On the endomorphism semigroup (and category) of bounded lattices
George Grätzer, Jiří Sichler · Pacific Journal of Mathematics · 1970
Recently, a large number of papers have been published on the representations of a semigroup with identity (i.e., a monoid) as the endomorphism semigroup of algebras and relational systems of various kinds.For instance Z. Hedrlίn and J. Lambek showed that every monoid can be represented as the endomorphism semigroup of a semigroup.Conspicuously missing from the list of algebras for which theorems of this type can be proved are lattices.The reason for this is that any constant map is a lattice endomorphism, and therefore endomorphism semigroups of lattices are very special.For partially ordered sets one can eliminate constant maps as endomorphisms by considering only those maps φ that satisfy x < y implies xφ < yφ (Z.Hedrlίn and R. H. McDowell); however, this would not be a very natural condition to impose on lattice endomorphisms.The approach of this paper 1 is to consider bounded lattices only, that is, lattices with smallest element 0 and largest element 1, and as endomorphisms to admit only those lattice endomorphisms that preserve 0 and 1 (i.e., keep 0 and 1 fixed; this amounts to considering 0 and 1 as nullary operations).Such endomorphisms are usually called {0, l}-endomorphisms but they are called simply endomorphisms in this paper.The first result is that every monoid is isomorphic to the monoid of all endomorphisms of a bounded lattice.One can also consider lattices with complementation , where ' is a complementation, that is, for every ae L, a A a' -0 and a v a' = 1.For such algebras an endomorphism is a lattice endomorphism φ that preserves ', that is, (aφ) f = a'ψ for all aeL.Every lattice with complementation is bounded, and any such endomorphism preserves 0 and 1.The second result is that every monoid is isomorphic to the endomorphism semigroup of a lattice with complementation.Both these results are consequences of much stronger theorems proved in this paper.