Commutative Monoid Research Articles

Graded almost pseudo-valuation domains

Let Γ be a nonzero commutative cancellative monoid (written additively), be a Γ-graded integral domain with for all , and H be the set of nonzero homogeneous elements of R. A homogeneous ideal P of R will be said to be strongly homogeneous primary if implies or for some integer , for every homogeneous elements x, y of RH . We say that R is a graded almost pseudo-valuation domain (gr-APVD) if each homogeneous prime ideal of R is strongly homogeneous primary. In this paper, we study some ring-theoretic properties of gr-APVDs and graded integral domains R such that is a gr-APVD for all homogeneous maximal ideals (resp., homogeneous maximal t-ideals) P of R.

Scalable monoids and quantity calculus

We define scalable monoids and prove their fundamental properties. Congruence relations on scalable monoids, direct and tensor products, subalgebras and homomorphic images of scalable monoids, and unit elements of scalable monoids are defined and investigated. A quantity space is defined as a commutative scalable monoid over a field, admitting a finite basis similar to a basis for a free abelian group. Observations relating to the theory of measurement of physical quantities accompany the results about scalable monoids. We conclude that the algebraic theory of scalable monoids and quantity spaces provides a rigorous foundation for quantity calculus.

Range updates and range sum queries on multidimensional points with monoid weights

Let P be a set of n points in Rd where each point p∈P carries a weight drawn from a commutative monoid (M,+,0). Given a d-rectangle rupd (i.e., an orthogonal rectangle in Rd) and a value Δ∈M, a range update adds Δ to the weight of every point p∈P∩rupd; given a d-rectangle rqry, a range sum query returns the total weight of the points in P∩rqry. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of O˜(n) space that handles an update in O˜(Tupd) time and a query in O˜(Tqry) time for arbitrary functions Tupd(n) and Tqry(n) satisfying Tupd⋅Tqry=n. The result holds for any fixed dimensionality d≥2. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture.

The noncommutative geometry of matrix polynomial algebras

This work investigates the noncommutative affine geometry of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix variables. A precise description of the spectrum (of maximal one-sided or bilateral ideals) of general matrix algebras is required. It results that the Zariski space of the irreducible representations of a matrix algebra is obtained by a natural gluing of the Zariski spaces of the irreducible representations of its diagonal components. An important step for the geometry of matrix polynomial algebras in commuting variables is achieved by a generalization of the Amitsur–Small Nullstellensatz, from which follows a precise description of their primitive quotients. We also characterize which of them are geometric algebras (in the sense of noncommutative deformation theory), reconstructible as algebras of observables from the scheme of irreducible representations. We then prove that each diagonal component of a matrix polynomial algebra in commuting variables is a Jacobson ring, whose non-Noetherian commutative geometry is efficiently described by the geometry of an affine essential subextension. And in the spirit of nonlocal algebraic geometry and addressing an open question by Charlie Beil, we obtain a class of non-Noetherian commutative monoid rings admitting closed points with positive geometric dimension.

Torsion factors of commutative monoid semirings

Torsion factors of commutative monoid semirings

Axiomatizability of the Class of Subdirectly Irreducible S-Acts over a Commutative Monoid

Axiomatizability of the Class of Subdirectly Irreducible S-Acts over a Commutative Monoid

The Yang–Baxter equation and Thompson’s group F

We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang–Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and Thompson’s group F. This raises the question of whether there are further connections between partial solutions and Thompson’s groups in general.

On Study of Monoids in Which Right Ideals are Generators

The main purpose of this paper is to study some classes of monoids in which every right ideal is a generator. We describe commutative monoids with this property and specially we show that for a commutative monoid S, every ideal is a generator if and only if S is a cancellative hereditary monoid which is equivalent to $$S=G$$ or $$S=G\times A$$ where G is a group and A is the additive monoid of non-negative integers. Also it is shown that for a non-trivial commutative monoid S without zero any ideal is a generator if and only if there exists a free S-act A such that for any subact B of $$A, trace~(B,A)=A$$ . Moreover for a regular monoid S it is shown any right ideal of S is a generator if and only if S has no two sided ideal.

A study of soft hyperideals in right regular LA-semihypergroups

Although a lot of work has been done by algebraist on soft set theory and hyperstructure theory, several fundamental results of soft intersection hyperideals in LA-semihypergroups remained untouched. This paper aims at studying some structural properties of LA-semihypergroups by applying soft set theory. In this paper, we define right regular LA-semihypergroups and introduce the concept of soft generalized bi-hyperideals in LA-semihypergroups. Further, we characterize right regular LA-semihypergroups in terms of different kinds of soft right hyperideals, soft left hyperideals and soft generalized bi-hyperideals. Idempotent soft hyperideals in LA-semihypergroups are introduced and some related results on them are obtained. It is proved that in an LA-semihypergroup with left identity, the collection of all soft left hyperideals, which are idempotent forms a commutative monoid. Examples are provided to illustrate the results.

On a class of poly-context-free groups generated by automata

This paper deals with graph automaton groups associated with trees and some generalizations. We start by showing some algebraic properties of tree automaton groups. Then we characterize the associated semigroup, proving that it is isomorphic to the partially commutative monoid associated with the complement of the line graph of the defining tree. After that, we generalize these groups by introducing the quite broad class of reducible automaton groups, which lies in the class of contracting automaton groups without singular points. We give a general structure theorem that shows that all reducible automaton groups are direct limits of poly-context-free groups which are virtually subgroups of the direct product of free groups; notice that this result partially supports a conjecture by T. Brough. Moreover, we prove that tree automaton groups with at least two generators are not finitely presented and they are amenable groups, which are direct limit of non-amenable groups.

On the Automorphism Group of Non-Necessarily Normal Affine Toric Varieties

Abstract Our main result is the following: let $X$ be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface $X^{\prime}$ with automorphism group isomorphic to the automorphism group of $X$ if and only if $X$ is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety $X$ of any dimension. Recall that normalized additive group actions on $X$ are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of $X$. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.

Graded amalgamated algebras along an ideal

Let [Formula: see text] and [Formula: see text] be two commutative rings graded by an arbitrary commutative monoid [Formula: see text], let [Formula: see text] be a homogeneous ideal of [Formula: see text] and [Formula: see text] a graded ring homomorphism. In this paper, we introduce a new graded ring construction called “the graded amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]” (denoted by [Formula: see text]). Then we investigate, through the properties of the homogeneous components that build the body of this “graded” study, the transfer of the property of graded-coherence to the graded amalgamation [Formula: see text].

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.

Complemented MacNeille completions and algebras of fractions

We introduce (ℓ-)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially ordered (lattice-ordered) set. Bimonoids form an appropriate framework for the study of a general notion of complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses in monoids. The central question of the paper is whether and how bimonoids can be embedded into complemented bimonoids, generalizing the embedding of cancellative commutative monoids into their groups of fractions and of bounded distributive lattices into their free Boolean extensions. We prove that each commutative (ℓ-)bimonoid embeds into a complete complemented commutative ℓ-bimonoid in a doubly dense way reminiscent of the Dedekind–MacNeille completion. Moreover, this complemented completion, which is term equivalent to a commutative involutive residuated lattice, sometimes contains a tighter complemented envelope analogous to the group of fractions. In the case of cancellative commutative monoids this algebra of fractions is precisely the familiar group of fractions, while in the case of Brouwerian (Heyting) algebras it is a (bounded) idempotent involutive commutative residuated lattice. This construction of the algebra of fractions in fact yields a categorical equivalence between varieties of integral and of involutive residuated structures which subsumes as special cases the known equivalences between Abelian ℓ-groups and their negative cones, and between Sugihara monoids and their negative cones.

Hopf monoids, permutohedral cones, and generalized retarded functions

The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having undecorated bosonic Fock space the combinatorial Hopf algebra of quasisymmetric functions. We construct a geometric realization of this Hopf monoid over the adjoint of the (essentialized) braid hyperplane arrangement, which identifies the monomial basis with signed characteristic functions of the interiors of permutohedral tangent cones. We show that the indecomposable quotient Lie coalgebra is obtained by restricting functions to chambers of the adjoint arrangement, i.e., by quotienting out the higher codimensions. The resulting functions are characterized by the Steinmann relations of axiomatic quantum field theory, demonstrating an equivalence between the Steinmann relations, tangent cones to (generalized) permutohedra, and having algebraic structure internal to species. Our results give a new interpretation of a construction appearing in the mathematically rigorous formulation of renormalization by Epstein–Glaser, called causal perturbation theory. In particular, we show that operator products of time-ordered products correspond to the H-basis of the cocommutative Hopf monoid of set compositions, and generalized retarded products correspond to a spanning set of its primitive part Lie algebra.

Asymptotic ω-Primality of Finitely Generated Cancelative Commutative Monoids

The computation of ω-primality has been object of study, mainly, for numerical semigroups due to its multiple applications to the Factorization Theory. However, its asymptotic version is less well known. In this work, we study the asymptotic ω-primality for finitely generated cancelative commutative monoids. By using discrete geometry tools and the Python programming language we present an algorithm to compute this parameter. Moreover, we improve the proof of a known result for numerical semigroups.

On the primality and elasticity of algebraic valuations of cyclic free semirings

A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number [Formula: see text], the additive monoid [Formula: see text] of the evaluation semiring [Formula: see text] is atomic. The atomic structure of both the additive and the multiplicative monoids of [Formula: see text] has been the subject of several recent papers. Here we focus on the monoids [Formula: see text], and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when [Formula: see text] is less than [Formula: see text], the atoms of [Formula: see text] are as far from being prime as they can possibly be. Then we establish some results about the elasticity of [Formula: see text], including that when [Formula: see text] is rational, the elasticity of [Formula: see text] is full (this was previously conjectured by Chapman, Gotti and Gotti).

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.

Chain conditions on (Rees) congruences of S-acts

In this paper, we introduce (Rees) artinian [Formula: see text]-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that (Rees) artinian [Formula: see text]-acts are those acts whose factor acts are all finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian [Formula: see text]-acts, namely, those acts whose (Rees) congruences are finitely generated. Moreover, as a useful tool for the investigation of chain conditions, we prove that the properties of being (Rees) noetherian and artinian are inherited in Rees short exact sequences. Next, we specifically consider the chain conditions on monoids. Finally, we prove that every right Rees artinian, commutative monoid is right Rees noetherian.

Applying monoid duality to a double contact process

In this paper we use duality techniques to study a coupling of the well-known contact process (CP) and the annihilating branching process. As the latter can be seen as a cancellative version of the contact process, we rebrand it as the cancellative contact process (cCP). Our process of interest will consist of two components, the first being a CP and the second being a cCP. We call this process the double contact process (2CP) and prove that it has (depending on the model parameters) at most one invariant law under which ones are present in both processes. In particular, we can choose the model parameters in such a way that CP and cCP are monotonely coupled. In this case also the above mentioned invariant law will have the property that, under it, ones (modeling “infected individuals”) can only be present in the cCP at sites where there are also ones in the CP. Along the way we extend the dualities for Markov processes discovered in our paper “Commutative monoid duality” to processes on infinite state spaces so that they, in particular, can be used for interacting particle systems.