Abstract. This paper presents an algebraic approach of some many-valued generalizations of modal logic. The starting point is the denition of the [0, 1]-valued Kripke models, where [0,1] denotes the well known MV-algebra.Two types of structures are used to dene validity of formulas: the class of frames and the class of n-valued frames. The latter structures are frames in which we specify in each world u the set (a subalgebra of n) of the allowed truth values of the formulas in u.We apply and develop algebraic tools (namely, canonical and strong canonical extensions) to generate complete modal n + 1-valued logics and we obtain many-valued counterparts of Shalqvist canonicity result.

We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median algebras and certain topological structures, we obtain descriptions of the median-preserving mappings between products of finitely many chains. Introduction and preliminariesIn this paper we are interested in certain algebraic structures called median algebras.A median algebra is a ternary algebra A = A, m that satisfies the following equations m(x, x, y) = x, m(x, y, z) = m(y, x, z) = m(y, z, x), m(m(x, y, z), t, u) = m(x, m(y, t, u), m(z, t, u)).Median algebras have been investigated by several authors (see [3,9] for early references on median algebras and see [2,10] for some surveys). For instance, it is shown in [14] that for each element a of a median algebra A, the relation ≤a defined on A byx ≤a y ⇐⇒ m(a, x, y) = x is a ∧-semilattice order with bottom element a. The associated operation ∧ is defined byx ∧ y = m(a, x, y). Semilattices constructed in this way are called median semilattices, and can be characterized as follows.

We investigate the associativity property for functions of indefinite arities and introduce and discuss the more general property of preassociativity, a generalization of associativity which does not involve any composition of functions

ABSTRACT. We investigate the barycentric associativity property for functions with indefinite arities and discuss the more general property of barycentric preassociativity, a generalization of barycentric associativity which does not involve any composition of functions. We also provide a generalization of Kolmogoroff-Nagumo's characterization of the quasi-arithmetic mean functions to barycentrically preassociative functions.

Sugeno integrals are aggregation operations involving a criterion weighting scheme based on the use of set functions called capacities or fuzzy measures. In this paper, we define generalized versions of Sugeno integrals on totally ordered bounded chains, by extending the operation that combines the value of the capacity on each subset of criteria and the value of the utility function over elements of the subset. We show that the generalized concept of Sugeno integral splits into two functionals, one based on a general multiplevalued conjunction (we call integral) and one based on a general multiple-valued implication (we call cointegral). These fuzzy conjunction and implication connectives are related via a so-called semiduality property, involving an involutive negation. Sugeno integrals correspond to the case when the fuzzy conjunction is the minimum and the fuzzy implication is Kleene-Dienes implication, in which case integrals and cointegrals coincide. In this paper, we consider a very general class of fuzzy conjunction operations on a finite setting, that reduce to Boolean conjunctions on extreme values of the bounded chain, and are nondecreasing in each place, and the corresponding general class of implications (their semiduals). The merit of these new aggregation operators is to go beyond pure lattice polynomials, thus enhancing the expressive power of qualitative aggregation functions, especially as to the way an importance weight can affect a local rating of an object to be chosen.