Locally factorizable transformation semigroups

Abstract

By the local subsemigroups of a semigroup S we mean the subsemigroups of S in the form eSe where e is an idempotent of S. A semigroup S is said to be factorizable if there exists a subgroup G of S such that S = GE(S) where E(S) is the set of all idempotents of S. A semigroup in which each local subsemigroup is factorizable is called a locally factorizable semigroup. Let X be a set. For a partial transformation α of X, the shift of α is defined to be the set S(α) = { X Ɛ Δα l Xα ≠ X} where Δα is the domain of α. A partial transformation α of X is said to be almost identical if and only if it had a finite Shift. In this thesis, we characterize locally factorizable transformation semigroups as follcws : THEOREM. The partial transformation semigroup on a set X is locally factorizable if and only if X is finite. COROLLARY. Let X be a set and let S be the full transformation semigroup on X or the symmetric inverse semigroup on X (the 1-1 partial transformation semigroup on X). Then the transformation semigroup S is locally factorizable if and if X is finite. THEOREM. For any set X, the semigroup of all almost identical partial transformations of X is finite. COROLLARY. For any set X, the semigroup of all almost identical transformations of X and the semigroup of all almost identical 1-1 partial transformations of X are locally factorizable. THEOREM. For any positive integer n and for any field F, the multiplicative semigroup of all nxn matrices over F is locally factorizable.