Abstract
Many-valued equalities emerge quite naturally in probabilistic microgeometry and theory of presheaves on complete Heyting algebras. This chapter presents a coherent and sound interpretation of values of many-valued equalities. The crucial step in this respect is a solution of the problem to attach a concrete mathematical expression to the intuitive idea of overlapping. The chapter also explains the role of many-valued equalities with global extent of existence in the theory of local equalities. If the square root of the extent of existence exists with respect to the underlying semi group operation, then it can be shown that all many-valued equalities have a universal representation by a pair. The localization of the extent of existence and its impact on the axiom of many-valued equalities is dealt in this chapter. A natural source of many-valued equalities is presheaves on complete Heyting algebras. The chapter illustrates the richness of the concept of many-valued equalities by various examples originated from the theory of Menger spaces and presheaves.
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