Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops

• DOI: 10.1007/s00153-005-0276-0
• Publication Date: 2005-05-02T06:41:48Z
• Authors: Carles Noguera; Francesc Esteva; Joan Gispert
• Keywords: Perfect algebra; Cancellative hoops; 512; MV-algebra; 02 engineering and technology; Disconnected rotation; Bipartite algebra; Prelinear semihoops; 0202 electrical engineering, electronic engineering, information engineering; Algebraizable logic; Local algebras; Prelinear semihoop; MV-algebras; Perfect algebras; Filter; IMTL-algebra; Bipartite algebras; Filters; Algebraizable logics; IMTL-algebras; Wajsberg hoops; Many-valued logic; Local algebra; Algebraizable logics; Bipartite algebras; Cancellative hoops; Disconnected rotation; Filters; IMTL-algebras; Local algebras; Many-valued logic; MV-algebras; Perfect algebras; Prelinear semihoops; Wajsberg hoops; Cancellative hoop

Peer Reviewed<br/>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. © Springer-Verlag 2005.<br/>The authors acknowledge partial support of the Spanish projects TIN2004-07933-C03-01, TIN2004-07933-C03-02 and MTM 2004-03102 and the Catalan project 2001SGR-0017 of DGR.<br/>