On a Characterization of Closure Operators by Identities on Semigroups

For a semigroup S it is well known, that the operator C on the power set P(S) of S, which assigns to every U ∈ P(S) the subsemigroup C(U) of S generated by U, is a closure operator (cf. [2], [4]). Let T ∈ P(S). Then U is called to be a generating set of T with respect to C if and only if U ⫅ T and T ⫅ C(U). Let GEN(T) be the system of all generating sets of T. To give a more detailed description of the generating sets of T the so-called isolated sets of T are considered. I is called to be isolated in T if and only if Ø ≠I ⫅ T and I ∩C(T\I) = Ø. Let ISO(T) be the system of all I such that I is isolated in T. Then there holds