Metric structures on some MTL-algebras and its applications

This paper focus on establishing in a unified way metric structures on some MTL-algebras. Let M be one of the R₀-algebra, MV -algebra, product algebra and G-algebra, ΩM be the set of all homomorphisms from M into the real unit interval [0; 1], and μ be a probability measure on ΩM. It is proved that these MTL-algebras (R₀-algebra, MV -algebra, product algebra and G-algebra) are standardly separable, it means that a ≤ b iff v(a) ≤ v(b) for any v ∈ Ω_M. The concepts of sizes of elements of M and similarity degrees of pairs of elements of M w.r.t. μ are introduced, and then a pseudo-metric on M is defined therefrom. As an applications, using metric MTL-algebras theories some more flexible approximate reasoning models of propositional logic could be established.