ON THE LATTICE OF FULL EVENTUALLY REGULAR SUBSEMIGROUPS
Peter R. Jones, Zihong Tian, Z. B. Xu · Communications in Algebra · 2005
We generalize to eventually regular (or ‘π-regular’) semigroups the study of the lattice of full regular subsemigroups of a regular semigroup, which has its most complete exposition in the case of inverse semigroups. By means of a judicious definition, it is shown that the full eventually regular subsemigroups of such a semigroup form a complete lattice LF, which projects onto the lattices of full regular subsemigroups of its regular principal factors. Our deepest results are obtained for those eventually regular semigroups in which the regular elements form a subsemigroup. In that case, LF also projects onto the lattice of full regular subsemigroups of that regular subsemigroup. In particular, we characterize such semigroups for which LF is distributive. A much more explicit description is obtained for the eventually regular semigroups in which the idempotents commute.