SEMIGROUPS CONTAINING COYERED ONE-SIDED IDEALS

In [3] a notion of a covered ideal was introduced. The aim of the present paper is to show some other properties and the mutual relation betwen covered ideals and bases of semigroups. Definition 1. A proper left ideal L of S is called a covered left ideal (briefly a CL-ideal) if LczS(S-L). Analogously a covered right ideal (CR-ideal) is defined. The case of two-sided ideals will be treated later. Clearly if S contains a zero element 0 and card | S | ^ 2, then 0 is a CL-ideal. Note that by definition S itself is not a CL-ideal. Lemma 1. If S contains two different left ideals Lx and L2 such that L{\JL2 = S, then none of the ideals Lu L2 is a CL-ideal. Proof. If L tuL2 = S, then S-L2czL,, and S-L,c=L2. Now L,czS(S-L1) mplies LxczSL2czL2 and L2czS(S-L2) implies L2czSLxczLu hence Li = L2 a:ontradiction. Corollary. If S contains more than one maximal left ideal, then none of them is a CL-ideal of S. If L is a left ideal of S and L^Sa, then L is certainly a CL-ideal. (For, in thi