Free Commutative Semigroup Research Articles

Structure of relatively free n-tuple semigroups

An n-tuple semigroup is an algebra defined on a set with n binary associative operations. This notion was considered by Koreshkov in the context of the theory of n-tuple algebras of associative type. The n>1 pairwise interassociative semigroups give rise to an n-tuple semigroup. This paper is a survey of recent developments in the study of free objects in the variety of n-tuple semigroups. We present the constructions of the free n-tuple semigroup, the free commutative n-tuple semigroup, the free rectangular n-tuple semigroup, the free left (right) k-nilpotent n-tuple semigroup, the free k-nilpotent n-tuple semigroup, and the free weakly k-nilpotent n-tuple semigroup. Some of these results can be applied to constructing relatively free cubical trialgebras and doppelalgebras.

Свободные прямоугольные n-кратные полугруппы

n-кратной полугруппой называется непустое множество G, снабженное n бинарными операциями $$\fbox{1}\,, \fbox{2}\,, ..., \fbox{n}\,,$$ удовлетворяющими аксиомам $$(x\fbox{r} \, y) \fbox{s}\, z=x\fbox{r}\,(y\fbox{s}\,z)$$ для всех $$x,y,z \in G$$ и $$r,s\in \{1,2,...,n\}.$$ Это понятие рассматривал Н.А.Корешков в контексте теории n-кратных алгебр ассоциативного типа. Доппельполугруппы являются 2-кратными полугруппами. n-кратные полугруппы имеют связи с интерассоциативными полугруппами, димоноидами, триоидами, доппельалгебрами, дуплексами, G-димоноидами и рестриктивными биполугруппами. Если операции n-кратной полугруппы совпадают, то она превращается в полугруппу. Таким образом, n-кратные полугруппы являются обобщением полугрупп. Класс всех n-кратных полугрупп образует многообразие. Недавно были построены свободная n-кратная полугруппа, свободная коммутативная n-кратная полугруппа, свободная k-нильпотентная n-кратная полугруппа и свободное произведение произвольных n-кратных полугрупп. Класс всех прямоугольных n-кратных полугрупп, то есть n-кратных полугрупп с n прямоугольными полугруппами, образует подмногообразие многообразия n-кратных полугрупп. В этой статье мы строим свободную прямоугольную n-кратную полугруппу и характеризуем наименьшую прямоугольную конгруэнцию на свободной n-кратной полугруппе.

Interassociativity Via Hyperidentities

In this paper the concept of interassociativity via hyperidentities of associativity is extended and describe the set of semigroups which are {i, j}-interassociative to the free semigroup and the free commutative semigroup, where i, j = 1, 2, 3.

Free Products of n-Tuple Semigroups

We construct a free product of arbitrary n-tuple semigroups, introduce the notion of n-bands of n-tuple semigroups and, in terms of this notion, describe the structure of the free product. We also construct a free commutative n-tuple semigroup of any rank and characterize one-generated free commutative n-tuple semigroups. Moreover, we describe the least commutative congruence on a free n-tuple semigroup and prove that the semigroups of the constructed free commutative n-tuple semigroup are isomorphic and that its automorphism group is isomorphic to a symmetric group.

Free commutative trioids

Loday and Ronco introduced the notion of a trioid and constructed the free trioid of rank 1. We construct the free commutative trioid of rank 1 and prove that the free commutative trioid of rank \(n>1\) is a subdirect product of the free commutative semigroup of rank n and the free commutative trioid of rank 1. We also characterize the least commutative congruence, the least commutative dimonoid congruences, and the least commutative semigroup congruence on the free trioid.

The Hopf Structure of Some Dual Operator Algebras

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury–Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.

Interassociativity on a free commutative semigroup

We describe all interassociates of a free commutative semigroup. Each interassociate of a free commutative semigroup is proved to be a variant of it or coincides with that semigroup. Conditions are found for the isomorphy of two variants of a free commutative semigroup.

Fixed point properties for semigroups of nonlinear mappings and amenability

In this paper we study fixed point properties for semitopological semigroup of nonexpansive mappings on a bounded closed convex subset of a Banach space. We also study a Schauder fixed point property for a semitopological semigroup of continuous mappings on a compact convex subset of a separated locally convex space. Such semigroups properly include the class of extremely left amenable semitopological semigroups, the free commutative semigroup on one generator and the bicyclic semigroup S1=〈a,b:ab=1〉.

Dimension polynomials of intermediate differential fields and the strength of a system of differential equations with group action

Let K be a differential field of zero characteristic with a basic set of derivations Δ = {δ 1, …, δ m } and let Θ denote the free commutative semigroup of all elements of the form \( \theta = \delta_1^{{k_1}} \cdots \delta_m^{{k_m}} \) where k i ∈ ℕ (1 ≤ i ≤ m). Let the order of such an element be defined as ord \( \theta = \sum\limits_{i = 1}^m {{k_i}} \), and for any r ∈ ℕ, let Θ(r) = {θ ∈ Θ | ord θ ≤ r}. Let L = K〈η 1, …, η s 〉 be a differential field extension of K generated by a finite set η = {η 1, …, η s } and let F be an intermediate differential field of the extension L/K. Furthermore, for any r ∈ ℕ, let \( {L_r} = K\left( {\bigcup\limits_{i = 1}^s {\Theta (r){\eta_i}} } \right) \) and F r = L r ∩ F. We prove the existence and describe some properties of a polynomial ϕ K,F,η (t) ∈ ℚ[t] such that ϕ K,F,η (r) = trdeg K F r for all sufficiently large r ∈ ℕ. This result implies the existence of a dimension polynomial that describes the strength of a system of differential equations with group action in the sense of A. Einstein. We shall also present a more general result, a theorem on a multivariate dimension polynomial associated with an intermediate differential field F and partitions of the basic set Δ.

Interassociates of the Free Commutative Semigroup on n Generators

The interassociates of the free commutative semigroup on n generators, for n > 1, are identified. For fixed n, let (S, ·) denote this semigroup. We show that every interassociate can be written in the form \((S, \ast_{\bar{k}})\), depending only on a n-tuple \(\bar{k}=( k_1, \ldots, k_n)\). Next, if \((S, \ast_{\bar{k}})\) and \((S, \ast_{\bar{\ell}})\) are isomorphic interassociates of (S, ·) such that \(\phi(x_i) = x_j\), for xii and xj in the generating set of S, then \(k_i = \ell_j\). Moreover, \((S, \ast_{\bar{k}})\cong(S,\ast_{\bar{\ell}})\) if and only if \({\{k_i\}_{i = 1}^n}\) is a permutation of \({\{\ell_i\}_{i = 1}^n}\).

A construction of commutative nilsemigroups

All commutative nilsemigroups are constructed from congruences on free commutative semigroups.

Semigroups ofI-Type

Assume thatSis a semigroup generated by {x1,…,xn}, and let U be the multiplicative free commutative semigroup generated by {u1,…,un}. We say thatSis ofI-typeif there is a bijectivev:U→Ssuch that for alla∈U, {v(u1a),…,v(una)}={x1v(a),…,xnv(a)}. This condition appeared naturally in the work on Sklyanin algebras by John Tate and the second author. In this paper we show that the condition for a semigroup to be ofI-type is related to various other mathematical notions found in the literature. In particular we show that semigroups ofI-type appear in the study of the set-theoretic solutions of the Yang–Baxter equation, in the theory of Bieberbach groups, and in the study of certain skew binomial polynomial rings which were introduced by the first author.

On some non-rational congruences on free commutative semigroups

On some non-rational congruences on free commutative semigroups

"Factorisatio numerorum" in arithmetical semigroups

Introduction. The problems of “factorisatio numerorum”, which go back more than 65 years, are concerned principally with (i) the total number f(n) of factorizations of a natural number n > 1 into products of natural numbers larger than 1, where the order of the factors is not counted, and (ii) the corresponding total number F (n) of factorizations when the order of the factors is counted. For example, f(12) = 4 while F (12) = 8. Some results and further references on such problems may be found in [1], [2], [5], [6] and [7] in particular. The object of this paper is to consider similar problems and results, with emphasis on the average numbers of factorizations of each kind, within the partly analogous but also quite distinct context of additive arithmetical semigroups. Such semigroups (to be defined below) are treated in the monographs [3], [4] within an abstract setting designed to conveniently cover (under minimal assumptions) concrete cases like (i) the multiplicative semigroup Gq of all monic polynomials in one indeterminate over a finite field Fq, (ii) semigroups of ideals in principal orders within algebraic function fields over Fq, (iii) semigroups formed under direct sum by the isomorphism classes of certain kinds of finite modules or algebras over such principal orders. Although the first main result below will be derived within a still more general framework, we shall formulate and state our second main conclusion within a context (see [4]) which conveniently covers the preceding concrete cases: An (additive) arithmetical semigroup will be understood to be a free commutative semigroup G with identity element 1, generated by a (countable) set P of “prime” elements, which admits an integer-valued “degree”

On the Jacobson radical of semigroup rings of commutative semigroups

Many authors have considered the radicals of semigroup rings of commutative semigroups. A list of the papers pertaining to this field is contained in [4]. In [1] Amitsur proved that, for any associative ring R and for every free commutative semigroup S, the equalities B(RS) = B(R)S and L(RS) = L(R)S hold, where B is the Baer radical and L is the Levitsky radical. A natural problem which arises is to describe semigroup rings RS such that π(RS) = π(R)S, where π is one of the most important radicals. For the Baer and Levitsky radicals and commutative semigroups a complete solution of the above problem follows from theorems 2·8 and 3·1 of [15].

The function field abstract prime number theorem

For arithmetical semigroups modelled on the positive integers, there is an ‘abstract prime number theorem’ (see, for example, [1]). In order to study enumeration problems in the several arithmetical categories whose prototype instead is the ring of polynomials in an indeterminate over a finite field of order q, Knopfmacher[2, 3] introduced the following modification. An additive arithmetical semigroup G is a free commutative semigroup with an identity, generated by a countable set of ‘primes’ P and admitting an integer-valued degree mapping ∂ with the properties(i) ∂(l) = 0,∂(p) > 0 for p∈P;(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b in G;(iii) the number of elements in G of degree n is finite. (This number will be denoted by G(n).)

A theorem on free envelopes

The free envelope of a finite commutative semigroup was defined by Grillet [Trans. Amer. Math. Soc. 149 (1970), 665-682] to be a finitely generated free commutative semigroup F ( S ) F(S) with identity and a homomorphism α : S → F ( S ) \alpha :\,S\, \to \,F(S) endowed with certain properties. Grillet raised the following question: does α ( S ) \alpha (S) always generate a pure subgroup of the free Abelian group with the same basis as F ( S ) F(S) ? We prove this is indeed the case. It follows as a result of two lemmas. Lemma 1: Given a full rank proper subgroup H of a finitely generated free Abelian group F and a basis X of F there exists a surjective homomorphism f : F → Z f:\,F\, \to \,{\textbf {Z}} such that f is positive on X and f | H {f_{\left | H \right .}} is not surjective. Lemma 2: A finitely generated totally cancellative reduced subsemigroup of a finitely generated free Abelian group F is contained in the positive cone of some basis of F. The following duality theorem is also proved. Let S ∗ ≅ Hom ⁡ ( S , N ) {S^{\ast }}\, \cong \,\operatorname {Hom} (S,\,N) where N is the nonnegative integers under addition. Then S ≅ S ∗ ∗ S\, \cong \,{S^{{\ast }{\ast }}} if and only if S is isomorphic to a unitary subsemigroup of a finitely generated free commutative semigroup with identity.

Directed colimits of free commutative semigroups

Directed colimits of free commutative semigroups

Rédei's characterization of congruences on finitely generated free commutative semigroups

Rédei's characterization of congruences on finitely generated free commutative semigroups

Translations of commutative unique factorization semigroups

Every commutative semigroup is a semilattice of archimedean subsemigroups. Given a semilattice Y and a collection of commutative semigroups S indexed by Y, the converse problem of constructing every commutative semigroup which is a semilattice (Y) of the semigroups S remains unsolved except in certain cases. If each S is an abelian group the result follows as a special case of Clifford's Theorem (see [33, Thm. 4.11). The authors [13 have given a solution for any semilattice where each S is assumed to be cyclic, and more recently, Tamura has solved the problem for exclusive semilattices of exclusive semigroups [63 For a characterization of the general semilattice decomposition when each Se is weakly reductive, see Theorem 7.8.13 of [41. An element x of a commutative semigroup S is said to be prime if x ~ S 2 . S is said to have unique factorization if each nonzero element of S can be written uniquely as a product of primes in the usual sense. A commutative semigroup has unique factorization if and only if it is free or a Rees quotient of a free commutative semigroup (E2~, Thm. i). Commutative nil unique factorization semigroups were used in the characterization of arbitrary commutative nil semigroups in [11 (see [53 for an alternate characterization). Since commutative unique factorization semigroups arise naturally in the general theory of commutative semigroups, it seems of interest