k(n+1k)f(x+ky)=0 via algebraic manipulations of the forward difference operator.

On the identification of finite non-group semigroups of a given order

Identifying finite non-group semigroups for every positive integer is significant because of many applications of such semigroups are functional in various branches of sciences such as computer science, mathematics and finite machines. The finite non-commutative monoids as a type of such semigroups were identified in 2014, for every positive integer. We here attempt to identify the finite commutative monoids and finite commutative non-monoids of a given integer n=p^alpha q^beta, for every integers alpha , beta ge 2 and different primes p and q. In order to recognize the commutative monoids, we present a class of 2-generated monoids of a given order, and for the commutative non-monoids of order n=p^alpha q^beta, we give the minimal generating set. Moreover, we prove that there are exactly (p^{alpha }-2)(q^{beta }-2) non-isomorphic commutative non-monoids of order p^alpha q^beta. The identification of non-group semigroups for the integers p^{2alpha } and 2p^alpha is achieved. The automorphism groups of these groups are specified as well. As a result of this study, an interesting difference between the abelian groups and the commutative semigroups of order p^2 is presented.