完全(α,β)-绝对纯幺半群

Completely(α,β)-absolutely pure monoid

完全右内射幺半群是一类具有重要研究价值的半群,完全α-绝对纯幺半群和完全右FC-内射(FSF-内射)幺半群是其两种不同的推广.通过引入(α,β)-绝对纯S-系的概念,将完全α-绝对纯幺半群和完全右FC-内射(FSF-内射)幺半群进一步推广为完全(α,β)-绝对纯幺半群,即所有S-系是(α,β)-绝对纯的幺半群.讨论了(α,β)-绝对纯S-系的性质,给出了完全(α,β)-绝对纯幺半群的理想-同余刻画,从而完全α-绝对纯幺半群和完全右FC-内射(FSF-内射)幺半群等的对应结论都可由此结果推出.

Completely right injective monoids are of great importance to the study of semigroups. Completely a-absolutely pure monoids and completely right FC-injective （FSF-injective） monoids are two different generalizations of completely right injective monoids. The concept of （α,β）-absolutely pure S-act is introduced, and then completely a-absolutely pure monoids and completely right FC-injective （FSF-injective） monoids are further extended to completely （α,β）-pure monoid, namely, all S-acts are （α,β）-absolutely pure. The properties of （α,β）-absolutely pure S-acts are investigated, and a characterization of completely （α,β）-pure monoid is presented in terms of right ideal and right congruence. Some conclusions about completely a-absolutely pure monoids and completely right FC-injective （FSF-injective） monoids are direct corollaries of the discussions.