Stone-type representations and dualities for varieties of bisemilattices

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces$^{\star}$. The categories of 2spaces and 2spaces$^{\star}$ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent

Interval-valued algebras and fuzzy logics

In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of ‘p implies q’ and ‘p and q’, given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter

About t-norms on type-2 fuzzy sets.

Walker et al. defined two families of binary operations on M (set of functions of [0,1] in [0,1]), and they determined that, under certain conditions, those operations are t-norms (triangular norm) or t-conorms on L (all the normal and convex functions of M). We define binary operations on M, more general than those given by Walker et al., and we study many properties of these general operations that allow us to deduce new t-norms and t-conorms on both L, and M

Convolution lattices

We propose two convolution operations on the set of functions between two bounded lattices and investigate the algebraic structure they constitute, in particular the lattice laws they satisfy. Each of these laws requires the restriction to a specific subset of functions, such as normal, idempotent or convex functions. Combining all individual results, we identify the maximal subsets of functions resulting in a bounded lattice, and show this result to be equivalent to the distributivity of the lattice acting as domain of the functions. Furthermore, these lattices turn out to be distributive as well. Additionally, we show that for the larger subset of idempotent functions, although not satisfying the absorption laws, the convolution operations satisfy the Birkhoff equation.This work has been supported by the Research Services of the Universidad Publica de Navarra, and by the research project TIN2016-77356-P from MINECO, AEI/FEDER, UE

Towards a linear algebra of programming

The Algebra of Programming (AoP) is a discipline for programming from specifications using relation algebra. Specification vagueness and nondeterminism are captured by relations. (Final) implemen- tations are functions. Probabilistic functions are half way between relations and functions: they express the propensity, or like- lihood of ambiguous, multiple outputs. This paper puts forward a basis for a Linear Algebra of Programming (LAoP) extending standard AoP towards probabilistic functions. Because of the quantitative essence of these functions, the allegory of binary relations which supports the AoP has to be extended. We show that, if one restricts to discrete probability spaces, categories of matrices provide adequate support for the extension, while preserving the pointfree reasoning style typical of the AoP.Fundação para a Ciência e a Tecnologia (FCT