On S-acts containing no maximal S-subacts

摘要

Let S be a semigroup and let A be an S-act. Some necessary and sufficient conditions that S-subacts of A are maximal S-subacts are given. A relation L which is similar to the Green relation in semigroups is defined. By the relation L, it is proved that a non-empty set L of A is a maximal S-subact if and only if AL is a (maximal) L-class. Finally, the concept of a C-subact is defined, some properties of C-subacts are discussed, and it is proved that A contains no maximal S-subacts if and only if every cyclic S-subact of A is a C-subact. Consequently, the results obtained by Imrich Fabrici that semigroups contain no maximal (left) ideals are the corollary of this paper.