Pseudo-MTL algebras or weak pseudo-BL algebras are non-commutative fuzzy structures which arise from pseudo-t-norms, namely, pseudo-BL algebras without the pseudo-divisibility condition. The aim of this paper is to investigate the properties of pseudo-BL algebras that also hold for pseudoMTL algebras. We will also study some classes of pseudo-MTL algebras such as good, local and Archimedean pse...

The study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i. e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras, B...

The aim of this work is to introduce the concepts of IMTL-filters and strong MTL-filters in MTL-algebras and show that these filters are related to IMTL-algebras and strong MTL-algebras, respectively. We also introduce the concepts of EIMTL and associative filters in MTL-algebras. The relationships between these filters and quotient algebras that are constructed via these filters are described....

we consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice $x$, $(x, forall)$ is a residuated lattice with a quantifier if and only if there is an $m$-relatively complete substructure of $x$. we also show that, for a strong residuated lattice $x$, $bigcap {p_{lambda} ,|,p_{lambda} {rm is an} m{rm -filter} } = {1}$ and hence that any strong re...

This paper introduces the implicative derivations and gives some of their characterizations on MTL-algebras. Furthermore, we provide representation MTL-algebras by obtain Boolean algebra via all derivations. Finally, explore relationship between derivation other operators show that there exists a bijection sets multiplier IMTL-algebras. The results this can common properties in t-norm-based fuz...

The class of all MTL-algebras is a variety, denoted MTL. Alternatively, an MTL-algebra is a representable, commutative, integral residuated lattice with a least element.

Discrete dualities are developed for n-potent MTL–algebras and for 2-potent BL–algebras. That is, classes of frames, or relational systems, are defined that serve as dual counterparts to these classes of algebras. The frames defined here are extensions of the frames that were developed for MTL– algebras in [25], [26]; the additional frame conditions required are given here and also the proofs t...

We consider properties of residuated lattices with universal quantifier and show that, for a residuated lattice $X$, $(X, forall)$ is a residuated lattice with a quantifier if and only if there is an $m$-relatively complete substructure of $X$. We also show that, for a strong residuated lattice $X$, $bigcap {P_{lambda} ,|,P_{lambda} {rm is an} m{rm -filter} } = {1}$ and hence that any strong re...

The problem of characterization of the structure of MTL-algebras, which form an equivalent algebraic semantics for Monoidal T-norm Based Logic (see [9]) in the sense of Blok and Pigozzi (see [3]), is still far from being solved. Since MTL-algebras are in fact subdirect products of chains, it suffices to investigate only the structure of MTL-chains if we want to characterize the structure of MTL...

In order to research the logical system whose propositional value is given in a lattice from the semantic viewpoint, Xu [7] proposed the concept of lattice implication algebras, and discussed some of their properties. Xu and Qin [8] introduced the notion of implicative filters in a lattice implication algebra, and investigated some of their properties. Turunen [5] introduced the notion of Boole...