Classes of residuated lattices. (English) Zbl 1119.03343

Ciungu, Lavinia Corina

An. Univ. Craiova, Ser. Mat. Inf. 33, 180-207 (2006).

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

Keywords:

residuated lattice; good residuated lattice; local residuated lattice; perfect residuated lattice; A