Homomorphisms and endomorphisms in varieties of pseudocomplemented distributive lattices (with applications to Heyting algebras)

Abstract

According to a result by K. B. Lee, the lattice of varieties of pseudocomplemented distributive lattices is the ω + 1 \omega + 1 chain B − 1 ⊂ B 0 ⊂ B 1 ⊂ ⋯ ⊂ B n ⊂ ⋯ ⊂ B ω {B_{ - 1}} \subset {B_0} \subset {B_1} \subset \cdots \subset {B_n} \subset \cdots \subset {B_\omega } in which the first three varieties are formed by trivial, Boolean, and Stone algebras respectively. In the present paper it is shown that any Stone algebra is determined within B 1 {B_1} by its endomorphism monoid, and that there are at most two nonisomorphic algebras in B 2 {B_2} with isomorphic monoids of endomorphisms; the pairs of such algebras are fully characterized both structurally and in terms of their common endomorphism monoid. All varieties containing B 3 {B_3} are shown to be almost universal. In particular, for any infinite cardinal κ \kappa there are 2 κ {2^\kappa } nonisomorphic algebras of cardinality κ \kappa in B 3 {B_3} with isomorphic endomorphism monoids. The variety of Heyting algebras is also almost universal, and the maximal possible number of nonisomorphic Heyting algebras of any infinite cardinality with isomorphic endomorphism monoids is obtained.