Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops |
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Authors: | Carles Noguera Francesc Esteva Joan Gispert |
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Institution: | (1) Research Institute in Artificial Intelligence, Spanish Council for Scientific Research,;(2) Department of Logic, University of Barcelona, |
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Abstract: | IMTL logic was introduced in 12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in 11] it was proved
to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is
still not known. Nevertheless, Jenei introduced in 20] a new way to obtain rotation-invariant semigroups and, in particular,
IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order
to give an algebraic interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra
used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly
those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated
by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational
axiomatizations for it. |
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Keywords: | Algebraizable logics Bipartite algebras Cancellative hoops disconnected rotation filters IMTL-algebras local algebras many-valued logic MV-algebras perfect algebras prelinear semihoops Wajsberg hoops |
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