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Classes of residuated lattices. (English) Zbl 1119.03343

Summary: The commutative residuated lattices were first introduced by M. Ward and R. P. Dilworth as generalization of ideal lattices of rings. Non-commutative residuated lattices, called sometimes pseudo-residuated lattices, biresiduated lattices or generalized residuated lattices are algebraic counterpart of substructural logics, that is, logics which lack some of the three structural rules, namely contraction, weakening and exchange. Complete studies on residuated lattices were developed by H. Ono, T. Kowalski, P. Jipsen and C. Tsinakis. The aim of this paper is to study some special classes of residuated lattices, such as local, perfect and Archimedean residuated lattices. As an important result of the paper we prove that, generally, the Archimedean residuated lattices are not commutative. Additionally, we study some properties of the lattice of filters of residuated lattice.

MSC:

03G10 Logical aspects of lattices and related structures
03G25 Other algebras related to logic
06D35 MV-algebras
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