Glrac Semigroups

Left restriction semigroups and left GC-lpp semigroups have been attracting their due attentions.The aim of this paper is to study a class of generalized regular semigroups,so-called glrac semigroups.Such semigroups are a common generalization of left restriction semigroups and left GC-lpp semigroups.We introduce the notion of -product and establish the -product construction theorem for glrac semigroups and of proper glrac semigroups.In particular,the construction of free glrac semigroups is established.It is pointed out that any free glrac semigroup is proper.It is proved that any glrac semigroup has proper -covers.We obtain the construction of proper S-covers for glrac semigroups.As their applications,we establish new constructions of left GC-lpp semigroups,left inverse semigroups,proper left GC-lpp semigroups,and proper left inverse semigroups.