Showing papers on "Bicyclic semigroup published in 1973"

Orthodox semigroups whose idempotents satisfy a certain identity

Abstract: An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy [xyx=yx] Bisimple left [right] inverse semigroups have been studied by Venkatesan [6] In this paper, we clarify the structure of general left [right] inverse semigroups Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx=xyzx In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup

Extensions of a semilattice by an inverse semigroup

Abstract: The structure of inverse semigroup extensions of one inverse semigroup R by any other is analyzed in the case where R is a semilattice. Both a representation and method of construction are given. A brief preliminary examination is made of a certain class of congruences, on inverse semigroups, which are intimately related to such extensions.

Note on a certain class of orthodox semigroups

Abstract: This is a continuation and also a supplement of the previous papers [5], [6] and [8] concerning orthodox semigroups1) In [8], it has been shown that a quasi-inverse semigroup is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup In this paper, we present a structure theorem for quasi-inverse semigroups and some relevant matters

Representations of inverse semigroups by one-to-one partial transformations of a set

Abstract: We show that each representation ϕ, say, of an inverse semigroup S, by means of transformations of a set X, determines a representation ϕ* by means of partial one-to-one transformations of X, in such a fashion that sϕ ↦ sϕ*, for s ∈ S, is an isomorphism of Sϕ upon Sϕ*. An immediate corollary is the classical faithful representation of an inverse semigroup as a semigroup of partial one-to-one transformations.

A property of projective ideals in semigroup algebras

Abstract: The condition that certain left ideals in a finite monoid generate projective left ideals in the semigroup algebra imposes a strong restriction on the intersection of principal left ideals in the semigroup. Let S be a finite monoid, k be a commutative ring with identity, and let Ic S be a left ideal in S. We demand that kI be projective as a left kSmodule and investigate the resulting restrictions on the structure of I. In particular we can look for necessary conditions on S for kS to be left hereditary. (A sufficient condition is obtained in [4] and [5].) Semigroup terminology below follows [1] and [2]. We first need the following facts, which are valid in any ring with identity. LEMMA 1. Let R be a ring with identity. Let Ic R be a left ideal which is projective as a left R-module. Let e E R be any idempotent. Then the left ideal I+ Re is projective if and only if JrflRe is a direct summand of L PROOF. We observe that we have the following two short exact sequences: 0 I(1 e) -* I + Re Re > 0, 0OI n Re-I--I I(1 -e)->*O, where the map on the right end of the first sequence is xH-*xe, which has kernel (I+Re) r)R(1 -e)=I(1 -e), and the map on the right end of the second sequence is xF-*x(1 -e). Since Re is projective, the first sequence always splits, so that I+Re is projective if and only if I(1 -e) is. On the other hand, since I is projective, 1(1 -e) is projective if and only if the second sequence splits. Received by the editors June 5, 1972. AMS (MOS) subject classifications (1970). Primary 20M10, 20M25; Secondary 16A32, 16A50, 16A60.

Applications of semigroup algebras to ideal extensions of semigroups

Abstract: Let D be a semigroup and Z the integers modulo p, P where p is a prime. In this article we show that information concerning the semigroup algebras Z D of D P over Z provide information concerning a class of P ideal extensions of D.

Classes of semigroups and classes of sets

Abstract: @ye A Among all such congruences there is a largest one, and the quotient monoid by this congruence is denoted by SA and is called the syntactic semigroup of A. This semigroup is finite if and only if the set A is recognizable (by a finite automaton). The semigroup SA can then easily be described using the minimal automaton of A. It is reasonable to expect that reasonable properties of the recognizable set A will be reflected by reasonable properties of the finite semigroups SA and vice-versa. In trying to establish such a dialog, one is handicapped by the fact that there are finite semigroups which are not syntactic monoids of any set. The objective of this note is to state a theorem showing that the above inconvenience disappears if one considers classes of sets (rather than individual sets) and classes of semigroups.