Atlas: Embedding Locally Inverse Semigroups into Rees Matrix Semigroups by Bernd Billhardt

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Colloquium on Semigroups
July 17-21, 2000
University of Szeged, Bolyai Institute
Szeged, Hungary

Organizers
Mária B. Szendrei, Eszter K. Horváth, István Szittyai, Géza Takách

Embedding Locally Inverse Semigroups into Rees Matrix Semigroups
by
Bernd Billhardt
University of Kassel, Kassel, Germany

Let S be a regular semigroup whose set of idempotents is denoted by E_S. S is called a locally inverse semigroup if eSe is an inverse subsemigroup of S for all e in E_S. If in addition E_S is a subsemigroup of S, then S is called a generalized inverse semigroup. We consider locally inverse semigroups S, satisfying the following condition:

(E): (i) S is a rectangular band I ×\Lambda of semigroups S_i\lambda,

: (ii) for all (i, \lambda) in I×\Lambda, e, f in S_{i \lambda} \cap E_S, and s, t in S, the equality seft = sfet holds.

Each straight locally inverse semigroup [2], i.e. locally inverse semigroup whose maximal subsemilattices are disjoint, and each weakly E-unitary locally inverse semigroup [1], i.e. locally inverse semigroup whose least completely simple congruence is idempotent pure, satisfies E.

It is shown that a locally inverse semigroup S satisfies condition (E) if and only if it is embeddable into a Rees matrix semigroup over a generalized inverse semigroup. This generalizes a result proven in [2], which states that each straight locally inverse semigroup is embeddable into a Rees matrix semigroup over an inverse semigroup.

References

[1]: B. Billhardt and M.B. Szendrei, Weakly E-unitary locally inverse semigroups, submitted.
[2]: F. Pastijn and M. Petrich, Straight locally inverse semigroups, Proc. London Math. Soc. (3) 49 (1984), 307 - 328.

Date received: March 27, 2000