Southeast Asian Bulletin of Mathematics On Epimorphisms and Left [Right] Regular Semigroups (original) (raw)
On epimorphisms and structurally regular semigroups
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.
Epimorphisms, Dominions, and Various Classes of Saturated Semigroups
Journal of Mathematics
In this paper, we discussed some saturated classes of ℋ -commutative semigroups, left (right) regular semigroups, medial semigroups, and paramedial semigroups. The results of this paper significantly extend the long standing result about normal bands that normal bands were saturated and, thus, significantly broaden the class of saturated semigroups.
Epimorphisms and dominions of regular semigroups
We show that a regular semigroups satisfying certain conditions in the containing semigroup is closed. As immediate corollaries, we have got that the special semigroup amalgam U = [{S, S }; U ; {i, α | U }] within the class of left [right] quasi-normal orthodox semigroups, R[L]-unipotent semigroups and left[right] Clifford semigroups is embeddable in a left [right] quasi-normal orthodox semigroup, R[L]-unipotent semigroup and left[right] Clifford semigroup respectively. Finally we have shown that the class of all semigroups satisfying the identity xyz = xz and the class of all semigroups satisfying the identity xy = xyx[yx = xyx] are closed within the class of all semigroups satisfying the identities xyz = xz and xy = xyx[yx = xyx] respectively.
Saturated (n, m)-Regular Semigroups
Mathematics
The aim of this paper is to determine several saturated classes of structurally regular semigroups. First, we show that structurally (n,m)-regular semigroups are saturated in a subclass of semigroups for any pair (n,m) of positive integers. We also demonstrate that, for all positive integers n and k with 1≤k≤n, the variety of structurally (0,n)-left seminormal bands is saturated in the variety of structurally (0,k)-bands. As a result, in the category of structurally (0,k)-bands, epis from structurally (0,n)-left seminormal bands is onto.
Cancellative Left (Right) Regular Semigroups
International Journal of Algebra and Statistics, 2012
A semigroup S is called regular semigroup if for every a ∈ S there exists x in S such that axa = a introduced by J. A. Green. In this paper, some preliminaries and basic concept of regular semigroups were presented. And proved that a cancellative semigroup S is left(right) regular semigroup if and only if it is a: (i) completely regular semigroup (ii) Clifford semigroup (iii) E-inversive semigroup (iv)-regular semigroup. Definition 1.1. An element a of a semigroup S is said to be regular if there exist an element x in S such that a = axa. Definition 1.2. A semigroup S is said to be regular semigroup if every element of S is regular. Example 1.3. (i) Every group is regular. (ii) Every inverse semigroup is regular. Definition 1.4. An element a is said to be an E-inversive of semigroup S if there exist an element x in S such that (ax) 2 = ax and (xa) 2 = xa. Definition 1.5. A semigroup S is said to be E-inversive semigroup if every element of S is E-inversive. Example 1.6. (i) Every regular semigroup is an E-inversive semigroup. (ii) Every inverse semigroup is an E-inversive semigroup.
Semigroups of left quotients—the uniqueness problem
Proceedings of the Edinburgh Mathematical Society, 1992
Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group -class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q.J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse ω-semigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations ℛ and ℒ in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise.The above result is then u...
Proceedings of the Edinburgh Mathematical Society, 1979
A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate w...