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

SOME GENERAL COMMENTS ON FUZZY SETS OF TYPE 2

This paper contains some general comments on the algebra of truth values of fuzzy sets of type-2. It details the precise mathematical relationship with the algebras of truth values of ordinary fuzzy sets and of interval-valued fuzzy sets. Subalgebras of the algebra of truth values, and t-norms on them are discussed. There is some discussion of finite type-2 fuzzy sets. Keywords: Type-2 fuzzy set, interval-valued fuzzy set, algebra of truth values, automorphism, t-norm, finite type-2 fuzzy set