Almost factorizable weakly ample semigroups
14/07/2007
Gracinda Maria dos Santos Gomes Moreira da Cunha
(CAUL)
Second International Congress in Algebras and Combinatorics, Xi'an University of Architecture and Technology, China
After recalling the concepts of "factorizable" and "almost factorizable" for inverse monoids and for inverse semigroups, respectively, we mention some well known results on these classes of semigroups, such as
a) $S$ is almost factorizable iff $S$ is an idempotent separating image of a semidirect product $G\ast Y$ of a semilattice $Y$ by a group $G$;
b) $S$ is isomorphic to some $G\ast Y$ iff $S$ is both $E$unitary and almost factorizable.
See for example [2].
We move on to present the correspondent theory for the wider classes of weakly ample monoids and weakly ample semigroups. The appropriate notions of "factorizable" and "almost (left) factorizable" are introduced, and the analogues of a) and b) are obtained. Here weakly ample semigroups are considered as special $(2,1,1)$algebras. In particular, we show that in b) the correspondent semidirect product $T\ast Y$, of a semilattice $Y$ by a unipotent monoid $T$ is no longer the convenient object. Here the rule of $G\ast Y$ is played by a particular subsemigroup $W(T,Y)$ of $Y\ast T$. This theory has been developed jointly with Mária B. Szendrei [4].
References:
[1] El Qallali, A. and Fountain, J.  Proper covers for left ample semigroups, Semigroup Forum 71 (2005), 411427.
[2] Lawson, M.  Inverse semigroups, World Scientific, 1998.
[3] Szendrei, Mária B.  Factorizability in certain classes over inverse semigroups, to appear in Semigroups and Formal Languages (Proceedings of the Conference Semigroups and Languages, Lisboa, Portugal, 1215 July 2005), World Scientific.
[4] Gomes, Gracinda M.S. and Szendrei, Mária B.  Almost factorizable weakly ample semigroups, to appear in Com. Algebra.
