Semigroups with Finiteness Conditions

A semigroup S is called locally finite if all its finitely generated subsemigroups are finite. Clearly, any locally finite semigroup is periodic, but finitely generated infinite periodic semigroups abound. The authors start by showing that the class of locally finite semigroups is closed under ideal extensions. They extend the assertion of Lemma 3 to a wider class of semigroups. Finally, they consider the condition M_R for the class of nil semigroups. The semigroups characterized by the equivalent conditions of Lemma 10 are called left T-nilpotent. The analog of Lemma 10 holds for right T-nilpotent semigroups defined symmetrically. Finally, the authors note that Proposition 13 can be applied to multiplicative subsemigroups of right noetherian rings because the a.c.c. on annihilator ideals is hereditary on subsemigroups.