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Logic Journal of IGPL Advance Access published online on September 26, 2007
Logic Journal of IGPL, doi:10.1093/jigpal/jzm039
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Monadic distributive lattices

Aldo V. Figallo 1, Inés Pascual and Alicia Ziliani

Departamento de Matemática, Universidad Nacional del Sur, 8000 Bahía Blanca, Argentina & Instituto de Ciencias Básicas, Universidad Nacional de San Juan, 5400 San Juan, Argentina. E-mails: avfigallo{at}gmail.com, aziliani{at}criba.edu.ar, inespascual756{at}hotmail.com


  Abstract

The purpose of this paper is to investigate the variety of algebras, which we call monadic distributive lattices, as a natural generalization of monadic Heyting algebras [16]. It is worth mentioning that the latter is a proper subvariety of the first one, as it is shown in a simple example. Our main interest is the characterization of simple and subdirectly irreducible monadic distributive lattices. In order to do this, a duality theory for these algebras is developed. The duality enables us to describe the lattice of congruences on monadic distributive lattices. Finally, our attention is focused upon the relationship between the category of dual spaces associatted with these algebras and the category of perfect Ono frames considered by Bezhanishvili in order to represent monadic Heyting algebras.

Key Words: Bounded distributive lattices • Priestley spaces • congruence relations • subdirectly irreducible algebras

1This work was partially supported by the Universidad Nacional del Sur, Bahía Blanca, Argentina.


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