Showing papers on "Cancellative semigroup published in 2011"

Limit Varieties Generated by Completely 0-Simple Semigroups

Abstract: A limit variety is a variety that is minimal with respect to being nonfinitely based. This paper presents a new infinite series of limit semigroup varieties, each of which is generated by a finite 0-simple semigroup with Abelian subgroups. These varieties exhaust all limit varieties generated by completely 0-simple semigroups with Abelian subgroups.

Pseudo-amenability of certain semigroup algebras

Abstract: In this paper, we investigate the pseudo-amenability of semigroup algebra l1(S), where S is an inverse semigroup with uniformly locally finite idempotent set. In particular, we show that for a Brandt semigroup \(S={\mathcal{M}}^{0}(G,I)\), the pseudo-amenability of l1(S) is equivalent to the amenability of G.

Topological monoids of monotone injective partial selfmaps of $\mathbb{N}$ with cofinite domain and image

Abstract: In this paper we study the semigroup $\mathscr{I}_{\infty}^{ earrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{ earrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology $\tau$ on $\mathscr{I}_{\infty}^{ earrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{ earrow}(\mathbb{N}),\tau)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{ earrow}(\mathbb{N}),\tau)$ in a topological semigroup.

Pullbacks, $C(X)$-algebras, and their Cuntz semigroup

Abstract: In this paper we analyse the structure of the Cuntz semigroup of certain $C(X)$-algebras, for compact spaces of low dimension, that have no $\mathrm{K}_1$-obstruction in their fibres in a strong sense. The techniques developed yield computations of the Cuntz semigroup of some surjective pullbacks of C$^*$-algebras. As a consequence, this allows us to give a complete description, in terms of semigroup valued lower semicontinuous functions, of the Cuntz semigroup of $C(X,A)$, where $A$ is a not necessarily simple C$^*$-algebra of stable rank one and vanishing $\mathrm{K}_1$ for each closed, two sided ideal. We apply our results to study a variety of examples.

Pseudo-contractibility and pseudo-amenability of semigroup algebras

Abstract: In this paper, we characterize pseudo-contractibility of l1(S), where S is a uniformly locally finite inverse semigroup. As a consequence, we show that for a Brandt semigroup \({S={\mathcal{M}}^{0}(G,I),}\) the semigroup algebra l1(S) is pseudo-contractible if and only if G and I are finite. Moreover, we study the notions of pseudo-amenability and pseudo-contractibility of a semigroup algebra l1(S) in terms of the amenability of S.

On the Cancellativity of AG-Groupoids

Abstract: An AG-groupoid is a non-associative groupoid in general in which the identity (ab)c = (cb)a holds. In this paper we study some struc- tural properties of AG-groupoids with respect to the cancellativity. We prove that cancellative and non-cancellative elements of an AG-groupoid S parti- tion S and the two classes are AG-subgroupoids of S if S has left identity e. Cancellativity and invertibility coincide in a nite AG-groupoid S with left identity e: For a nite AG-groupoid S with left identity e having at least one non-cancellative element, the set of non-cancellative elements form a maximal ideal. We also prove that for an AG-groupoid S; the conditions (i) S is left cancellative (ii) S is right cancellative (iii) S is cancellative, are equivalent.

Geometric Semigroup Theory

Abstract: Geometric semigroup theory is the systematic investigation of finitely-generated semigroups using the topology and geometry of their associated automata. In this article we show how a number of easily-defined expansions on finite semigroups and automata lead to simplifications of the graphs on which the corresponding finite semigroups act. We show in particular that every finite semigroup can be finitely expanded so that the expansion acts on a labeled directed graph which resembles the right Cayley graph of a free Burnside semigroup in many respects.

Distributive and neutral elements of the lattice of commutative semigroup varieties

Abstract: We completely describe commutative semigroup varieties that are distributive, standard, or neutral elements of the lattice of all commutative semigroup varieties. In particular, we prove that the properties of being a distributive element and of being a standard element in this lattice are equivalent.

Irreducibility in the set of numerical semigroups with fixed multiplicity

Abstract: In this paper we introduce the notion of m-irreducibility that extends the standard concept of irreducibility of a numerical semigroup when the multiplicity is fixed. We analyze the structure of the set of m-irreducible numerical semigroups, we give some properties of these numerical semigroups and we present algorithms to compute the decomposition of a numerical semigroup with multiplicity m into m-irreducible numerical semigroups.

Fuzzy Good Congruences on Left Semiperfect Abundant Semigroups

Abstract: Motivated by studying fuzzy congruences in groups, semigroups, and ordered semigroups, and as a continuation of N. Kuroki and Y. Tan's works in regular semigroups in terms of fuzzy subsets, in this article we introduce the notions of a fuzzy good congruence relation, a fuzzy cancellative congruence relation on abundant semigroups, and give some properties, and characterizations of fuzzy good congruences on such semigroups. Furthermore, we characterize fuzzy good congruences of left semiperfect abundant semigroups, and get sufficient and necessary conditions for an abundant semigroup to be left semiperfect.

Approximations in a semigroup by using a neighbourhood system

Abstract: Let S be a semigroup. We consider the relation γ and its transitive closure γ*. The relation γ is the smallest equivalence relation on S so that S/γ* is a commutative semigroup. Based on the relation γ, we define a neighbourhood system for each element of S, and we present a general framework of the study of approximations in semigroups. The connections between semigroups and operators are examined.

Commutative semigroups with cancellation law: a representation theorem

Abstract: Any commutative, cancellative semigroup S with 0 equipped with a uniformity can be embedded in a topological group \(\widetilde{S}\). We introduce the notion of semigroup symmetry T which enables us to turn \(\widetilde{S}\) into an involutive group. In Theorem 2.8 we prove that if S is 2-torsion-free and T is 2-divisible then the decomposition of elements of \(\widetilde{S}\) into a sum of elements of the symmetric subgroup \(\widetilde{S}_{s}\) and the asymmetric subgroup \(\widetilde{S}_{a}\) is polar. In Theorem 3.7 we give conditions under which a topological group \(\widetilde{S}\) is a topological direct sum of its symmetric subgroup \(\widetilde{S}_{s}\) and its asymmetric subgroup \(\widetilde{S}_{a}\). Theorem 2.8 and Theorem 3.7 are designed to be useful tools in studying Minkowski–Radstrom–Hormander spaces (and related topological groups \(\widetilde{S}\)), which are natural extensions of semigroups of bounded closed convex subsets of real Hausdorff topological vector spaces.

Codistributive Elements of the Lattice of Semigroup Varieties

Abstract: We prove that if a semigroup variety is a codistributive element of the lattice SEM of all semigroup varieties, then it either coincides with the variety of all semigroups or is a variety of semigroups with completely regular square. We completely classify strongly permutable varieties that are codistributive elements of SEM.We prove that a semigroup variety is a costandard element of the lattice SEMif and only if it is a neutral element of this lattice. In view of results obtained earlier, this gives a complete description of costandard elements of the lattice SEM.

Graph inverse semigroups: their characterization and completion

Abstract: Graph inverse semigroups generalize the polycyclic inverse monoids and play an important role in the theory of C*-algebras. This paper has two main goals: first, to provide an abstract characterization of graph inverse semigroups; and second, to show how they may be completed, under suitable conditions, to form what we call the Cuntz-Krieger semigroup of the graph. This semigroup is the ample semigroup of a topological groupoid associated with the graph, and the semigroup analogue of the Leavitt path algebra of the graph.

To the theory of infinitely differentiable semigroups of operators

Abstract: Given a linear relation (multivalued linear operator) with certain growth restrictions on the resolvent, an infinitely differentiable semigroup of operators is constructed. It is shown that the initial linear relation is a generator of this semigroup. The results obtained are intimately related to certain results in the monograph “Functional analysis and semi-groups” by Hille and Phillips. §

Genus of numerical semigroups generated by three elements

Abstract: In this paper we study numerical semigroups generated by three elements. We give a characterization of pseudo-symmetric numerical semigroups. Also, we will give a simple algorithm to get all the pseudo-symmetric numerical semigroups with give Frobenius number.

Filters in (Quasiordered) Semigroups and Lattices of Filters

Abstract: A filter in a semigroup is a subsemigroup whose complement is an ideal. (Alternatively, in a quasiordered semigroup, a slightly more general definition can be given.) We prove a number of results related to filters in a semigroup and the lattice of filters of a semigroup. For instance, we prove that every complete algebraic lattice can be the lattice of filters of a semigroup. We prove that every finite semigroup is a homomorphic image of a finite semigroup whose lattice of filters is boolean and which belongs to the pseudovariety generated by the original semigroup. We describe filter lattices of some well-known semigroups such as full transformation semigroups of finite sets (which are three-element chains) and free semigroups (which are boolean).

Class semigroups and t-class semigroups of integral domains

Abstract: The class (resp., t-class) semigroup of an integral domain is the semigroup of the isomorphy classes of the nonzero fractional ideals (resp., t-ideals) with the operation induced by ideal (t-) multiplication. This paper surveys recent literature which studies ring-theoretic conditions that reflect reciprocally in the Clifford property of the class (resp., t-class) semigroup. Precisely, it examines integral domains with Clifford class (resp., t-class) semigroup and describes their idempotent elements and the structure of their associated constituent groups.

Positive matrix semigroups with binary diagonals

Abstract: We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in $${\mathcal{S}}$$ satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.

Abundant semigroups with quasi-ideal s-adequate transversals

Abstract: In this paper, the connection of the inverse transversal with the adequate transversal is explored. It is proved that if S is an abundant semigroup with an adequate transversal , then S is regular if and only if is an inverse semigroup. It is also shown that adequate transversals of a regular semigroup are just its inverse transversals. By means of a quasi-adequate semigroup and a right normal band, we construct an abundant semigroup containing a quasi-ideal S-adequate transversal and conversely, every such a semigroup can be constructed in this manner. It is simpler than the construction of Guo and Shum [9] through an SQ-system and the construction of El-Qallali [5] by W(E, S).

Irregular abelian semigroups with weakly amenable semigroup algebra

Abstract: In this paper we give counterexamples for the open problem, posed by Blackmore (Semigroup Forum 55:359–377, 1987) of whether weak amenability of a semigroup algebra l1(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but l1(S) is weakly amenable. Examples are then given.

Distributive hyperidentities in semigroups

Abstract: The paper gives a description of the class of semigroups in which the hyperidentity of left distributivity is essentially valid. It is proved that the class of all semigroups, in which this hypeidentity is essentially valid, is a union of three finitely based semigroup varieties, and the basic identities of all varieties are given in explicit forms.

Sub-classes of the monoid of left cancellative languages

Abstract: A language A is left cancellative if from AB=AC, it follows that B=C, for any two languages B and C. Semi-singular and inf-singular languages are two disjoint sub-sets of left cancellative languages and are introduced by Hsieh and Shyr [Left cancellative elements in the monoid of languages, Soochow J. Math. 4 (1978), pp. 7-15]. In this paper, we further study them. It is shown that all non-dense and all maximal left cancellative languages are semi-singular while all right dense left cancellative languages are inf-singular. Finally, a theorem shows that there is a left cancellative language which is neither semi-singular nor inf-singular.

On semigroups admitting ring structure

Abstract: A right-chain semigroup is a semigroup whose right ideals are totally ordered by set inclusion. The main result of this paper says that if S is a right-chain semigroup admitting a ring structure, then either S is a null semigroup with two elements or sS=S for some s∈S. Using this we give an elementary proof of Oman’s characterization of semigroups admitting a ring structure whose subsemigroups (containing zero) form a chain. We also apply this result, along with two other results proved in this paper, to show that no nontrivial multiplicative bounded interval semigroup on the real line ℝ admits a ring structure, obtaining the main results of Kemprasit et al. (ScienceAsia 36: 85–88, 2010).

On the closure of the extended bicyclic semigroup

Abstract: In the paper we study the semigroup $\mathscr{C}_{\mathbb{Z}}$ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup $\mathscr{C}_{\mathbb{Z}}$ and prove that every non-trivial congruence $\mathfrak{C}$ on the semigroup $\mathscr{C}_{\mathbb{Z}}$ is a group congruence, and moreover the quotient semigroup $\mathscr{C}_{\mathbb{Z}}/\mathfrak{C}$ is isomorphic to a cyclic group. Also we show that the semigroup $\mathscr{C}_{\mathbb{Z}}$ as a Hausdorff semitopological semigroup admits only the discrete topology. Next we study the closure $\operatorname{cl}_T(\mathscr{C}_{\mathbb{Z}})$ of the semigroup $\mathscr{C}_{\mathbb{Z}}$ in a topological semigroup $T$. We show that the non-empty remainder of $\mathscr{C}_{\mathbb{Z}}$ in a topological inverse semigroup $T$ consists of a group of units $H(1_T)$ of $T$ and a two-sided ideal $I$ of $T$ in the case when $H(1_T) eq\varnothing$ and $I eq\varnothing$. In the case when $T$ is a locally compact topological inverse semigroup and $I eq\varnothing$ we prove that an ideal $I$ is topologically isomorphic to the discrete additive group of integers and describe the topology on the subsemigroup $\mathscr{C}_{\mathbb{Z}}\cup I$. Also we show that if the group of units $H(1_T)$ of the semigroup $T$ is non-empty, then $H(1_T)$ is either singleton or $H(1_T)$ is topologically isomorphic to the discrete additive group of integers.

Universal bounds for matrix semigroups

Abstract: We show that any compact semigroup of n × n matrices is similar to a semigroup bounded by √ n. We give examples to show that this bound is best possible and consider the effect of the minimal rank of matrices in the semigroup on this bound.