We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.For these three classes of semigroups, type-II is equal to type-II construct.
Inverse semigroups and varieties of finite semigroups
