scholarly journals An embedding theorem for free inverse semigroups

1981 ◽  
Vol 22 (2) ◽  
pp. 217-222
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Embedding Theorem ◽  
Inverse Semigroups ◽  
Free Inverse Semigroup ◽  
Free Generators

In this note it is shown that if S is a free inverse semigroup of rank at least two and if e, f are idempotents of S such that e > f then S can be embedded in the partial semigroup eSe/fSf. The proof makes use of Scheiblich's construction for free inverse semigroups [7, 8] and of Reilly's characterisation of a set of free generators in an inverse semigroup [4, 5].

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.

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 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 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 GRÖBNER–SHIRSHOV BASES FOR FREE INVERSE SEMIGROUPS

2009 ◽  
Vol 19 (02) ◽  
pp. 129-143 ◽  
Author(s):  
L. A. BOKUT ◽  
YUQUN CHEN ◽  
XIANGUI ZHAO
Keyword(s):  
Normal Form ◽  
Inverse Semigroup ◽  
Normal Forms ◽  
Inverse Semigroups ◽  
Free Inverse Semigroup ◽  
New Construction

A new construction for free inverse semigroups was obtained by Poliakova and Schein in 2005. Based on their result, we find Gröbner–Shirshov bases for free inverse semigroups with respect to the deg-lex order of words. In particular, we give the (unique and shortest) normal forms in the classes of equivalent words of a free inverse semigroup together with the Gröbner–Shirshov algorithm to transform any word to its normal form.

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 Free generators in free inverse semigroups: Corrigenda

1973 ◽  
Vol 9 (3) ◽  
pp. 479-480 ◽  
Author(s):  
N.R. Reilly
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inverse Subsemigroup ◽  
Free Generators ◽  
Necessary And Sufficient

In [1], Theorem 2.2, a necessary and sufficient condition is given for a subset of an inverse semigroup to generate a free inverse subsemigroup. However one very obvious further condition is omitted. The result should read as follows.

GROWTH AND EXISTENCE OF IDENTITIES IN A CLASS OF FINITELY PRESENTED INVERSE SEMIGROUPS WITH ZERO

1996 ◽  
Vol 06 (01) ◽  
pp. 105-121 ◽  
Author(s):  
L.M. SHNEERSON ◽  
D. EASDOWN
Keyword(s):  
Inverse Semigroup ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Inverse Semigroups ◽  
Semigroup Identity ◽  
Free Inverse Semigroup ◽  
Nontrivial Identity ◽  
Rees Quotient ◽  
Finitely Presented

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial or exponential growth, and that the type of growth is algorithmically recognizable. We prove that such a semigroup has polynomial growth if and only if it satisfies a certain semigroup identity. However we give an example of such a semigroup which has exponential growth and satisfies some nontrivial identity in signature with involution.

scholarly journals Ideals of free inverse semigroups

1980 ◽  
Vol 30 (2) ◽  
pp. 157-167
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Semigroups ◽  
Proper Ideal ◽  
Principal Ideals ◽  
Free Inverse Semigroup ◽  
Necessary And Sufficient

AbstractIt is shown that no proper ideal of a free inverse semigroup is free and that every isomorphism between ideals is induced by a unique automorphism of the whole semigroup. In addition, necessary and sufficient conditions are given for two principal ideals to be isomorhic.

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.

Sign in / Sign up

Export Citation Format

Share Document