scholarly journals Proper Dubreil-Jacotin inverse semigroups

1975 ◽  
Vol 16 (1) ◽  
pp. 40-51 ◽  
Author(s):  
R. McFadden
Keyword(s):  
Inverse Semigroup ◽  
Algebraic Structure ◽  
Inverse Semigroups ◽  
Partial Ordering ◽  
Group Congruence ◽  
Complete Class ◽  
Ordered Semigroups ◽  
Partially Ordered ◽  
Minimum Group ◽  
Integrally Closed

This paper is concerned mainly with the structure of inverse semigroups which have a partial ordering defined on them in addition to their natural partial ordering. However, we include some results on partially ordered semigroups which are of interest in themselves. Some recent information [1, 2, 6, 7,11] has been obtained about the algebraic structure of partially ordered semigroups, and we add here to the list by showing in Section 1 that every regular integrally closed semigroup is an inverse semigroup. In fact it is a proper inverse semigroup [10], that is, one in which the idempotents form a complete class modulo the minimum group congruence, and the structure of these semigroups is explicitly known [5].

scholarly journals Strongly E-reflexive inverse semigroups

1977 ◽  
Vol 20 (4) ◽  
pp. 339-354 ◽  
Author(s):  
L. O'Carroll
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Group Congruence ◽  
Minimum Group

Let S be an inverse semigroup with semilattice of idempotents E. We denote by σ the minimum group congruence on S (6), and by τ the maximum idempotent-determined congruence on S (2). (Recall that the congruence η on S is called idempotent-determined if (e, x)∈ η and e ∈ E imply that x ∈ E.) In general τ ⊆ σ.

scholarly journals Inverse semigroups as extensions of semilattices

1975 ◽  
Vol 16 (1) ◽  
pp. 12-21 ◽  
Author(s):  
Liam O'Carroll
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Group Congruence ◽  
Minimum Group

Let S be an inverse semigroup with semilattice of idempotents E, and let ρ be a congruence on S. Then ρ is said to be idempotent-determined [2], or I.D. for short, if (a, b) ∈ р and a∈E imply that b ∈ E. If, further, ρ is a group congruence, then clearly ρ is the minimum group congruence on S, and in this case S is said to be proper [8]. Let T = S/ρ.

scholarly journals A rank for right congruences on inverse semigroups

2007 ◽  
Vol 76 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Victoria Gould
Keyword(s):  
Inverse Semigroup ◽  
Maximal Subgroup ◽  
Inverse Semigroups ◽  
The Other ◽  
Brandt Semigroup ◽  
Group Congruence ◽  
Existentially Closed ◽  
Bicyclic Monoid ◽  
A Chain ◽  
Lattice Of Congruences

The S-rank (where ‘S’ abbreviates ‘sandwich’) of a right congruence ρ on a semigroup S is the Cantor-Bendixson rank of ρ in the lattice of right congruences ℛ of S with respect to a topology we call the finite type topology. If every ρ ϵ ℛ possesses S-rank, then S is ranked. It is known that every right Noetherian semigroup is ranked and every ranked inverse semigroup is weakly right Noetherian. Moreover, if S is ranked, then so is every maximal subgroup of S. We show that a Brandt semigroup 0(G, I) is ranked if and only if G is ranked and I is finite.We establish a correspondence between the lattice of congruences on a chain E, and the lattice of right congruences contained within the least group congruence on any inverse semigroup S with semilattice of idempotents E(S) ≅ E. Consequently we argue that the (inverse) bicyclic monoid B is not ranked; moreover, a ranked semigroup cannot contain a bicyclic -class. On the other hand, B is weakly right Noetherian, and possesses trivial (hence ranked) subgroups.Our notion of rank arose from considering stability properties of the theory Ts of existentially closed (right) S-sets over a right coherent monoid S. The property of right coherence guarantees that the existentially closed S-sets form an axiomatisable class. We argue that B is right coherent. As a consequence, it follows from known results that TB is a theory of B-sets that is superstable but not totally transcendental.

EXTENSIONS OF REGULAR ORTHOGROUPS BY INVERSE SEMIGROUPS

1995 ◽  
Vol 05 (03) ◽  
pp. 317-342 ◽  
Author(s):  
BERND BILLHARDT
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Normal Extension ◽  
Group Congruence ◽  
Completely Regular ◽  
Orthodox Semigroups

Let V be a variety of regular orthogroups, i.e. completely regular orthodox semigroups whose band of idempotents is regular. Let S be an orthodox semigroup which is a (normal) extension of an orthogroup K from V by an inverse semigroup G, that is, there is a congruence ρ on S such that the semigroup ker ρ of all idempotent related elements of S is isomorphic to K and S/ρ≅G. It is shown that S can be embedded into an orthodox subsemigroup T of a double semidirect product A**G where A belongs to V. Moreover T itself can be chosen to be an extension of a member from V by G. If in addition ρ is a group congruence we obtain a recent result due to M.B. Szendrei [16] which says that each orthodox semigroup which is an extension of a regular orthogroup K by a group G can be embedded into a semidirect product of an orthogroup K′ by G where K′ belongs to the variety of orthogroups generated by K.

scholarly journals Certain congruences on a completely regular semigroup

1974 ◽  
Vol 15 (2) ◽  
pp. 109-120 ◽  
Author(s):  
Thomas L. Pirnot
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Group Congruence ◽  
Completely Regular Semigroup ◽  
Rees Matrix Semigroup ◽  
Semilattice Congruence ◽  
Completely Regular ◽  
Maximal Group ◽  
Minimum Group ◽  
Matrix Semigroup

Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences ρ on a completely regular semigroup S such that S/ρ is a semilattice of groups. We shall call such a congruence an SG-congruence.

scholarly journals Certain fundamental congruences on a regular semigroup

1966 ◽  
Vol 7 (3) ◽  
pp. 145-159 ◽  
Author(s):  
J. M. Howie ◽  
G. Lallement
Keyword(s):  
Inverse Semigroup ◽  
Algebraic Theory ◽  
Primary Concern ◽  
Group Congruence ◽  
Semilattice Congruence ◽  
Regular Semigroups ◽  
Recent Developments ◽  
Minimum Group ◽  
Band Congruence ◽  
Lattice Of Congruences

In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by α β and η respectively.

scholarly journals The lattice of inverse subsemigroups of a reduced inverse semigroup

1976 ◽  
Vol 17 (2) ◽  
pp. 161-172 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Group Congruence ◽  
Minimum Group ◽  
Inverse Subsemigroups

An inverse semigroup R is said to be reduced (or proper) if ℛ∩σ= i (where σ is the minimum group congruence on R). McAlister has shown ([3], [4]) that every reduced inverse semigroup is isomorphic with a “P-semigroup” P(G, , ), for some semilattice , poset containing as an ideal, and group G acting on by order-automorphisms; (and, conversely, every P-semigroup is reduced). In [4], he also found the morphisms between P-semigroups, in terms of morphisms between the respective groups, and between the respective posets.

scholarly journals E-unitary inverse semigroups over semilattices

1978 ◽  
Vol 19 (1) ◽  
pp. 1-12 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Inverse Semigroups ◽  
Partially Ordered ◽  
Image Position

An inverse semigroup is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper [4], the author showed that any E-unitary inverse semigroup is isomorphic to a semigroup constructed from a triple (G, ℋ, ) consisting of a down-directed partially ordered set ℋ, an ideal and subsemilattice of ℋ and a group G acting on ℋ, on the left, by order automorphisms in such a way that ℋ = G. This semigroup is denoted by P(G, ℋ, ); it consists of all pairs (a, g)∈ × G such that g−1a ∈ , under the multiplication

scholarly journals E-unitary congruences on inverse semigroups

1976 ◽  
Vol 17 (1) ◽  
pp. 57-75 ◽  
Author(s):  
N. R. Reilly ◽  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Structure Theorem ◽  
Inverse Semigroups ◽  
Partially Ordered ◽  
Important Structure

By an E-unitary inverse semigroup we mean an inverse semigroup in which the semilattice is a unitary subset. Such semigroups, elsewhere called ‘proper’ or ‘reduced’ inverse semigroups, have been the object of much recent study. Free inverse semigroups form a subclass of particular interest.An important structure theorem for E-unitary inverse semigroups has been obtained by McAlister [4, 5]. From a triple (G, ) consisting of a group G, a partially ordered set and a subset , satisfying certain conditions, he constructs an E-unitary inverse semigroup P(G, ). A semigroup of this type is called a P-semigroup. The structure theorem states that every E-unitary inverse semigroup is, to within isomorphism, of this form. A second theorem asserts that every inverse semigroup is isomorphic to a quotient of a Psemigroup by an idempotent-separating congruence. We refer below to these results as McAlister's Theorems A and B respectively. A triple (C, ) of the type used to construct a P-semigroup is here termed a “McAlister triple”. It is shown further, in [5], that there is essentially only one such triple corresponding to a given E-unitary inverse semigroup.

scholarly journals A NOTE ON THE HOWSON PROPERTY IN INVERSE SEMIGROUPS

2016 ◽  
Vol 94 (3) ◽  
pp. 457-463 ◽  
Author(s):  
PETER R. JONES
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Necessary And Sufficient

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.

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