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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Authors: M. E. Adams, V. Koubek and J. Sichler
Journal: Trans. Amer. Math. Soc. 285 (1984), 57-79
MSC: Primary 06D15; Secondary 06D20
DOI: https://doi.org/10.1090/S0002-9947-1984-0748830-6
MathSciNet review: 748830
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Abstract: According to a result by K. B. Lee, the lattice of varieties of pseudocomplemented distributive lattices is the $ \omega + 1$ chain $ {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}$ by its endomorphism monoid, and that there are at most two nonisomorphic algebras in $ {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}$ are shown to be almost universal. In particular, for any infinite cardinal $ \kappa $ there are $ {2^\kappa }$ nonisomorphic algebras of cardinality $ \kappa $ in $ {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.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0748830-6
Article copyright: © Copyright 1984 American Mathematical Society