Showing papers on "Cancellative semigroup published in 1980"

On semisimple semigroup rings

Abstract: ABSTRACr Let qr be a property of rings that satisfies the conditions that (i) homomorphic images of I-rings are Ir-rings and (ii) ideals of Ir-rings are Ir-rings Let S be a semilattice P of semigroups S If each semigroup ring R[SG] (a E P) is IT-semisimple, then the semigroup ring R[SG] is also Ir-semisimple Conditions are found on P to insure that each R[SG] (a E P) is Ir-semisimple whenever S is a strong semilattice P of semigroups S and R[S] is Ir-semisimple Examples are given to show that the conditions on P cannot be removed These results and examples answer several questions raised by J Weissglass

Embedding inverse semigroups in coset semigroups

Abstract: The setK(G) of all cosets X of a group G, modulo all subgroups of G, forms an inverse semigroup under the multiplication X*Y=smallest coset that constains XY. In this note we show that each inverse semigroup S can be embedded in some coset semigroupK(G). This follows from a result which shows that symmetric inverse semigroups can be embedded in the coset semigroups of suitable symmetric groups. We also give necessary and sufficient conditions on an inverse semigroup S in order that it should be isomorphic to someK(G).

On a pull-back diagram for orthodox semigroups

Abstract: In the present paper we deal with two problems concerning orthodox semigroups. M. Yamada raised the questions in [6] whether there exists an orthodox semigroup T with band of idempotents E and greatest inverse semigroup homomorphic image S for every band E and inverse semigroup S which have the property that Open image in new window is isomorphic to the semilattice of idempotents of S, and if T exists then whether it is always unique up to isomorphism. T. E. Hall [1] has published counter-examples in connection with both questions and, moreover, he has given a necessary and sufficient condition for existence. Now we prove a more effective necessary and sufficient condition for existence and deal with uniqueness, too. On the other hand, D. B. McAlister's theorem in [4] saying that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup is generalized for orthodox semigroups. The proofs of these results are based on a theorem concerning a special type of pullback diagrams. In verifying this theorem we make use of the results in [5] which we draw up in Section 1. The main theorems are stated in Section 2. For the undefined notions and notations the reader is referred to [2].

Arithmetical and semihereditary semigroup rings

Abstract: In this paper conditions on the commutative ring R (with identity) and the commutative semigroup ring S (with identity) are found which characterize those semigroup rings R[S] which are reduced or have weak global dimension at most one. Likewise, those semigroup rings R[S] which are semihereditary are completely determined in terms of R and S.

Right inverse semigroups

Abstract: In a recent paper of the author the well-known Vagner-Preston Theorem on inverse semigroups was generalized to include a wider class of semigroups, namely right normal right inverse semigroups. In an attempt to generalize the theorem to include all right inverse semigroups, the notion of μ – μ i transformations is introduced in the present paper. It is possible to construct a right inverse band B M ( X ) of μ – μ i transformations. From this a set A M ( X ) for which left and right units are in B M ( X ) and satisfying certain conditions is constructed. The semigroup A M ( X ) so constructed is a right inverse semigroup. Conversely every right inverse semigroup can be isomorphically embedded in a right inverse semigroup constructed in this way. 1980 Mathematics subject classification (Amer. Math. Soc.) : 20 M 20.

The free elementary * orthodox semigroup

Abstract: A * orthodox semigroup is a unary semigroup ( S , ·, *) which satisfies the axioms (1) x ** = x , (2) xx * x = x , (3) ( xy )* = y * x *, and xx * yy * zz * ∈ E ( S ) = E . Such a semigroup is orthodox in the usual sense that EE ⊂ E . Since * orthodox semigroups are equationally defined, they form a variety. This paper characterizes the free * orthodox semigroup F on a single generator x . An idempotent e is a projection provided e * = e . There is a unique projection in each R class and in each L class. Each element w ∈ F is given a canonical form by locating the projection in its R class and its L class. The characterization is achieved by describing the multiplication between these canonical forms. As a corollary, the band E ( F ) is described. This band is regular in the sense that it satisfies the regularity equation axaya = axya.

Generalized inverse semigroups and amalgamation

Abstract: Given that free products with amalgamation of inverse semigroups exist without “collapse”, or equivalently that any amalgam of inverse semigroups is strongly embeddable in an inverse semigroup, it is natural to ask likewise if free products with amalgamation of generalized inverse semigroups exist without collapse. We note the result of Imaoka [1976b], that free products exist for the class of generalized inverse semigroups. As yet we are unable to answer this question. The main result of this paper is that any amalgam of generalized inverse semigroups is strongly embeddable in a semigroup. Of course this gives some hope that our question above will have an affirmative answer. Our proof is via three representation extension properties. We show that any generalized inverse semigroup has the representation extension property in any containing generalized inverse semigroup; and that any right generalized inverse semigroup has the free (and hence the strong) representation extension property in any containing right generalized inverse semigroup. Our results have been obtained independently by Teruo Imaoka (private communication).

Direct products of cyclic semigroups

Abstract: It will be proved that a finite semigroup which is decomposable into a direct product of cyclic semigroups which are not groups is uniquely so decomposable. It will then be determined when a finite semigroup has such a decomposition and how its non-group cyclic direct factors, if they exist, can be found.

Semigroups and graphs

Abstract: Various semigroups are associated with a graph. We start with the semigroup of paths, then from this form its maximal inverse semigroup morphic image, the inverse semigroup of the graph. We also consider the free inverse semigroup generated by the arrows of the graph and show that the inverse semigroup of the graph is a Rees quotient of this free inverse semigroup. What is sometimes called the fundamental groupoid of the graph is identified with the maximal primitive morphic image of the inverse semigroup of the graph.

On the structure of regular semigroups in which the maximal subgroups form a band of groups

Abstract: A regular semigroup S is said to be quasi-orthodox if and only if there exist an inverse semigroup I and a surjective homomorphism f : S → I such that ef −1 is a completely simple subsemigroup of S for each idempotent e of I . If a regular semigroup S satisfies the following property P , then S is necessarily quasi-orthodox: ( P ) The maximal subgroups of S form a band of groups. Such a semigroup S is called a quasi-orthodox semigroup with ( P ). In this paper, the structure of quasi-orthodox semigroups with ( P ) is studied. Structure theorems are established for the class of general quasi-orthodox semigroups and for some special classes of quasi-orthodox semigroups. In particular the concept of spined product of orthodox semigroups with ( P ) is introduced, and it is shown that an orthodox semigroup S is isomorphic to the spined product of an H -degenerated orthodox semigroup and an H -compatible inverse semigroup if and only if S has the property ( P ).

The isomorphism semigroup of a periodic abelian group

Abstract: The isomorphism semigroup S ( G ) of a group G is the semigroup of isomorphic mappings between subgroups of G , with composition its operation. This paper will show that a periodic abelian group is uniquely determined by its isomorphism semigroup.

On quasi-transitive semigroup actions

Abstract: A semigroup S acts quasi-transitively on a space X if the orbits form a partition of X. Some results are proved giving characterisations of normal quasi-transitive acts and quasi-transitive acts for which the action maps satisfy various other conditions. Finally a result concerning the effect of quasi-transitive action by a continuum semigroup on a subspace of the plane is proved.