Showing papers on "Cancellative semigroup published in 1973"
01 Jan 1973
TL;DR: In this article, a characterization of the free inverse semigroup I on a non-empty set X is presented. But the characterization is restricted to the case X = 0 and X = 1.
Abstract: This note announces a characterization of the free inverse semigroup I on a non-empty set X.
67 citations
TL;DR: The structure of general left-right inverse semigroups has been investigated in this paper, where 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 semiigroup and a right inverse semigroup.
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
34 citations
20 citations
01 Feb 1973
TL;DR: Theorem 3 relates this class of semigroups to left amenable semigroup as mentioned in this paper and further the class of all semiigroups with this property, where the convolution powers p(n) of P implied by the above property are studied.
Abstract: Let S be a discrete semigroup, P a probability measure on S and SF S with limsupn(P0,)(s))1/n=1. We study limit theorems for the convolution powers p(n) of P implied by the above property and further the class of all semigroups with this property. Theorem 3 relates this class of semigroups to left amenable semigroups.
17 citations
14 citations
TL;DR: In this article, the structure of inverse semigroup extensions by any other is analyzed in the case where R is a semilattice and a brief preliminary examination is made of a certain class of congruences, on inverse semigroups, which are intimately related to such extensions.
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.
13 citations
12 citations
TL;DR: A construction for finding other subgroups of the power semigroup is presented which directly generalizes the formation of factor groups.
10 citations
9 citations
TL;DR: In this article, a structure theorem for quasi-inverse semigroups is presented, which is a continuation and also a supplement of the previous papers [5], [6] and [8] concerning orthodox semiigroups.
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
9 citations
TL;DR: In this article, the structure of left cancellative semigroups is described, and the existence of idempotents is investigated, and it is shown that the existence is related to maximal proper right ideals.
Abstract: In this note we shall describe the structure of left cancellative semigroups. We shall also investigate the existence of idempotents and show that the existence of idempotents is related to the existence of maximal proper right ideals.
01 Feb 1973
TL;DR: Theorem 2.1 as mentioned in this paper shows that if S is a commutative semigroup and S is polish, then S can be embedded as an open subsemigroup of a locally compact group.
Abstract: Let S be a commutative semigroup which is a topological space such that the translations are both continuous and open maps. The main result states that if (1) either S is Suslin such that there is at least one point of continuity for the semigroup mapping S x S-S or S is polish and (2) 3 a nontrivial Radon measure on S such that y(V)=,u(x+ V) for V open cS and x c S, then S can be embedded as an open subsemigroup of a locally compact group. It is also shown that if S is polish and a cancellation semigroup then S can be embedded as an open subsemigroup of a group. In [6], R. Rigelhof proved that if S is a locally compact abelian semigroup such that the translations are both continuous and open and 3 a Radon measure p, such that V x c S, V nonvoid open set Vc S, j(x+V)=y(V)>O, then S can be embedded as an open subsemigroup of a locally compact group. In the context of our main result (stated in the abstract) we first show that V nonvoid open set Vc S, g(V)>0. We then follow the methods of [6] to get a topology on the abelian group G generated by S. At this stage, we cannot conclude that G is a topological group. We proceed as follows to show that G is a locally compact space. We first construct a translation invariant Radon measure i on G which extends the measure ,u on S. Then, modifying an idea of D. Montgomery [5] we conclude that the images of certain 'good' subsets of G under inversion are universally measurable, enabling us to get a compact set KC:G with the property 2(K)>O and -K is compact. Then, as in [3], we show that G is locally compact. A small modification of the above method (Lemmas 2 and 6) and a result of Wu [9] yield an embedding theorem (Theorem 2) for polish commutative cancellation semigroups in which translations are also open maps. This embedding theorem gives an alternative and somewhat simpler proof of the main result for the case of polish semigroups. In the sequel S will be a semigroup and It a Radon measure on it satisfying the conditions stated in the main result. Received by the editors May 16, 1972 and, in revised form, August 24, 1972. AMS (MOS) subject classifications (1969). Primary 2205; Secondary 2875.
TL;DR: In this article, it was shown that each representation ϕ, say, of an inverse semigroup S, by means of transformations of a set X, determines a representationϕ* by using partial one-to-one transformations of X, in such a fashion that sϕ ↦ s ϕ*, for s ∈ S, is an isomorphism of S ϕ upon Sϕ*.
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.
TL;DR: In this article, a congruence ϱ on an N -semigroup S is called an N-congruence on S if S/g9 is an n-semigroup.
01 Jan 1973
TL;DR: In this article, it was shown that the following two statements concerning a bounded real function f on S are equivalent: (i) f -FL(S) c FL(S); and (ii) f is S-convergent to a constant x, i.e.
Abstract: Let S be a countably infinite left amenable cancellative semigroup, FL(S) the space of left almost-convergent functions on S. The purpose of this paper is to show that the following two statements concerning a bounded real function f on S are equivalent: (i) f -FL(S) c FL(S); (ii) there is a constant cx such that for each ->0 there exists a set A (S satisfying (a) p(XA)=0 for each left invariant mean 9 on S and (b) If(x)cx 0 and qp(lf)=qp(f) for seS and fenm(S), where l,fe m(S) is defined by (lsf)(t)=f(st), t E S. The set of left invariant means on S is denoted by ML(S). If ML(S) is nonempty, then S is said to be left amenable [2]. A bounded real function f on a left amenable semigroup is called left almost-convergent if (f) equals a fixed constant d(f) as 79 runs through ML(S) [2]. The set of all left almost-convergent functions, denoted by FL(S), is a vector subspace of m(S) and it contains constant functions. But, in general, it is not closed under multiplication. The purpose of this paper is to study this aspect of FL(S) and our main result is the following. THEOREM. Let S be a countable left-cancellative left amenable semigroup withoutfinite left ideals. Then the following two statements concerning a function f E m(S) are equivalent: (i) f is a multiplier of FL(S), i.e., f * FL(S) cFL(S); (ii) f is S-convergent to a constant x, i.e.,for a given e>0 there exists a set A c S such that (a) 79(XA)=0 for each 9p E ML(S), and (b) If(x)-xl
01 Feb 1973
TL;DR: The condition that certain left ideals in a finite monoid generate projective ideals in the semigroup algebra imposes a strong restriction on the intersection of principal left ideals as mentioned in this paper, and the resulting restrictions on the structure of I. In particular, they require that kI be projective as a left kSmodule and investigate the resulting restriction on I. (A sufficient condition is obtained in [4] and [5].)
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.
TL;DR: In this paper, it was shown that information concerning the semigroup algebras Z D of D P over Z provides information concerning a class of P ideal extensions of D.
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.