Atlas: Algebraic Gentzen systems applied to varieties of residuated lattices by Peter Jipsen

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Conference on Ordered Algebraic Structures
March 7-9, 2002
Vanderbilt University
Nashville, TN, USA

Organizers
Peter Jipsen, Constantine Tsinakis

Algebraic Gentzen systems applied to varieties of residuated lattices
by
Peter Jipsen
Vanderbilt University
Coauthors: Hiroakira Ono (Japan Advanced Institute of Science and Technology)

Gentzen systems have been used very effectively in logic to prove completeness, decidability, interpolation and disjunction properties for various classical and non-classical logics. Via algebraic logic, these results translate to interesting algebraic properties for the corresponding varieties of algebras. But it is also useful to translate Gentzen systems into an algebraic framework so that these methods may be applied directly to patially ordered algebras.

We present an algebraic version of Gentzen systems as a set of quasi-inequalities. Applying these ideas to residuated lattices, we show how they imply interpolation and the disjunction property for various subvarieties as well as the variety of all residuated lattices. From the disjunction property it follows that these subvarieties are join-irreducible in the lattice of all subvarieties, and since they are generated by their finite members, one can show that they are not completely join-irreducible. We also discuss connections between interpolation and the amalgamation property.

Date received: January 31, 2002