scispace - formally typeset
Search or ask a question

Showing papers on "Bicyclic semigroup published in 1998"


Journal ArticleDOI
TL;DR: In this article, the structure of a categorical dual I*_X to the symmetric inverse monoid is described and representations of an inverse semigroup in this dual symmetric monoid are discussed, and conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of a semigroup S in certain other inverse algebras (that is, inverse monoids in which all finite infima======
Abstract: There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid I_X , that is, a monoid of partial one-to-one self-maps of a set X. The present paper describes the structure of a categorical dual I*_X to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in I_X and I*_X, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

78 citations


Journal ArticleDOI
TL;DR: In this article, the structure of a semigroup S generated by a permutation group G of units and an idempotent ϵ is studied and it is shown that S is a regular semigroup.
Abstract: It is well known that the semigroup of all transformations on a finite set X of order n is generated by its group of units, the symmetric group, and any idempotent of rank n − 1. Similarly, the symmetric inverse semigroup on X is generated by its group of units and any idempotent of rank n − 1 while the analogous result is true for the semigroup of all n × n matrices over a field. In this paper we begin a systematic study of the structure of a semigroup S generated by its group G of units and an idempotent ϵ . The first section consists of preliminaries while the second contains some general results which provide the setting for those which follow. In the third section we shall investigate the situation where G is a permutation group on a set X of order n and ϵ is an idempotent of rank n − 1. In particular, we shall show that any such semigroup S is regular. Furthermore we shall determine when S is an inverse or orthodox semigroup or completely regular semigroup. The fourth section deals with a special ca...

76 citations


Journal ArticleDOI
TL;DR: In this article, a new representation of the generator of a subordinate semigroup as a limit of bounded operators is presented, where the generator is viewed as a function of a generator of the original semigroup.
Abstract: We prove a new representation of the generator of a subordinate semigroup as limit of bounded operators. Our construction yields, in particular, a characterization of the domain of the generator. The generator of a subordinate semigroup can be viewed as a function of the generator of the original semigroup. For a large class these functions we show that operations at the level of functions has its counterpart at the level of operators.

75 citations


Journal ArticleDOI
TL;DR: For a semigroup S and a set B, the relative rank of S modulo A is the minimal cardinality of a setB such that S can be generated.
Abstract: For a semigroup S and a set the relative rank of S modulo A is the minimal cardinality of a setB such that generates S. We show that the relative rank of an infinite full transformation semigroup modulo the symmetric group, and also modulo the set of all idempotent mappings, is equal to 2. We also characterise all pairs of mappings which, together with the symmetric group or the set of all idempotents, generate the full transformation semigroup.

70 citations


Journal ArticleDOI
TL;DR: In this paper, the authors established conditions under which (Tt)t≥0 is similar to a contraction semigroup, i.e., there exists an isomorphism S E B (H) such that (S-1 Tt S)t ≥ 0 is a contraction semiigroup.
Abstract: )t≥0 on a Hilbert space H, we establish conditions under which (Tt)t≥0 is similar to a contraction semigroup, i.e., there exists an isomorphism S E B (H) such that (S-1 Tt S)t≥0 is a contraction semigroup. In the case when the generator -A of (Tt)t≥0 is one-to-one, we obtain that (Tt)t≥0 is similar to a contraction semigroup if and only if A admits bounded imaginary powers. This characterizes one-to-one operators of type strictly less than π/2 on H which belong to BIP (H).

60 citations



Journal ArticleDOI

29 citations


Journal Article
TL;DR: In this paper, the authors describe how the graded minimal resolution of certain semigroup algebras is related to the combinatorics of simplicial complexes and obtain characterizations of the Cohen-Macaulay and Gorenstein conditions.
Abstract: We describe how the graded minimal resolution of certain semigroup algebras is related to the combinatorics of some simplicial complexes We obtain characterizations of the Cohen-Macaulay and Gorenstein conditions The Cohen- Macaulay type is computed from combinatorics As an application, we compute explicitly the graded minimal resolution of monomial both affine and simplicial projective surfaces

26 citations





Journal ArticleDOI
TL;DR: In this article, the authors characterize the semigroups with CEP by a set of conditions on their structure, and show that every such semigroup is a semilattice of nil extensions of rectangular groups.
Abstract: on a semigroup of T of S extends to the semigroup S, if there exists a congruence ρ on s such that ρ|T= ρT. A semigroup S has the congruence extension property, CEP, if each congruence on each semigroup extends to S. In this paper we characterize the semigroups with CEP by a set of conditions on their structure (by this we answer a problem put forward in [1]). In particular, every such semigroup is a semilattice of nil extensions of rectangular groups.


Journal ArticleDOI
TL;DR: This work gives an algorithm solving SYNTACTIC MONOID for a large class of finite monoids in O(¦M¦ 3 ) time and shows that a slight generalization of SYNT ACTICMONOID is NP-complete.

Journal ArticleDOI
TL;DR: In this article, the authors characterized a -inverse semigroup S for which Sub S is 0-distributive or 0-modular, and showed that the latter is not stronger than the former.
Abstract: A semigroup S is called -regular if for every element a of S there exists m 2 Z (the set of positive integers) such that a is regular. Let us denote by r(a) the least positive integer m such that a is regular of S and call it the regular index of an element a: If every regular element of a -regular semigroup S possesses a unique inverse, then S is called -inverse [1]. Let A be a subsemigroup of a -inverse semigroup S: We say that A is a -inverse subsemigroup of S if for any a 2 A; a 2 RegA (the set of all regular elements of A) [2]. Obviously, A is a -inverse subsemigroup of S if and only if for any a 2 A and every m 2 Z; a 2 RegS implies a 2 RegA: For a -inverse semigroup S; a set Sub S of all -inverse subsemigroups (including the empty set) of S forms a lattice with respect to intersection denoted as usual by T and union denoted by h; i , where hA;Bi is the -inverse subsemigroup generated by the union of subsets A;B of S: A lattice L( V ; W ) with zero is 0-distributive [0-modular ] if for any a; b; c 2 L , a V b = a V c = 0[a V b = 0 and a c] implies a V (b W c) = 0[a V (b W c) = c]: In order to avoid misunderstanding, we remark that, in contrast with relation between distributivity and modularity, 0-distributivity is not stronger than 0-modularity. Moreover, for the lattices under examination in this paper, the latter turns out to be stronger than the former (see Corollaries 2.3 and 2.4 below). As to general case, there is no implicative relation between these two conditions. Indeed, it is easy to verify that the 5-element lattice called pentagon is 0-distributive but not 0-modular, and the 5-element lattice called diamond is 0-modular but not 0-distributive. The aim of this paper is to characterize a -inverse semigroup S for which Sub S is 0-distributive or 0-modular. In [2, 3], the author characterized a -inverse semigroup S for which Sub S is modular or complemented, respectively. Suppose S is a semigroup and A is a subset of S: We will denote by ES the set of all idempotents of S; by GrS the set of all group elements of S; by RegS the set of all regular elements of S; and by hAi the subsemigroup of S generated by A: We will denote by Ge the maximal subgroup of S with the idempotent e as identity element. For e 2 ES we put

Journal ArticleDOI
J. Banasiak1
TL;DR: In this article, it was shown that a given pair (A, B) generates a B-bounded semigroup if and only if in a certain extrapolation space related to the operator B, the closure of A generates a semigroup.
Abstract: In [3]A. Bellini-Morante defined and analysed a new one-parameter family of bounded operators which he called a B-bounded semigroup. The definition was motivated by an example from the transport theory where the evolution generated by an operator A was in a certain sense controlled by another operator B. In this paper we show that a given pair (A, B) generates a B-bounded semigroup if and only if in a certain extrapolation space related to the operator B, the closure of A generates a semigroup and we also address some related topics.

Journal ArticleDOI
TL;DR: The Mobius semigroup studied in this paper arises naturally geometrically as the (compression) subsemigroup of the group of Mobius transformations which carry some fixed open Mobius ball into itself.
Abstract: The Mobius semigroup studied in this paper arises very naturally geometrically as the (compression) subsemigroup of the group of Mobius transformations which carry some fixed open Mobius ball into itself. It is shown, using geometric arguments, that this semigroup is a maximal subsemigroup. A detailed analysis of the semigroup is carried out via the Lorentz representation, in which the semigroup resurfaces as the semigroup carrying a fixed half of a Lorentzian cone into itself. Close ties with the Lie theory of semigroups are established by showing that the semigroup in question admits the structure of an Ol'shanskii semigroup, the most widely studied class of Lie semigroups.


Journal ArticleDOI
TL;DR: In this article, it was shown that the subset of freek-generated subsemigroups of the semigroup of all automaton transformations over a finite alphabet is a second category set (in the sense of the Baire category approach).
Abstract: It is established that the subset of freek-generated subsemigroups of the semigroup of all automaton transformations over a finite alphabet is a second category set (in the sense of the Baire category approach) in the set of allk-generated subsemigroups. A continuum series of pairs of automaton transformations each of which generates a free semigroup of rank two is indicated. A criterion is established for this semigroup to be a finite-automaton group.


Journal ArticleDOI
TL;DR: In this paper, the structure of the semigroup S is investigated when RS admits a compact topology and it is proved that the last principal factors of S have finitely many of right or left ideals.
Abstract: The structure of the semigroup S is investigated when the semigroup ring RS admits a compact topology. It is proved that, in case of the semisimple semigroup S , the ”last” principal factors of S have finitely many of right or left ideals. It is shown that it is not true for other factors.


Journal ArticleDOI
TL;DR: In this article, the authors studied the semigroup generated by the set of the kemel-trace operators on the congruence lattice of a primitive regular semigroup, which is best described as an orthogonal sum of completely 0-simple semigroups.
Abstract: The aim of this paper is the study of the semigroup generated by the set of the kemel-trace operators on the congruence lattice of a primitive regular semigroup. Since the structure of the nontrivial primitive regular semigroups is best described as an orthogonal sum of completely 0-simple semigroups, the problem of describing the previous semigroup is solved by determining the semigroups associated with any semigroup of the orthogonal sum. To do the latter, use is made of the Rees Theorem.