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Logic Journal of IGPL Advance Access first published online on September 25, 2007
This version published online on September 25, 2007
Logic Journal of IGPL, doi:10.1093/jigpal/jzm038
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Pure Hilbert algebras with infimum

Aldo Figallo, Jr

Instituto de Ciencias Básicas, Universidad Nacional de San Juan & Departamento de Matemática, Universidad Nacional del Sur. E-mail: alfiga{at}uns.edu.ar


  Abstract

In [6], iH-algebras were introduced in order to indicate an equational version of the class of Hilbert algebras where each pair of elements has infimum. These authors also proved that this variety has the class of Curry's implicative semilattices ([9]) as a proper subvariety. On the other hand, in [4] a special class of Hilbert algebras associated with ordered sets, which they called order algebras, were investigated. These algebras were also studied in [1] under the name of pure Hilbert algebras.

Bearing in mind the above results, in this paper we introduce the notion of pure Hilbert algebras with infimum (or ipH-algebras, for short). Furthermore, we characterize the lattice of ipH-congruences and we determine the subdirectly irreducible ipH-algebras. Besides, we prove that subdirectly irreducible ipH-algebras are also subdirectly irreducible iH-algebras.

Received for publication 12 October 2006.
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