Maximal cancellative subsemigroups and cancellative congruences

Abstract

A subsemigroup T T of a commutative semigroup S S is called a mild ideal if for any a ∈ S , a T ∩ T ≠ ϕ a \in S,aT \cap T \ne \phi . It is shown here that any maximal cancellative subsemigroup T T of a commutative, idempotent-free, archimedean semigroup S S must be a mild ideal of S S . Maximal cancellative subsemigroups exist in abundance due to Zorn’s lemma. It is also shown that if T T is mild ideal of a commutative semigroup S S , then every cancellative congruence of T T has a unique extension to a cancellative congruence of S S .