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 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 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.

BICYCLIC SUBSEMIGROUPS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

2010 ◽  
Vol 20 (01) ◽  
pp. 89-113 ◽  
Author(s):  
EMANUELE RODARO
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Inverse Semigroups ◽  
Amalgamated Free Product ◽  
Free Product With Amalgamation ◽  
Necessary And Sufficient ◽  
Completely Semisimple

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max {|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the [Formula: see text]-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the [Formula: see text]-classes to be finite.

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.

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 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.

Residual finiteness of free products of combinatorial strict inverse semigroups

1994 ◽  
Vol 124 (1) ◽  
pp. 137-147 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Inverse Semigroups ◽  
Residual Finiteness ◽  
Free Products ◽  
Residually Finite

It is shown that the free product of two residually finite combinatorial strict inverse semigroups in general is not residually finite. In contrast, the free product of a residually finite combinatorial strict inverse semigroup and a semilattice is residually finite.

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.

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