Abstract
This chapter discusses the infinite image homomorphisms of distributive bounded lattices. The first result concerning the universality of varieties of lattices is because of Birkhoff. He showed that each group is isomorphic to the automorphism group of a distributive lattice. A result of Schein is in contrast with this fact—he proved that distributive lattices are determined up to antiisomorphism by their endomorphism monoids. This result was generalized by McKenzie and Tsinakis for bounded distributive lattices and (0,1)-endomorphism monoids. As these monoids contain left zeros, Birkoff's theorem cannot be generalized to monoids. Infinite (0,1)-endomorphisms of bounded distributive lattices do not fulfil an analogous theorem. The chapter presents a theorem that states that the category T of all totally order disconnected compact spaces together with all continuous order preserving mappings is dually isomorphic to the variety BD (= BDo) of all bounded distributive lattices.
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