Semigroups with Ascending Chain Condition
M. Satyanarayana · Journal of the London Mathematical Society · 1972
A semigroup S is said to satisfy the ascending chain condition (a.c.c.) on right ideals if every chain of right ideals under set-inclusion terminates at a finite stage. The class of semigroups with a.c.c. on right ideals includes the class of finite semigroups. In this paper we discuss some of the properties enjoyed by semigroups with a.c.c. and in particular those properties which the semigroups with a.c.c. share with semi-groups which are finite. It is well known that a 0-simple semigroup has no zero-divisors if and only if it has no proper nilpotent elements [1; 71]. Naturally one may ask what type of 0-simple semigroup has no zero-divisors. We prove in §1 that a class of semigroups satisfying a.c.c. has no zero divisors. The property mentioned above for 0-simple semigroups can also be proved for non-O-simple semigroups in which every right ideal is principal. Right simple semigroups and simple semigroups may or may not have idempotents, but if they are finite they have idempotents. We prove in §2 that certain types of right simple and simple semigroups with a.c.c. have idempotents. Finally, we prove that right simple semigroups in which every left ideal is principal are groups.