Generalized semigroups of quotients

Abstract

For S a semigroup with 0 and M S {M_S} a right S-set, certain classes of sub S-sets called right quotient filters are defined. A study of these right quotient filters is made and examples are given including the classes of intersection large and dense sub S-sets respectively. The general semigroup of right quotients Q corresponding to a right quotient filter on a semigroup S is developed and basic properties of this semigroup are noted. A nonzero regular semigroup S is called primitive dependent if each nonzero right ideal of S contains a 0-minimal right ideal of S. The theory developed in the paper enables us to characterize all primitive dependent semigroups having singular congruence the identity in terms of subdirect products of column monomial matrix semigroups over groups.