Vague Ordered Fields: towards an Axiomatic Theory of Vague Real Line

The notion of fuzzy function based on many-valued equivalence relations (many-valued similarity relations (equalities) [17, 18, 19], fuzzy equivalence relations [4, 6, 7, 21, 22, 26], similarity relations [1, 2, 3, 15, 28], indistinguishability operators [27], etc.) has been introduced by several authors, and applied to category theory [5], approximate reasoning and fuzzy control theory [8, 10, 15, 22]. The author of this talk [8, 9, 10] later proposed other versions of this kind of fuzzy function, known as strong fuzzy function and perfect fuzzy function, which have more desirable and powerful representation properties than the others. Many-valued equivalence relation-based fuzzy orderings have been studied by Höhle-Blanchard [16] and Bodenhofer [1, 2, 3] w.r.t. different special integral, commutative cqm-lattices. Later on, these fuzzy orderings are generalized on the basis of a fixed and a general integral, commutative cqm-lattice M = (L,≤, ∗) under the name M -vague orderings [13, 14]. For a given nonempty set X and an M -equivalence relation E on it, an M -vague ordering on X is a special L-fuzzy relation on X satisfying some further properties by means of E. Strong (perfect) fuzzy functions [8, 9, 10] form the elementary tools of vague algebra [9, 11, 12] and vague lattices [13, 14]. In contrast to fuzzy algebra [23] and fuzzy lattices [25], vague algebra and vague lattices basically involve vaguely defined binary operations (M -vague binary operations [9, 11, 12]) and vaguely defined ordering relations (M -vague orderings), where the integral, commutative cqm-lattice M = (L,≤, ∗) [10, 20] denotes the many-valued logical basis of these studies. A vague binary operation ◦̃ on X can be roughly described as a special L-fuzzy relation (more precisely, a special strong fuzzy function) from X×X to X with some reasonable properties formulated in terms of E [9, 11, 12]. Strong (perfect) fuzzy functions propose a new approach to the fuzzy setting of numerous different branches of mathematics. Vague algebra and vague lattices are only two important cases of such an approach. The development of a sound theory of real line equipped with M -vague orderings, M -vague addition operations and M -vague multiplication operations [9, 12], which will be called vague real line, lies at the