Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Let Ω Ω be the semigroup of all mappings of a countably infinite set Ω . If U and V are subsemigroups of Ω Ω , then we write U ≈ V if there exists a finite subset F of Ω Ω such that the subsemigroup generated by U and F equals that generated by V and F . The relative rank of U in Ω Ω is the least cardinality of a subset A of Ω Ω such that the union of U and A generates Ω Ω . In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω . migroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈ . Moreover such semigroups have relative rank 0 , 1 , 2 , or d in Ω Ω where d is the minimum cardinality of a dominating family for N N . We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in Ω Ω are 0 , 1 , 2 , and d . w that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2 ℵ 0 .