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

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.

Free nilpotent minimum algebras

“…In the terminology of [17] and [18], the previous corollary is asserting that directly indecomposable NM-algebras without a fixpoint are perfect IMTL-algebras and directly indecomposable NM-algebras with fixpoint are perfect IMTL-algebras plus fixpoint.…”

States on finite linearly ordered IMTL-algebras

“…Definition 2.5 (Noguera et al 2005). Let L be an IMTLalgebra, x 2 L. The order of x is the least positive integer m such that x m ¼ 0, denoted by ordðxÞ.…”

On triangular norm based axiomatic extensions of the weak nilpotent minimum logic

“…On the one hand, it is clear that for every finite WNM-chain C, if, and only if, C has no negation fixpoint. On the other hand, in the papers [22,23], while studying the varieties generated by perfect IMTL-algebras and perfect MTL-algebras, the authors proved that a WNM-chain is perfect if, and only if, it has no negation fixpoint. Thus, the equation for perfect MTL-chains will be enough to obtain an equational base for the variety we are considering now: (¬(¬x) 2 ) 2 ≈ ¬(¬x 2 ) 2 .…”