Journal article
A factorization of dual prehomomorphisms and expansions of inverse semigroups
Publication Details
Authors: | Billhardt, B. |
Publication year: | 2004 |
Journal: | Studia Scientiarum Mathematicarum Hungarica |
Pages range : | 295-308 |
Volume number: | 41 |
ISSN: | 0081-6906 |
Abstract
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 iota(S) : S --> S(S) such that for each dual prehomomorphism 0 from S into an inverse semigroup T there exists a unique homomorphism theta* : S(S) --> T with iota(S)theta* = theta. If we restrict the class of dual prehomomorphisms under consideration to order preserving ones, S(S) may be replaced by a certain homomorphic image (S) over cap (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 (S) over cap (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.
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 iota(S) : S --> S(S) such that for each dual prehomomorphism 0 from S into an inverse semigroup T there exists a unique homomorphism theta* : S(S) --> T with iota(S)theta* = theta. If we restrict the class of dual prehomomorphisms under consideration to order preserving ones, S(S) may be replaced by a certain homomorphic image (S) over cap (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 (S) over cap (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.
