SEMIGRUP REGULER DAN SIFAT-SIFATNYA

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete ∩ \cap homomorphism but not a ∨ \vee -homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no minimum element in a kernel class. The kernel mapping preserves neither intersections nor joins. The set of axioms presented in [7] for right kernel systems is reviewed. A new set of axioms is obtained as a consequence of the fact that a right congruence is the intersection of the principal right congruences on the idempotent classes. Finally, it is shown that even though a congruence on a regular semigroup is the intersection of the principal congruences on the idempotent classes, the situation is not the same for right congruences on a regular semigroup. Right congruences on a regular, even orthodox, semigroup are not, in general, determined by their idempotent classes.

A regular semigroup S S is locally inverse if each local submonoid e S e eSe , e e an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism θ \theta which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.

A method for proving the embeddability of semigroup amalgams is introduced. After providing necessary and sufficient conditions in terms of representations for the weak embeddability of a semigroup amalgam, it successfully deals with the embedding of inverse semigroup amalgams into inverse semigroups and the embedding of an amalgam of regular semigroups whose core is full in each member.

SynopsisA regular semigroup S is called V-regular if for any elements a, b ∈ S and any inverse (ab)′ of ab, there exists an inverse a′ of a and an inverse b′ of b such that (ab)′ = b′a′. A characterization of a V-regular semigroup is given in terms of its partial band of idempotents. The strongly V-regular semigroups form a subclass of the class of V-regular semigroups which may be characterized in terms of their biordered set of idempotents. It is shown that the class of strongly V-regular semigroups comprises the elementary rectangular bands of inverse semigroups (including the completely simple semigroups), a special class of orthodox semigroups (including the inverse semigroups), the strongly regular Baer semigroups (including the semigroups that are the multiplicative semigroup of a von Neumann regular ring), the full transformation semigroup on a set, and the semigroup of all partial transformations on a set.

In this survey we consider three kinds of algorithmic questions concerning varieties of semigroups. We are interested in identity problems, in the solvability of a system of equations and in the structure of all solutions of a given system. We study them in significant varieties of semigroups, monoids, groups, completely simple semigroups, completely regular semigroups (in particular semigroups satisfying the identity xr≈x) and involutary semigroups (in particular star regular semigroups and inverse semigroups).

For any permutation σ on N = {1,2,…,n}: (i) Green’s relations are characterized in the centralizer C(σ) of σ (relative to the semigroup PTn of partial transformations on N); and (ii) a criterion is given for C(σ) to be a regular semigroup (inverse semigroup, union of groups).

Rees Matrix Covers and Semidirect Products of Regular Semigroups

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band ES of idempotents of S such that F∋e, ES∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of FS.

An element x of a semigroup [ring] A is said to regular if there is an element y of A such that x = xyx, and A is called a regular semigroup [(Von Neumann) regular ring] if every element of A is regular. A quasi-ideal of a semigroup [ring] A is a subsemigroup [subring] Q of A such that AQ intersection QA [is a subset of] Q,and a bi-ideal of A is a subsemingroup [subring] B of A such that BAB [is a subset of] B. Recall that for nonempty subsets X and Y of a ring A, XY denotes the set of all finite sums of the form sigma x[subscript i] y[subscript i] where x[subscript i] [is an element of set] X and Y[subscript i] [is an element of set] Y. We say that a semigroup or a ring has the BQ-property if its quasi-ideals and bi-ideals coincide. It is know that every regular semigroup and every regular ring has the BQ-property. For a nonempty set X, let T(X) denote the full transformation semigroup on X, and for [is an empty set] is not equal to Y[is a subset of] X, let T (X,Y) and T [bar] (X, Y) be the subsemigroups of T(X) defined by T(X, Y)={ alpha is an element of a set T(X) l ran alpha is a subset of Y} and T[bar] (X, Y)={ alpha is an elemaent of a set T(X) l Yalpha is a subset of Y}. Symons and Magill introduced and studied T(X, Y) and T[bar] (X, Y) in 1975 and 1966, respectively. If V is a vector space over a field F, let L[subscript F](V) be the set of all linear transformations alpha : V vector V. For a subspace W of V, define L[subscript F] (V, W) and L[bar][subscript F] (V, W) analogously as follows : L[subscript F] (V, W) = {alpha is an element of a set L[subscriptF(V) l ran alpha is a subset of W} and L[bar][subscript F](V, W) = {alpha is an element of a set L[subscritp F] (V) l { Walpha is a subset of W},and we also consider K[subscript F](V, W) = {alpha L[subscriptF] (V) l W is a subset of ker alpha}. Then L[subscriptF](V, W), L[bar] [subscript F](V, W) and K[subscript F] (V, W) are subsemigroups of (L[subscript F] (V), omicron) and subrings of (L[subscript F] (V),+,omicron) where omicron and+are the composition and usual addition of linear transformations, respectively. This research consists of two major parts. In the first part, we give necessary and sufficient conditions for the elements of these semigroups to be regular. As a consequence, characterizations determinging when these semigroups are regular are given. In the second part, we provide necessary and sufficient conditions for these semigroups and rings to have the BQ-property.

SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.

An adequate semigroup S is said to be ample if for any e2 = e, a ∈ S, ae = (ae)†a and ea = a(ea)*. It is well known that inverse semigroups are ample semigroups. The purpose of this paper is to study matrix representations of an ample semigroup. Some properties of ample semigroups are obtained. It is proved that any indecomposable good matrix representations of an ample semigroup can be constructed by using those of weak Brandt semigroups.

The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

In this paper we study epimorphisms, dominions and related properties for some classes of structurally (n,m)-regular semigroups for any pair (n,m) of positive integers. In Section 2, after a brief introduction of these semigroups, we prove that the class of structurallly (n,m)-generalized inverse semigroups is closed under morphic images. We then prove the main result of this section that the class of structurally (n,m)-generalized inverse semigroups is saturated and, thus, in the category of all semigroups, epimorphisms in this class are precisely surjective morphisms. Finally, in the last section, we prove that the variety of structurally (o, n)-left regular bands is saturated in the variety of structurally (o, k)-left regular bands for all positive integers k and n with 1 ≤ k ≤ n.