昇降鎖律をみたす正則半群についての諸注意

The class of semigroups satisfying the descending chain conditions on principal (left, right) ideals has been studied by Green [2], Munn [4] and some other papers. In this paper, we shall give a survey on the structure of the class of regular semigroups satisfying chain conditions.
Firstly, several kinds of both ascending chain conditions (a.c.c. for short) and descending chain conditions (d.c.c. for short) on (left, right) ideals or on Green's relations L, R, J are introduced, and the regular semigroups with O, each of which satisfies some of these chain conditions, are studied. Secondly, the concepts of "L(R, or J)-bounded" and "locally L(R, or J)-bounded" are also introduced, and the connection between these properties and the chain conditions are discussed.