Vol. 35, No. 3, 1970

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On the endomorphism semigroup (and category) of bounded lattices

George Grätzer and J. Sichler

Vol. 35 (1970), No. 3, 639–647

Recently, a large number of papers have been published on the representations of a semigroup with identity (i.e., a momoid) as the endomorphism semigroup of algebras and relational systems of various kinds. For instance Z. Hedrlín 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. Hedrlin and R. H. McDowell); however, this would not be a very natural condition to impose on lattice endomorphisms.

The approach of this paperl 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 {0,1}-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 L;,,′⟩, where is a complementation, that is, for every a L,aa= 0 and aa= 1. For such algebras an endomorphism is a lattice endomorphism φ that preserves , that is, ()= afφ for all a L. 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.

Mathematical Subject Classification
Primary: 06.30
Secondary: 20.00
Received: 21 April 1970
Published: 1 December 1970
George Grätzer
J. Sichler