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.

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.

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 Matrix Representations of Inverse Semigroups

1969 ◽  
Vol 9 (1-2) ◽  
pp. 29-61 ◽  
Author(s):  
G. B. Preston
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Irreducible Matrix ◽  
Matrix Representations ◽  
Alternative Account ◽  
Necessary And Sufficient ◽  
Brandt Semigroups ◽  
Complete Account

In his paper [1], W. D. Munn determines the irreducible matrix representations of an arbitrary inverse semigroup. Munn also gives a necessary and sufficient condition upon a 0-simple inverse semigroup for it to have a non-trivial matirx representation and for such semigroups gives a complete account of their representations. Munn's results rest upon the earlier work of Clifford [2] in which the representations of Brandt semigroups were determined. An alternative account of such representations was given by Munn in [3]. This earlier work is presented in Sections 5.2 and 5.4 of [4].

XII.—The Lattice of Congruences on a Bisimple ω-Semigroup

1967 ◽  
Vol 67 (3) ◽  
pp. 175-184
Author(s):  
W. D. Munn
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Natural Numbers ◽  
Necessary And Sufficient ◽  
Lattice Of Congruences

SynopsisA necessary and sufficient condition is determined for the modularity of the lattice of congruences on a bisimple inverse semigroup whose semilattice of idempotents is order-anti-isomorphic to the set of natural numbers.

scholarly journals On finite generation and presentability of Schützenberger products

2007 ◽  
Vol 83 (3) ◽  
pp. 357-368 ◽  
Author(s):  
Peter Gallagher ◽  
Nik Ruškucs
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Generation ◽  
Necessary And Sufficient ◽  
Finitely Presented

AbstractThe finite generation and presentation of Schützenberger products of semigroups are investigated. A general necessary and sufficient condition is established for finite generation. The Schützenberger product of two groups is finitely presented as an inverse semigroup if and only if the groups are finitely presented, but is not finitely presented as a semigroup unless both groups are finite.

scholarly journals ON GENERATORS AND PRESENTATIONS OF SEMIDIRECT PRODUCTS IN INVERSE SEMIGROUPS

2009 ◽  
Vol 79 (3) ◽  
pp. 353-365 ◽  
Author(s):  
E. R. DOMBI ◽  
N. RUŠKUC
Keyword(s):  
Semidirect Product ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Semidirect Products ◽  
Necessary And Sufficient ◽  
Finitely Presented

AbstractIn this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.

EMBEDDING THEOREMS FOR GROUPS PRESENTED VIA PARTIAL AUTOMORPHISMS: A GENERALIZATION OF SEMIDIRECT PRODUCTS AND HNN EXTENSIONS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 51-84 ◽  
Author(s):  
AKIHIRO YAMAMURA
Keyword(s):  
Inverse Semigroups ◽  
Embedding Problem ◽  
Sufficient Condition ◽  
Embedding Theorems ◽  
Necessary And Sufficient Condition ◽  
Semidirect Products ◽  
Geometric Methods ◽  
Hnn Extensions ◽  
Van Kampen Diagrams ◽  
Necessary And Sufficient

We examine a certain embedding problem for groups that have a presentation described by partial automorphisms. Semidirect products and HNN extensions have such a presentation. The embedding problem is closely related to systems of partial automorphisms, which are formalized by the concept of inverse semigroups. A necessary and sufficient condition for a group to be embedded in a certain sense is obtained by geometric methods using van Kampen diagrams.

scholarly journals Naturally ordered regular semigroups with maximum inverses

1989 ◽  
Vol 32 (1) ◽  
pp. 33-39 ◽  
Author(s):  
Tatsuhiko Saito
Keyword(s):  
Regular Semigroup ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inverse Transversal ◽  
Regular Semigroups ◽  
Inverse Subsemigroup ◽  
Necessary And Sufficient

Let S be a regular semigroup. An inverse subsemigroup S° of S is called an inverse transversal if S° contains a unique inverse of each element of S. An inverse transversal S° of S is called multiplicative if x°xyy° is an idempotent of S° for every x, y∈S, where x° denotes the unique inverse of x∈S in S°. In Section 1, we obtain a necessary and sufficient condition in order for inverse transversals to be multiplicative.

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