An introduction to vague complemented ordered sets

The purpose of this paper is to introduce a theory of fuzzily defined complement operations on nonempty sets equipped with fuzzily defined ordering relations. Many-valued equivalence relation-based fuzzy ordering relations (also called vague ordering relations) provide a powerful and a comprehensive mathematical modelling of fuzzily defined partial ordering relations. For this reason, starting with a nonempty set X equipped with a many-valued equivalence relation and a vague ordering relation, a fuzzily defined complement operation (called a vague complement operation) on X will be formulated by means of the underling many-valued equivalence relation and vague ordering relation. Because of the fact that the practical implementations of vague complement operations basically depend on their representation properties, a considerable part of this paper is devoted to the representations of vague complement operations. In addition to this, the present paper provides various nontrivial examples for vague complements, and introduces a many-valued logical interpretation of quantum logic as a real application of vague complements.