We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x∨¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To pr...

The notion of a state is an analogue of a probability measure and was first introduced by Kôpka and Chovanec for MV-algebras and by Riec̆an for BLalgebras. The states have also been studied for different types of non-commutative fuzzy structures such as pseudo-MV algebras, pseudo-BL algebras, bounded R`monoids, residuated lattices and pseudo-BCK semilattices. In this paper we investigate the sta...

Nonassociative Lambek Calculus (NL) is a pure logic of residuation, involving one binary operation (product) and its two residual operations defined on a poset [26]. Generalized Lambek Calculus GL involves a finite number of basic operations (with an arbitrary number of arguments) and their residual operations [7]. In this paper we study a further generalization of GL which admits operations wh...

We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-∗. An investigation of congruence properties (epermutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Fin...

Bosbach states represent a way of probabilisticly evaluating the formulas from various (commutative or non-commutative) many-valued logics. They are defined on the algebras corresponding to these logics with values in [0, 1]. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of [0, 1], in this paper we introduce Bosbach sta...

The main ‘philosophical’ outcome of this article is to demonstrate that the structural description of residuated lattices requires the use of the co-residuated setting. A construction, called skew symmetrization, which generalizes the well-known representation of an ordered Abelian group obtained from the positive (or negative) cone of the algebra is introduced here. Its definition requires lea...

An important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic, is that of a filter. Filters can be used, amongst others, to define congruence relations. Specific kinds of filters include Boolean filters and prime filters. In this paper, we define several different filters of residuated lattices and triangle algebras and examine their mutual...

In this note we give a characterization of meet-projections in simple atomistic lattices that generalizes results on the aggregation of partitions in cluster analysis. Documents de Travail du Centre d'Economie de la Sorbonne 2010.58 1 AGGREGATION and RESIDUATION Bruno LECLERC 1 and Bernard MONJARDET 2 Abstract In this note we give a characterization of meet-projections in simple atomistic latti...

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation pro...