A Theory of Refinement Structure of Hedge Algebras and Its Applications to Fuzzy Logic

In [13], an algebraic approach to the natural structure of domains of linguistic variables was introduced. In this approach, every linguistic domain can be interpreted as an algebraic structure called a hedge algebra. In this paper, a refinement structure of hedge algebras based on free distributive lattices generated by linguistic hedge operations will be examined in order to model structure of linguistic domains more properly. In solving this question, we restrict our consideration to the specific hedge algebras called PN-homogeneous hedge algebras. It is shown that any PN-homogeneous hedge algebra can be refined to a refined hedge algebra (RHA, for short) and every RHA with a chain of the primary generators is a distributive lattice. Especially, we shall examine RHAs with exactly two distinct generators, which will be called symmetrical RHAs. Furthermore, in the symmetrical RHAs of the linguistic truth variable, we are able to define negation and implication operation, which, according to their properties, may be interpreted as logical negation and implication in a kind of fuzzy logic called linguisticvalued logic. Some elementary properties of these operations will be also examined. This yields a possibility to construct a method in linguistic reasoning, which is based on linguistic-valued fuzzy logic corresponding to the symmetrical RHAs of the linguistic truth variable.