Primitive Inverse Semigroup Research Articles

New orders in primitive inverse semigroup

New orders in primitive inverse semigroup

The structure of 0-bisimple R-unipotent semigroups

We generalize theory of Lawson for 0 -bisimple inverse monoids to wider classes of 0 -bisimple regular semigroups. We give a necessary and sufficient condition for a 0 -bisimple R-unipotent semigroup to admit a 0-restricted idempotent-pure prehomomorphism to a primitive inverse semigroup. Several illustrations of the theory are obtained as an application of the results in this paper.

Some notes on strongly E*-unitary inverse semigroups

In this paper, we give a direct proof that every strongly \(\user1{E}^ * --\user1{unitary}\) inverse semigroup can be embedded into a 0-semidirect product of a semilattice with zero by a group. As a corollary, we obtain a new proof of the structure theory of strongly \(\user1{E}^ * --\user1{unitary}\) inverse semigroups described in [1]. We also prove that the strongly \(\user1{E}^ * --\user1{unitary}\) inverse semigroups are precisely \(\user1{E}^ * -\user1{unitary}\) inverse semigroups equipped with a \(\user1{0 - restricted}\), idempotent pure prehomomorphism to a primitive inverse semigroup.

Semigroups which Have a minimum primitive inverse congruence

We investigate primitive inverse semigroups and Brandt semigroups as analogues of groups for semigroups with zero. We give a description of the minimum primitive inverse congruence on a categorical E *-dense E-semigroup. We show that a categorical semigroup S with a primitive inverse congruence has a minimum such congruence if $$\tilde D\left( S \right)$$ is *-dense in S. Here $$\tilde D\left( S \right)$$ is the least full, weakly self-conjugate, *-unitary and *-reflexive subsemigroup of S. Our results are analogous to those of Fountain, Pin and Weil for general semigroups.

Primitive inverse congruences on categorical semigroups

AbstractWe give an abstract description of the kernel of a proper primitive inverse congruence on a categorical semigroup. More specifically, we show that it is a *-reflexive, *-unitary, *-dense subsemigroup, and that on a given categorical semigroup there is a one–one correspondence between such subsemigroups and the proper primitive inverse congruences. Our results allow us to give a description of the minimum proper primitive inverse semigroup congruence on a strongly E*-dense categorical semigroup.

The Structure of 0-E-unitary inverse semigroups I: the monoid case

This is the first of three papers in which we generalise the classical McAlister structure theory for E-unitary inverse semigroups to those 0-E-unitary inverse semigroups which admit a 0-restricted, idempotent pure prehomomorphism to a primitive inverse semigroup. In this paper, we concentrate on finding necessary and sufficient conditions for the existence of such prehomomorphisms in the case of 0-E-unitary inverse monoids. A class of inverse monoids which satisfy our conditions automatically are those which are unambiguous except at zero, such as the polycyclic monoids.

Semigroups with zero whose idempotents form a subsemigroup

The structure of a categorical, E*-dense, E*-unitary E-semigroup S is elucidated in terms of a ‘B-quiver’, where B is a primitive inverse semigroup. In the case where S is strongly categorical, B is a Brandt semigroup. A covering theorem is also proved, to the effect that every categorical E*-dense E-semigroup has a cover which is a categorical, E*-dense, E*-unitary E-semigroup.

The structure of pseudo-inverse semigroups

A regular semigroup S is called a pseudo-inverse semigroup if eSe is an inverse semigroup for each e= e2 C S. We show that every pseudo-inverse semigroup divides a semidirect product of a completely simple semigroup and a semilattice. We thereby give a structure theorem for pseudo-inverse semigroups in terms of groups, semilattices and morphisms. The structure theorem which is presented here generalizes several structure theorems which have been given for particular classes of pseudo-inverse semigroups by several authors, and thus contributes to a unification of the theory. Completely (0-) simple semigroups and inverse semigroups form the first prototypes for the study of pseudo-inverse semigroups. We therefore can say that the theory of regular semigroups began with the study of pseudo-inverse semigroups [40, 45]. We may distinguish four successful trends in the papers which since then have dealt with some wider classes of pseudo-inverse semigroups: 1. the subdirect products of completely 0-simple and completely simple semigroups, 2. the generalized inverse semigroups (orthodox pseudo-inverse semigroups, 3. the normal band compositions of inverse semigroups, and 4. Rees matrix semigroups over inverse semigroups (with zero). Subdirect products of completely 0-simple semigroups and completely simple semigroups were initiated in [13, Chapter 2] and studied in great detail in [18] (see also ?4 of [14]); this class contains several interesting subclasses: (a) the trees of completely 0-simple semigroups [18] which include the primitive regular semigroups [7, Vol. II, 16, 39, 44, 46], (b) the regular locally testable semigroups [50] which include the normal bands [36] and the combinatorial completely 0-simple semigroups, (c) the normal bands of groups [37] which include the semilattices of groups [7, Vol. I], (d) the subdirect products of Brandt semigroups which include the locally testable semigroups which are inverse semigroups [50] and the primitive inverse semigroups [39]. The generalized inverse semigroups were introduced in [48] as a special class of orthodox semigroups; they include (a) the inverse semigroups, (b) the orthodox completely 0-simple semigroups [9] and the rectangular groups, (c) the Received by the editors October 25, 1979 and, in revised form, March 25, 1981 and June 18, 1981. AMS (MOS) subject classifications (1970). Primary 20M10.

Extensions of One Primitive Inverse Semigroup by Another

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.