Additive Monoid Research Articles

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.

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.

Archimedean classes in additive monoids

Summand absorbing submodules are common in modules over (additively) idempotent semirings, for example, in tropical algebra. A submodule [Formula: see text] of [Formula: see text] is summand absorbing, if [Formula: see text] implies [Formula: see text] for any [Formula: see text]. This paper proceeds the study of these submodules, and more generally of additive monoids, with emphasis on their archimedean classes and quotient structures.

Factorization in additive monoids of evaluation polynomial semirings

For a positive real α, we can consider the additive submonoid M of the real line that is generated by the nonnegative powers of α. When α is transcendental, M is a unique factorization monoid. However, when α is algebraic, M may not be atomic, and even when M is atomic, it may contain elements having more than one factorization (i.e., decomposition as a sum of irreducibles). The main purpose of this paper is to study the phenomenon of multiple factorizations inside M. When α is algebraic but not rational, the arithmetic of factorizations in M is highly interesting and complex. In order to arrive to that conclusion, we investigate various factorization invariants of M, including the sets of lengths, sets of Betti elements, and catenary degrees. Our investigation gives continuity to recent studies carried out by Chapman et al. in 2020 and by Correa-Morris and Gotti in 2022.

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.

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).

On the additive structure of algebraic valuations of polynomial semirings

In this paper, we study factorizations in the additive monoids of positive algebraic valuations N0[α] of the semiring of polynomials N0[X] using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when N0[α] is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid N0[α], the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for N0[α] to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of α. The properties of being finitely generated, half-factorial, and length-factorial are also investigated along the way.

Factorization in monoids by stratification of atoms and the Elliott problem

In an additive factorial monoid each element can be represented as a linear combination of irreducible elements (atoms) with uniquely determined coefficients running over all natural numbers. In this paper we develop for a wide class of non-factorial monoids a concept of stratification for atoms which allows to represent each element as a linear combination of atoms where the coefficients are uniquely determined when restricted in a particular way. This wide class includes inside factorial monoids and in particular simplicial affine monoids. In the latter case the question of uniqueness is related to a problem studied by E. B. Elliott in a paper from 1903. For the monoid of all nonnegative solutions of a certain linear Diophantine equation in three variables, Elliott considers “simple sets of solutions” (atoms of the monoid) and looks for a method that gives “every set once only”. We show that for simplicial affine monoids in two dimensions a stratification is always possible, which answers to Elliott’s problem also for the cases he left open. The results in this paper on the stratification of atoms for monoids in general may be seen also as an answer to a “generalized Elliott problem”. This manuscript employs very much the concept of extraction monoid which, as many other tools, stems from discrete geometry but it is less well-known.

Об эндоморфизмах аддитивного моноида подсетей двухслойной нейронной сети

Previously, for each multilayer neural network of direct signal propagation (hereinafter, simply a neural network), finite commutative groupoids were introduced, which were called additive subnet groupoids. These groupoids are closely related to the subnets of the neural network over which they are built. A grupoid is a monoid if and only if it is built over a two-layer neural network. Earlier, endomorphisms and their properties were studied for these groupoids. Some endomorphisms were constructed, but an exhaustive element-by-element description was not received. It was shown that every finite monoid is isomorphic to some submonoid of the monoid of all endomorphisms of a suitable additive subnet groupoid for some suitable neural network. In this paper, we study endomorphisms of additive groupoids of subnets of twolayer neural networks. The main result of the work is an element-wise description of the monoid of all endomorphisms of additive monoids of subnets built over a two-layer neural network. The item-by-item description is obtained by constructing a general form of endomorphism. The general view of an endomorphism is parameterized by the endomorphisms of suitable booleans with respect to the union operation. Therefore, endomorphisms of these Booleans were studied in this work. In particular, the semirings of endomorphisms of these Booleans with respect to the union were studied. In addition, to describe the general form of the endomorphism of the additive monoid of subnets, homomorphisms of one Boalean into another (with respect to union) were used.

Classification of congruences of twisted partition monoids

The twisted partition monoid PnΦ is an infinite monoid obtained from the classical finite partition monoid Pn by taking into account the number of floating components when multiplying partitions. The main result of this paper is a complete description of the congruences on PnΦ. The succinct encoding of a congruence, which we call a C-pair, consists of a sequence of n+1 congruences on the additive monoid N of natural numbers and a certain (n+1)×N matrix. We also give a description of the inclusion ordering of congruences in terms of a lexicographic-like ordering on C-pairs. This is then used to classify congruences on the finite d-twisted partition monoids Pn,dΦ, which are obtained by factoring out from PnΦ the ideal of all partitions with more than d floating components. Further applications of our results, elucidating the structure and properties of the congruence lattices of the (d-)twisted partition monoids, will be the subject of a future article.

Sums of binomial coefficients evaluated at [formula omitted], and applications

The additive monoid R+(x) is defined as the set of all nonnegative integer linear combinations of binomial polynomials (xn) for n∈Z+. This paper is concerned with the inquiry into the structure of R+(α) for complex numbers α. Particularly interesting is the case of algebraic α which are not non-negative integers. This question is motivated by the study of functors between Deligne categories Rep(St) (and also Rep(GLt)) for t∈C﹨Z+. We prove that this object is a ring if and only if α is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by α, i.e. all elements of OQ(α), are also contained in this ring. We also give two explicit representations of R+(α) for both algebraic integers and general algebraic numbers α. One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of R+(α) for both quadratic algebraic numbers and roots of unity.

Bi-atomic classes of positive semirings

A subsemiring S of $$\mathbb {R}$$ is called a positive semiring provided that S consists of nonnegative numbers and $$1 \in S$$ . Here we study factorizations in both the additive monoid $$(S,+)$$ and the multiplicative monoid $$(S\backslash \{0\}, \cdot )$$ . In particular, we investigate when, for a positive semiring S, both $$(S,+)$$ and $$(S\backslash \{0\}, \cdot )$$ have the following properties: atomicity, the ACCP, the bounded factorization property (BFP), the finite factorization property (FFP), and the half-factorial property (HFP). It is well known that in the context of cancellative and commutative monoids, the chain of implications HFP $$\Rightarrow $$ BFP and FFP $$\Rightarrow $$ BFP $$\Rightarrow $$ ACCP $$\Rightarrow $$ atomicity holds. Here we construct classes of positive semirings wherein both the additive and multiplicative structures satisfy each of these properties, and we also give examples to show that, in general, none of the implications in the previous chain is reversible.

From generic partition refinement to weighted tree automata minimization

Partition refinement is a method for minimizing automata and transition systems of various types. Recently,we have developed a partition refinement algorithm that is generic in the transition type of the given systemand matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministicautomata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved bymodelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the runtime analysis of our algorithm to cover additional instances, notably weighted automata and, more generally,weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids suchas (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the bestknown algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concretesystem types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular,and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors.Experiments show that even for complex system types, the tool is able to handle systems with millions oftransitions.

Twisted Polynomial and Power Series Rings

Let R be a commutative ring with identity and $${\mathbb {N}}_0$$ be the additive monoid of nonnegative integers. We say that a function $$t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R$$ is a twist function on R if t satisfies the following three properties for all $$n, m, q \in {\mathbb {N}}_0$$ : (i) $$t(0,q) = 1$$ , (ii) $$t(n,m) = t(m,n)$$ , and (iii) $$t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)$$ . Let $$R[\![X]\!]$$ (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For $$f = \sum _{n=0}^{\infty } a_nX^n$$ and $$g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]$$ , let $$f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n$$ , $$f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n$$ . Then, $$R^t[\![X]\!]:= (R[\![X]\!], +, *_t)$$ and $$R^t[X] := (R[X], +, *_t)$$ are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of $$R^t[\![X]\!]$$ and $$R^t[X]$$ with focus on divisibility properties including UFDs and GCD-domains. We also show how these two rings are related to the usual power series and polynomial rings.

Standard bases for modules over polynomial subalgebras

In this article, we present the theory of standard bases for modules over polynomial subalgebras based on arbitrary gradings (induced by an additive monoid Γ) which we called SΓ-bases. This paper deals with the problem of construction of SΓ-bases for submodules of free module over polynomial subalgebras. For this purpose, we present an algorithm based on SΓ-normal form and discuss it for some special gradings.

Atomic and AP semigroup rings $F[X;M]$, where $M$ is a submonoid of the additive monoid of nonnegative rational numbers

We investigate the atomicity and the AP property of the semigroup rings $F[X;M]$, where $F$ is a field, $X$ is a variable and $M$ is a submonoid of the additive monoid of nonnegative rational numbers. The main notion that we introduce for the purpose of the investigation is the notion of essential generators of $M$.

Homotopy Types of Frobenius Complexes

Let Λ be a submonoid of the additive monoid \({\mathbb{N}}\). There is a natural order on Λ defined by \({\lambda \leq \lambda +\mu}\) for \({\lambda,\mu \in \Lambda}\). A Frobenius complex of Λ is defined to be the order complex of an open interval of Λ. Suppose \({r \geq 2}\) and let \({\rho}\) be a reducible element of Λ. We construct the additive monoid \({\Lambda[\rho/r]}\) obtained from Λ by adjoining a solution to the equation \({r\alpha=\rho}\). We show that any Frobenius complex of \({\Lambda[\rho/r]}\) is homotopy equivalent to a wedge of iterated suspensions of Frobenius complexes of Λ. As a consequence, we derive a formula for the multi-graded Poincare series associated to \({\Lambda[\rho/r]}\). As an application, we determine the homotopy types of the Frobenius complexes of some additive monoids. For example, we show that if Λ is generated by a finite geometric sequence, then any Frobenius complex of Λ is homotopy equivalent to a wedge of spheres.

On Some Properties and Comparative Analysis for Different OWA Monoids

We study the properties of OWA multiplication monoid. By introducing and-accumulation vectors of OWA operators, we consider the set of all n-dimensional OWA operators as a lattice. Then, we analyze some monotonicity properties of OWA operators based on an ordering induced by and-accumulation vectors. We also show that the lattice-theoretical operations are a kind of counterpart of the OWA multiplication monoid (and its dual, additive monoid). An example of using the OWA multiplication monoid and the lattice-theoretical structure in decision making problem is provided.

PS-Modules over Ore Extensions and Skew Generalized Power Series Rings

A rightR-moduleMRis called a PS-module if its socle,SocMR, is projective. We investigate PS-modules over Ore extension and skew generalized power series extension. LetRbe an associative ring with identity,MRa unitary rightR-module,O=Rx;α,δOre extension,MxOa rightO-module,S,≤a strictly ordered additive monoid,ω:S→EndRa monoid homomorphism,A=RS,≤,ωthe skew generalized power series ring, andBA=MS,≤RS,≤, ωthe skew generalized power series module. Then, under some certain conditions, we prove the following: (1) IfMRis a right PS-module, thenMxOis a right PS-module. (2) IfMRis a right PS-module, thenBAis a right PS-module.

Symbolic Dynamics: Entropy = Dimension = Complexity

Let d be a positive integer. Let G be the additive monoid ? d or the additive group ? d . Let A be a finite set of symbols. The shift action of G on A G is given by S g (x)(h) = x(g+h) for all g, h ? G and all x ? A G . A G-subshift is defined to be a nonempty closed set X ⊆ A G such that S g (x)?X for all g ? G and all x ? X. Given a G-subshift X, the topological entropy ent(X) is defined as usual (Ruelle Trans. Am. Math. Soc. 187, 237---251, 1973). The standard metric on A G is defined by ?(x, y) = 2 ? | F n | $2^{-|F_{n}|}$ where n is as large as possible such that x?F n = y?F n . Here F n = {0, 1, ? , n} d if G = ? d , and F n = {?n, ? , ?1, 0, 1, ? , n} d if G = ? d . For any X ⊆ A G the Hausdorff dimension dim(X) and the effective Hausdorff dimension effdim(X) are defined as usual (Hausdorff Math. Ann. 79, 157---179 1919; Reimann 2004; Reimann and Stephan 2005) with respect to the standard metric. It is well known that effdim(X) = sup x?X lim inf n K(x?F n )/|F n | where K denotes Kolmogorov complexity (Downey and Hirschfeldt 2010). If X is a G-subshift, we prove that ent(X) = dim(X) = effdim(X), and ent(X) ? limsup n K(x?F n )/|F n | for all x ? X, and ent(X) = lim n K(x?F n )/|F n | for some x ? X.