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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,a∧a′ = 0 and a∨a′ = 1. For such algebras
an endomorphism is a lattice endomorphism φ that preserves ′, that is, (aφ)′ = 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.
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