α-Cuts and models of fuzzy logic

Abstract

It is well known that any fuzzy set X in a classical set A with values in a complete (residuated) lattice Ω can be identified with a system of α-cuts X _α, α ∈ Ω. In this paper analogical results are presented for sets with similarity relations with values in Ω (e.g. Ω-sets) which are objects of two special categories Set(Ω) and SetR(Ω) of Ω-sets and for fuzzy sets defined as morphisms from Ω-set into a special Ω-set (Ω, ↔ ). It is proved that also such fuzzy sets can be defined equivalently as special cut systems (C _α)_α. Finally, models of first-order fuzzy logic based on these cut systems are defined and relationships between interpretations of formulae in classical models (based on Ω-sets) and in models based on these cut systems are investigated.

Keywords:

• similarity relation
• fuzzy sets in sets with similarity relations
• cut systems
• fuzzy sets in cut systems
• models of a fuzzy logic
• categories of Ω-sets

Acknowledgement

This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).