A Homomorphism Theorem for Semigroups
D. B. McAlister · Journal of the London Mathematical Society · 1968
If S = S ° is a semigroup which has a O-restricted completely 0-simple homomorphic image then it is known that S need not have a maximal completely 0-simple homo-morphic image. The main theorem of this paper shows that although this is true for homomorphisms onto it is not the case when we consider homomorphisms into. Specifically, we have the following result. (Theorem 1.5). Let and a O-restricted homomorphism Y \\ ofS into #(S) such that thefollowing are satisfied. (i) If 9 is a O-restricted homomorphism of S into a member Toftf then there exists a O-restricted homomorphism (\\> of^(S) into T such that the diagram commutes. (ii) The homorphism <\\) in (i) is unique with this property. The method of proof of Theorem 1.5 is to give an explicit construction for and an entirely analogous construction can be used to show that if # is the class of all semigroups having some set of implicational and existential properties, the analog of (i) holds. (Theorem 3.1.) In particular this is true if # is (a) the class of all simple semigroups; (b) the class of all regular semigroups; (c) the class of all regular bisimple semigroups. In general, and in these three cases in particular, (ii) does not hold. However, if # is one of the following classes, both (i) and (ii) are satisfied. (d) the class of all inverse semigroups; (e) the class of all semigroups which are unions of groups; (f) the class of all semilattices of groups; (g) the class of all completely simple semigroups; (h) the class of all groups. Thus in these cases #(S) is the co-reflection of S in the category of members of # and their homomorphisms and hence is unique up to isomorphism.