Showing papers on "Bicyclic 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.

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).

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.

A note on strongly $E$-reflexive inverse semigroups

Abstract: In contrast to the semilattice of groups case, an inverse semigroup S which is the union of strongly E-reflexive inverse subsemigroups need not be strongly E-reflexive. If, however, the union is saturated with respect to the Green's relation 6D, and in particular if the union is a disjoint one, then S is indeed strongly E-reflexive. This is established by showing that 6) -saturated inverse subsemigroups have certain pleasant properties. Finally, in contrast to the E-unitary case, it is shown that the class of strongly E-reflexive inverse semigroups is not closed under free inverse products. The reader is referred to [1], [2] for the basic theory of inverse semigroups, including the theory of free inverse products. Recall from [4], [5] that an inverse semigroup S is said to be strongly E-reflexive whenever S is a semilattice of E-unitary inverse semigroups, or alternatively, whenever there exists a semilattice of groups congruence on S such that only idempotents are linked to idempotents under -. In [4], [5] we studied this class of semigroups and showed that many of the properties of semilattices of groups and of E-unitary inverse semigroups generalise to this class, albeit sometimes in a weaker form. We continue this line of investigation here. In what is by now a classic theorem, Clifford showed that an inverse semigroup which is a union of groups is a semilattice of groups. We ask to what extent this is true for strongly E-reflexive inverse semigroups. It is already known that a semilattice of strongly E-reflexive inverse semigroups is again strongly E-reflexive [5]. The following simple example shows that we cannot hope for a full generalisation of Clifford's theorem. Consider the bisimple inverse o-semigroup S(G, a), where the endomorphism a of the group G is not injective. As noted in [4, p. 341], S(G, a) is not strongly E-reflexive. However, using [1, Lemma 1.31], it is easily seen that S(G, a) is a union of its maximal subgroups and copies of the bicyclic semigroup, and these are all E-unitary. The restriction we require will now be given, and the example just noted would seem to indicate that it is the weakest possible. Let S be an inverse semigroup with semilattice of idempotents E. Let U be an inverse subsemigroup of S which is 6D -saturated in the sense that x 6D y' E U implies x E U, where 6D denotes the usual Green's relation on S. The maximal group homomorphic image of U is denoted by U with udenoting the image of u (u E U). Let U' = {x E SIx > u for some u E U); note that U' may equal S. Received by the editors March 23, 1979. AMS (MOS) subject classifications (1970). Primary 20M10. ? 1980 American Mathematical Society 0002-9939/80/0000-0301 /$0 1.75 352 This content downloaded from 207.46.13.148 on Sun, 11 Sep 2016 04:20:48 UTC All use subject to http://about.jstor.org/terms STRONGLY E-REFLEXIVE INVERSE SEMIGROUPS 353 The first result shows that U' has some pleasant properties. PROPOSITION. (i) U' is an inverse subsemigroup of S which contains U, and xy E U'implies x E U'andy E U'. (ii) The rule: xp = uif x > u E U and x4 = 0 otherwise, gives a well-defined homomorphism 0: S U? such that k1 U is the canonical homomorphism onto U. PROOF. (i) xy > U E U = xx > xyy 1x uu X1 x > uu x R. u =X x E U', since U is 6D -saturated and 6R C 6D. Dually, y E U'. The remainder of the result is easily proven. (ii) Suppose x E U' with x > u E U and x > v E U. Then u = ex, v = fx where e=uuEUnE,f=vv-1EUnE. Hence efu=efv, and ef6EEn U, so that u= . It is then almost immediate that 0 is well-defined. The rest of the result involves a little routine calculation, using (i). REMARK. Taking S to be a semilattice with more than two elements, we see that U need not be an ideal of U' in Proposition 1. The proposition enables us to prove our main result. THEOREM. Let S be a union of 6D -saturated strongly E-reflexive inverse subsemigroups Si, i E I. Then S is strongly E-reflexive. PROOF. Each Si is a semilattice Ai of E-unitary inverse semigroups T7, X E Ai. It is easily shown that each T/x is 6D-saturated in S. Hence we may suppose without loss of generality that each Si is E-unitary. For each i E I, let Oi: S -> S?o be the homomorphism defined as in (ii) above, and let T be the direct product of the Si?. Then the pi induce a homomorphism 0: S -T with s0 having ith component s5ci, i E I. Now So is a semilattice of groups, since T is. Suppose that x4 = eo for some e E E, where x E Si say. Then x4i is the identity element of Si, and since S5 is E-unitary it follows that x E E; whence the result. COROLLARY. Let S be a disjoint union of strongly E-reflexive inverse subsemigroups. Then S itself is strongly E-reflexive. PROOF. Clearly each of the inverse subsemigroups in question is 6D -saturated in S. REMARK. The elementary theory of inverse semigroups shows that an inverse semigroup S which is a union of groups is a disjoint union of its maximal subgroups He, e E E, and that this is the 6D -decomposition of S. Hence S is the union of the 6D -saturated E-unitary inverse subsemigroups He. It is easy to show that the homomorphism 4 in the proof of the theorem is injective in this case. Hence S is a subdirect product of the He with zero added possibly. From this one can deduce, again by elementary means, that S is a semilattice of groups with the multiplication defined by linking homomorphisms. Thus, modulo some elementary results, our theory restricts to Clifford's classic theorems. Now let E be the semilattice { e, f, g} where e > g, f > g, and e, f are incomparable. Let S be the semilattice of groups Ge U Gf U Gg where Ge' Gg are trivial and Gf is the cyclic 2-group; let T be the semilattice of groups He U Hf U Hg This content downloaded from 207.46.13.148 on Sun, 11 Sep 2016 04:20:48 UTC All use subject to http://about.jstor.org/terms

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.