On Recovering a Bounded Distributive Lattice from Its Endomorphism Monoid

The sub algebra functor Sub A is faithful for Boolean algebras (Sub A •Sub B implies A • B, see D. Sachs [7] ), but it is not faithful for bounded distributive lattices or unbounded distributive lattices. The automorphism functor Aut A is highly unfaithful even for Boolean algebras. The endomorphism functor End A is the most faithful of all three. B. M. Schein [8] and K. D.' Magill [5] established its faithfulness for Boolean algebras. See also C. J. Maxson [6]. In addition Schein [8] established the faithfulness of the endomorphism functor for lower bounded distributive lattices and upper bounded distributive lattices. He further showed that if A and B are any unbounded distributive lattices then any isomorphism 1⁄2: End A-• End B is induced by a unique isomorphism or dual isomorphism between A and B. H. J. Bandelt [1 ] established the corresponding result for median algebras. See also Bandelt [2]. The aforementioned results leave open the question whether an analogous result holds for bounded distributive lattices. It is the purpose of this paper to give an affirmative answer to this question. We prove: THEOREM. Let A and B be two bounded distributive lattices each of which has at least two elements. If 1⁄2: End A • End B is an isomorphism, then there exists a unique isomorphism or dual-isomorphism •r: A -• B such that (*) 1⁄2(g) =rr.g-rr -1for all g C End A.