scholarly journals Lattice isomorphisms of inverse semigroups

1978 ◽  
Vol 21 (2) ◽  
pp. 149-157 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Free Groups ◽  
Inverse Semigroups ◽  
Strong Sense ◽  
Lattice Isomorphism ◽  
Free Inverse Semigroup ◽  
Subgroup Lattices ◽  
The Subject ◽  
Inverse Subsemigroups

A largely untouched problem in the theory of inverse semigroups has been that of finding to what extent an inverse semigroup is determined by its lattice of inverse subsemigroups. In this paper we discover various properties preserved by lattice isomorphisms, and use these results to show that a free inverse semigroup ℱℐx is determined by its lattice of inverse subsemigroups, in the strong sense that every lattice isomorphism of ℱℐx upon an inverse semigroup T is induced by a unique isomorphism of ℱℐx upon T. (A similar result for free groups was proved by Sadovski (12) in 1941. An account of this may be found in Suzuki's monograph on the subject of subgroup lattices (14)).

scholarly journals Inverse semigroups determined by their lattices of inverse subsemigroups

1981 ◽  
Vol 30 (3) ◽  
pp. 321-346 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Inverse Semigroups ◽  
Lattice Isomorphism ◽  
Free Inverse Semigroup ◽  
Inverse Subsemigroups ◽  
Completely Semisimple

AbstractIn a previous paper ([14]) the author showed that a free inverse semigroup is determined by its lattice of inverse subsemigroups, in the sense that for any inverse semigroup T, implies . (In fact, the lattice isomorphism is induced by an isomorphism of upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and E-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.

scholarly journals A note on free inverse semigroups

1974 ◽  
Vol 19 (1) ◽  
pp. 17-23 ◽  
Author(s):  
L. O'Carroll
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Alternative Proof ◽  
Group Of Automorphisms ◽  
Free Inverse Semigroup ◽  
Explicit Constructions ◽  
Inverse Subsemigroups

Recently Scheiblich (7) and Munn (3), amongst others, have given explicit constructions for FIA, the free inverse semigroup on a non-empty set A. Further, Reilly (5) has investigated the free inverse subsemigroups of FIA. In this note we generalise two of Reilly's lesser results, and also characterise the surjective endomorphisms of FIA. The latter enables us to determine the group of automorphisms of FIA, and to show that if A is finite then FIA is Hopfian (a result proved independently by Munn (3)). Finally, we give an alternative proof of Reilly's main theorem, which uses Munn's theory of birooted trees.

scholarly journals Lattice isomorphisms of modular inverse semigroups

1988 ◽  
Vol 31 (3) ◽  
pp. 441-446 ◽  
Author(s):  
K. G. Johnston
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Lattice Isomorphism ◽  
Lattice Of Subalgebras ◽  
Survey Article ◽  
Structural Isomorphism ◽  
Inverse Subsemigroups

For an inverse semigroup S we will consider the lattice of inverse subsemigroups of S, denoted L(S). A major problem in algebra has been that of finding to what extent an algebra is determined by its lattice of subalgebras. (See, for example, the survey article [9]). By a lattice isomorphism (L-isomorphism, structural isomorphism, or projectivity) of an inverse semigroup S onto another T we shall mean an isomorphism Φ of L(S) onto L(T). A mapping φ from S to T is said to induce Φ if AΦ = Aφ for all A in L(S). We say that S is strongly determined by L(S) if every lattice isomorphism of S onto T is induced by an isomorphism of S onto T.

scholarly journals Inverse semigroups whose full inverse subsemigroups form a chain

1981 ◽  
Vol 22 (2) ◽  
pp. 159-165 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Analogous Problem ◽  
A Chain ◽  
Inverse Subsemigroups

The structure of semigroups whose subsemigroups form a chain under inclusion was determined by Tamura [9]. If we consider the analogous problem for inverse semigroups it is immediate that (since idempotents are singleton inverse subsemigroups) any inverse semigroup whose inverse subsemigroups form a chain is a group. We will therefore, continuing the approach of [5, 6], consider inverse semigroups whose full inverse subsemigroups form a chain: we call these inverse ▽-semigroups.

scholarly journals A note on shortly connected inverse semigroups

1991 ◽  
Vol 43 (3) ◽  
pp. 463-466 ◽  
Author(s):  
S.M. Goberstein
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
The Other ◽  
Inverse Subsemigroups ◽  
Completely Semisimple

Shortly connected and shortly linked inverse semigroups arise in the study of inverse semigroups determined by the lattices of inverse subsemigroups and by the partial automorphism semigroups. It has been shown that every shortly linked inverse semigroup is shortly connected, but the question of whether the converse is true has not been addressed. Here we construct two examples of combinatorial shortly connected inverse semigroups which are not shortly linked. One of them is completely semisimple while the other is not.

scholarly journals Homogeneity of inverse semigroups

2018 ◽  
Vol 28 (05) ◽  
pp. 837-875 ◽  
Author(s):  
Thomas Quinn-Gregson
Keyword(s):  
Inverse Semigroup ◽  
Finite Groups ◽  
Abelian Groups ◽  
Semigroup Theory ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Homogeneous Groups ◽  
Inverse Subsemigroups

An inverse semigroup [Formula: see text] is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if [Formula: see text] then there exists a unique [Formula: see text] such that [Formula: see text] and [Formula: see text]. We say that a countable inverse semigroup [Formula: see text] is a homogeneous (inverse) semigroup if any isomorphism between finitely generated (inverse) subsemigroups of [Formula: see text] extends to an automorphism of [Formula: see text]. In this paper, we consider both these concepts of homogeneity for inverse semigroups, and show when they are equivalent. We also obtain certain classifications of homogeneous inverse semigroups, in particular periodic commutative inverse semigroups. Our results may be seen as extending both the classification of homogeneous semilattices and the classification of certain classes of homogeneous groups, such as homogeneous abelian groups and homogeneous finite groups.

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 THE ÉTALE GROUPOID OF AN INVERSE SEMIGROUP AS A GROUPOID OF FILTERS

2013 ◽  
Vol 94 (2) ◽  
pp. 234-256 ◽  
Author(s):  
M. V. LAWSON ◽  
S. W. MARGOLIS ◽  
B. STEINBERG
Keyword(s):  
Functional Analysis ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Groupoid ◽  
Linear Representations ◽  
Inverse Subsemigroups ◽  
Étale Groupoid

AbstractPaterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.

scholarly journals Free generators in free inverse semigroups

1972 ◽  
Vol 7 (3) ◽  
pp. 407-424 ◽  
Author(s):  
N.R. Reilly
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Idempotent Element ◽  
Necessary And Sufficient Condition ◽  
Countable Number ◽  
Number Of Generators ◽  
Free Inverse Semigroup ◽  
Free Generators ◽  
Necessary And Sufficient

Using the characterization of the free inverse semigroup F on a set X, given by Scheiblich, a necessary and sufficient condition is found for a subset K of an inverse semigroup S to be a set of free generators for the inverse sub semigroup of S generated by K. It is then shown that any non-idempotent element of F generates the free inverse semigroup on one generator and that if |X| > 2 then F contains the free inverse semigroup on a countable number of generators. In addition, it is shown that if |X| = 1 then F does not contain the free inverse semigroup on two generators.

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