一类拟正则半群的性质和特征

Properties and characterizations of a class of quasi-regular semigroups

定义了一类拟正则半群,即拟右半群.利用拟正则半群和左中心幂等元的性质,证明了S为拟右半群时,(1)S为拟完全正则半群;(2)RegS为完全正则半群;(3)R*为S上的最小半格同余;(4)RegS上的每个R-类Tα为右群;(5)TαGα×Eα,其中Gα为群,Eα为右零半群.在此基础上得到了3个等价命题:若S为具有左中心幂等元半群,则(1)S为拟右半群;(2)S为拟完全正则的,RegS为S的理想;(3)S为右群强半格的诣零理想扩张.

A class of quasi-regular semigroup is defined, namely quasi-right semigroup. By using properties of quasi-regular semigroups and left central idempotents, some statements are proved. Let S be a quasi-right semigroup, then （1） S is a quasi-completely regular semigroup; （2） RegS is a completely regular semigroup; （3） R^＊ is the smallest semilattice congruence on S; （4） Each R-class Ta on RegS is a right group; （5） Ta≌Ga×Ea, where Ga is a group,Ea is a right zero semigroup. On the basis, three equivalent statements are obtained. Let S be a semigroup with left central idempotents, then （1） S is a quasi-right semigroup; （2） S is a quasi-completely regular,and RegS is an ideal; （3） S is a nil-extension of strong semilattice of right semigroup.