A factorization of dual prehomomorphisms and expansions of inverse semigroups

2004 ◽  
Vol 41 (3) ◽  
pp. 295-308
Author(s):  
B. Billhardt
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Natural Generalization ◽  
Inverse Semigroups ◽  
Universal Property ◽  
Alternative Description ◽  
Unique Homomorphism

For any inverse semigroup S we construct an inverse semigroup S(S), which has the following universal property with respect to dual prehomomorphisms from S: there is an injective dual prehomomorphism ιS: S → S(S) such that for each dual prehomomorphism θ from S into an inverse semigroup T there exists a unique homomorphism θ* : S(S) → T with ιSθ* = θ. If we restrict the class of dual prehomomorphisms under consideration to order preserving ones, S(S) may be replaced by a certain homomorphic image Ŝ(S) which can be viewed as a natural generalization of the Birget-Rhodes prefix expansion for groups [4] to inverse semigroups. Recently, Lawson, Margolis and Steinberg [8] have given an alternative description of Ŝ(S) which is based on O’Carroll’s theory of idempotent pure congruences [11]. It should be noted that our ideas can be used to simplify some of their arguments.

scholarly journals Inverse semigroups and their natural order

1978 ◽  
Vol 19 (1) ◽  
pp. 59-65 ◽  
Author(s):  
H. Mitsch
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Homomorphic Image ◽  
Point Of View ◽  
Inverse Semigroups ◽  
Natural Order ◽  
Theoretical Point ◽  
Trivial Group

The natural order of an inverse semigroup defined by a ≤ b ⇔ a′b = a′a has turned out to be of great importance in describing the structure of it. In this paper an order-theoretical point of view is adopted to characterise inverse semigroups. A complete description is given according to the type of partial order an arbitrary inverse semigroup S can possibly admit: a least element of (S, ≤) is shown to be the zero of (S, ·); the existence of a greatest element is equivalent to the fact, that (S, ·) is a semilattice; (S, ≤) is directed downwards, if and only if S admits only the trivial group-homomorphic image; (S, ≤) is totally ordered, if and only if for all a, b ∈ S, either ab = ba = a or ab = ba = b; a finite inverse semigroup is a lattice, if and only if it admits a greatest element. Finally formulas concerning the inverse of a supremum or an infimum, if it exists, are derived, and right-distributivity and left-distributivity of multiplication with respect to union and intersection are shown to be equivalent.

Inverse semigroups generated by a pair of subgroups

1977 ◽  
Vol 77 (1-2) ◽  
pp. 9-22 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Homomorphic Image ◽  
Main Part ◽  
Structure Theorem ◽  
Inverse Semigroups ◽  
Infinite Cyclic Group ◽  
The Third ◽  
Group A ◽  
Maximum Group

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

scholarly journals Free inverse semigroups

1973 ◽  
Vol 16 (4) ◽  
pp. 443-453 ◽  
Author(s):  
G. B. Preston
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Canonical Forms ◽  
Principal Ideals ◽  
Free Inverse Semigroup ◽  
Alternative Description ◽  
Alternative Procedures ◽  
The Way

In an important recent paper H. E. Scheiblich gave a construction of free inverse semigroups that throws considerable light on their structure [1]. In this note we give an alternative description of free inverse semigroups. What Scheiblich did was to construct a free inverse semigroup as a semigroup of isomorphisms between principal ideals of a semilattice E, say, thus realising free inverse semigroups as inversee subsemigroupss of the semigroup TE, a kind of inverse semigroup introduced and exploited by W. D. Munn [2]. We go instead directly to canonical forms for the elements of a free inverse semigroup. The connexion between our construction and that of Scheiblich's will be clear. There are several alternative procedures possible to reach our construction on which we comment on the way.

scholarly journals Finite semigroups with commuting idempotents

1987 ◽  
Vol 43 (1) ◽  
pp. 81-90 ◽  
Author(s):  
C. J. Ash ◽  
T. E. Hall
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Finite Semigroups

AbstractWe show that every such semigroup is a homomorphic image of a subsemigroup of some finite inverse semigroup. This shows that the pseudovariety generated by the finite inverse semigroups consists of exactly the finite semigroups with commuting idempotents.

scholarly journals Inverse semigroups generated by nilpotent transformations

1984 ◽  
Vol 99 (1-2) ◽  
pp. 153-162 ◽  
Author(s):  
John M. Howie ◽  
M. Paula O. Marques-Smith
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Nilpotent Elements ◽  
Inverse Subsemigroup ◽  
Symmetric Inverse Semigroup ◽  
Infinite Cardinality ◽  
Index 2 ◽  
Rees Quotient ◽  
The Ideal

SynopsisLet X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=<B 1B>, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = <N2>, where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.

scholarly journals Division theorems for inverse and pseudo-inverse semigroups

1981 ◽  
Vol 31 (4) ◽  
pp. 415-420
Author(s):  
F. Pastijn
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Division Theorem ◽  
Inverse Subsemigroup ◽  
Pseudo Inverse

AbstractWe show that every inverse semigroup is an idempotent separating homomorphic image of a convex inverse subsemigroup of a P-semigroup P(G, L, L), where G acts transitively on L. This division theorem for inverse semigroups can be applied to obtain a division theorem for pseudo-inverse semigroups.

scholarly journals Some covering and embedding theorems for inverse semigroups

1976 ◽  
Vol 22 (2) ◽  
pp. 188-211 ◽  
Author(s):  
D. B. McAllister
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Main Question ◽  
Embedding Theorems

AbstractAn inverse semigroup S is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper the author showed that every inverse semigroup is an idempotent separating homomorphic image of an E-unitary inverse semigroup. The main question considered in this paper is the following. Given an inverse semigroup S give a method for constructing E-unitary inverse semigroups P and idempotent separating homomorphisms φ P → S in such a way that the structure of P as a P-semigroup is evident.

scholarly journals Modular inverse semigroups

1987 ◽  
Vol 43 (1) ◽  
pp. 47-63 ◽  
Author(s):  
Katherine G. Johnston ◽  
Peter R. Jones ◽  
T. E. Hall
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
A Chain ◽  
Inverse Subsemigroups ◽  
Maximum Group

AbstractAn inverse semigroup S is said to be modular if its lattice 𝓛𝓕 (S) of inverse subsemigroups is modular. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a semigroup S is modular if and only if (I) S is combinatorial, (II) its semilattice E of idempotents is “Archimedean” in S, (III) its maximum group homomorphic image G is locally cyclic and (IV) the poset of idempotents of each 𝓓-class of S is either a chain or contains exactly one pair of incomparable elements, each of which is maximal. Thus in view of earlier results of the second author a simple modular inverse semigroup is “almost” distributive. The bisimple modular inverse semigroups are explicitly constructed. It is remarkable that exactly one of these is nondistributive.

Rees matrix semigroups over inverse semigroups

1986 ◽  
Vol 102 (1-2) ◽  
pp. 61-90 ◽  
Author(s):  
F. J. Pastijn ◽  
Mario Petrich
Keyword(s):  
Inverse Semigroup ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Rees Matrix Semigroup ◽  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Special Cases ◽  
Matrix Semigroup ◽  
Matrix Semigroups ◽  
Completely Simple

SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

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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