An algebraic approach to linguistic hedges in Zadeh's fuzzy logic

A unified algebraic approach to the natural structure of domains of linguistic variables, which was proposed by Ho and Wechler, is given and it is shown that a class of finite symmetrical RH_algebras has a rich enough algebraic structure.

Abstract

The paper addresses the mathematical modelling of domains of linguistic variables, i.e. term-sets of linguistic variables, in order to obtain a suitable algebraic structure for the set of truth values of Zadeh's fuzzy logic. We shall give a unified algebraic approach to the natural structure of domains of linguistic variables, which was proposed by Ho and Wechler (Fuzzy Sets and Systems 35 (1990) 281) and, then, by Ho and Nam (Proc. NCST Vietnam 9 (1) (1997) 15; Logic, Algebra and Computer Science, Vol. 46, Banach Center Publications, PWN, Warsaw, 1999, p. 63). In this approach, every linguistic domain can be considered as an algebraic structure called hedge algebra, because properties of its unary operations reflect semantic characteristics of linguistic hedges. Many fundamental properties of refined hedge algebras (RH_algebras) are examined, especially it is shown that every RH_algebra of a linguistic variable with a chain of the primary terms is a distributive lattice. RH_algebras with exactly two distinct primary terms, one being an antonym of the other, will also be investigated and they will be called symmetrical RH_algebras. It is shown that a class of finite symmetrical RH_algebras has a rich enough algebraic structure.