Showing papers on "Cancellative semigroup published in 1972"

On (n, m)-rings

Abstract: The following paper is concerned with the extension of the usual ring concept to the case where the underlying group and semigroup are respectively ann-ary group and anm-ary semigroup. Most attention has been paid to homomorphism theorems and to some idealtheoretic aspects.

One-parameter inverse semigroups

Abstract: This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroup in a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free one-parameter inverse semigroups and then describe their congruences. 0. Let G be a subgroup of the multiplicative group of positive real numbers and let P denote the subsemigroup of G consisting of all x E G with x ? 1. Denote by Wp the class of all inverse semigroups H for which there isa homomorphism f: P -> H such that f(P) generates H (no proper inverse subsemigroup of H contains f(P)). We shall call such semigroups H one-parameter inverse semigroups and denote by W= UP Wp the class of all one-parameter inverse semigroups. The class W contains well-known semigroups. For example, each homomorphic image of a subgroup of R, the positive real numbers, is a member of W. Also the bicyclic semigroup B is a member of W, as is seen by noting that B is generated by a copy of the nonnegative integers. Indeed, if H is any elementary inverse semigroup, then H1 is generated by a homomorphic image of the nonnegative integers, and so is a one-parameter inverse semigroup. The main purpose of this paper is to describe all one-parameter inverse semigroups. In the process of doing this, we shall construct what we term the free oneparameter inverse semigroups Fp, one for each subgroup G of R and its associated semigroup P. The semigroup Fp is the only inverse semigroup (up to isomorphism) generated by a subsemigroup isomorphic with P which has the property that each homomorphism f: P -> S, an inverse semigroup, extends uniquely to a homomorphism f: Fp -> S. In particular, every H E Wp is a homomorphic image of Fp. We thus adopt the point of view that by describing Fp and the lattice of congruences of Fp for arbitrary P, we will have described all one-parameter inverse semigroups. We shall assume a certain familiarity with the algebraic theory of semigroups, particularly inverse semigroups. (See Clifford and Preston [1].) The existence and uniqueness of Fp is a consequence of a theorem due to McAlister [3, Theorem 33]. We were greatly aided in the actual description of Fp Presented to the Society, August 27, 1969; received by the editors June 10, 1969. AMS 1970 subject classifications. Primary 20M10; Secondary 20M05, 22A15.

Regular semigroups satisfying certain conditions on idempotents and ideals

Abstract: The structure of regular semigroups is studied (1) whose poset of idempotents is required to be a tree or to satisfy a weaker condition concerning the behavior of idempotents in different !D-classes, or (2) all of whose ideals are categorical or satisfy a variation thereof. For this purpose the notions of Dmajorization of idempotents, where D is a ED-class, !-majorization, E-categorical ideals, and completely semisimple semigroups without contractions are introduced and several connections among them are established. Some theorems due to G. Lallement concerning subdirect products and completely regular semigroups and certain results of the author concerning completely semisimple inverse semigroups are either improved or generalized.

Bisimple left inverse semigroups

Abstract: A semigroup S is called left inverse if every principal right ideal of S has a unique idempotent generator. We investigate D-classes of regular semigroups and of left inverse semigroups, characterizing those which are subsemigroups. We give a description of a bisimple left inverse semigroup S with identity element e as a quotient of the cartesian product Le×Re of the L-class Le of S and the R-class Re of S containing e. We also describe the maximal inverse semigroup homomorphic image of S.

The maximum idempotent-separating congruence on a regular semigroup

Abstract: The expression (1) for \i suffers from two maladies: it provides us with no information about \x which is not immediately deducible from Lallement's theorem and it is clearly not the sort of expression which may be readily used to decide if two given elements a and b of S are related under ft. In (2), J. M. Howie gave and alternative expression for \i in the case where S is an inverse semigroup. He determined that the maximum idempotentseparating congruence on an inverse semigroup S is given by

Embedding in compact uniquely divisible semigroups

Abstract: D. R. Brown and M. Friedberg have conjectured that each compact abelian semigroup can be embedded in a compact divisible semigroup. V. R. Hancock proved that each abelian algebraic semigroup can be embedded in a divisible abelian algebraic semigroup. In this paper we provide a partial solution to the conjecture of Brown and Friedberg by employing a topological version of Hancock's method as part of our construction. A theorem giving sufficient conditions for the Bohr compactification of weakly reductive semigroups to be injective is proved and used in the proof of our main result.

On the semigroup of a linear nonsingular automaton

Abstract: This paper is concerned with the structure of the semigroup of a linear nonsingular automaton and gives necessary and sufficient conditions for any semigroup to be isomorphic with the semigroup of a linear non-singular subautomaton.

Semigroups of right quotients of a semigroup which is a semilattice of groups

Abstract: Let S be a semigroup with zero which is a semilattice of groups. In [6], McMorris showed that the semigroup of quotients Q=Q(S) corresponding to the filter of “dense” right ideals of the semigroup S is also a semilattice of groups. He accomplished this by noting that Q is a regular semigroup in which all idempotents are central, an equivalent formulation of a semilattice of groups. In this paper we develop the semigroup of quotients Q corresponding to an arbitrary right quotient filter on S (as defined herein) and note the above result in this more general setting by explicitly constructing a semigroup which is isomorphic to Q. We also see that the underlying semilattice for Q in this case is isomorphic to a semigroup of quotients of the original semilattice for the semigroup S.

A class of semigroups embeddable in groups.

Abstract: Simple conditions are given in this paper that are sufficient to ensure the embeddability of a semigroup in a group. It is shown that the class of semigroups satisfying the conditions properly contains the class of all cancellative left quasi-reversible semigroups.

Building semigroups from groups (and reduced semigroups)

Abstract: Any semigroup can be quite explicitly rebuilt from its quotient by any congruence contained in ℋ and certain groups and mappings. This fact underlies most of the classical and recent structure theorems in semigroup theory, and yields new ones, e.g. concerning finite commutative semigroups.

Congruences on generalized inverse semigroups

Abstract: A generalized inverse semigroup is a regular semigroup whose idempotents satisfy a permutation identity X1 X2...Xn=Xp1 Xp2...Xpn, where (P1, P2..., Pn) is a nontrivial permutation of (1, 2,..., n). Yamada [4] has given a complete classification of generalized inverse semigroups in terms of inverse semigroups, left normal bands, and right normal bands. In this paper we show that every congruence on a generalized inverse semigroup is uniquely determined by a congruence on its associated inverse semigroup, left normal band, and right normal band. A converse is also provided.

Extensions of One Primitive Inverse Semigroup by Another

Abstract: Every inverse semigroup containing a primitive idempotent is an ideal extension of a primitive inverse semigroup by another inverse semigroup. Consequently, in developing the theory of inverse semigroups, it is natural to study ideal extensions of primitive inverse semigroups (cf. [3; 7]). Since the structure of any primitive inverse semigroup is known, an obvious type of ideal extension to consider is that of one primitive inverse semigroup by another. In this paper, we will construct all such extensions and give an abstract characterization of the resulting semigroup. The problem of extending one primitive inverse semigroup by another can be essentially reduced to that of extending one Brandt semigroup by another Brandt semigroup. The latter problem has been solved by Lallement and Petrich in [3] in case the first Brandt semigroup has only a finite number of idempotents.

3.—Some Structure Semigroup Results for Arens-Singer Semigroups *

Abstract: In this paper we discuss the structure semigroup of the L 1 -algebra of an Arens-Singer semigroup. Arising from this study we provide a complete and rather unexpected description of a nontrivial structure semigroup. We then link the above ideas with that of the almost periodic compactification of the semigroup.

Inverse semigroups of partial transformations and -classes

Abstract: If S is an inverse semigroup and 0 is the relation on the lattice A(S) of congruences on S defined by saying that two congruences ρl9p2 are ^-equivalent if and only if they induce the same partition of the idempotents then Θ is a congruence on Λ(S) and each #-class is a complete modular sublattice of A(S). If X is a partially ordered set then Jx denotes the inverse semigroup of one-to-one partial transformations of X which are order isomorphisms of ideals of X onto ideals of X, while if X is a semilattice, Tz denotes the inverse subsemigroup of Jx consisting of those elements a whose domain Δ(a) and range f (a) are principal ideals. It is shown that any inverse semigroup is isomorphic to an inverse subsemigroup of Jx for some semilattice X.

Sur les ideaux des demi-groupes nilpotents

Abstract: In this paper, we establish several properties of the lattice of ideals of a nilpotent semigroup. In particular, we characterize the finite nilpotent semigroups by the length of this lattice, and we generalize a theorem of T. Tamura and M. Yamada [5], concerning the ideal extension of a nilpotent semigroup by a null semigroup.