EQ-algebras were introduced by Novak (2006) as an algebraic structure of truth values for fuzzy-type theory (FFT). Novák and De Baets (2009) various kinds such good, residuated, IEQ-algebras. In this paper, we define the notion (pre)ideal in bounded (BEQ-algebras) investigate some properties. Then, introduce a congruence relation on good BEQ-algebras using ideals, then, solve open problem Paad ...

Sheaves of structures are useful to give constructions in universal algebra and model theory. We can describe their logical behavior terms Heyting-valued structures. In this paper, we first provide a systematic treatment sheaves from the viewpoint categorical logic. then prove form {\L}o\'s's theorem for also characterization which holds with respect any maximal filter.

We examine the notion of bisimulation and its ramifications, in the context of the family of Heyting-valued modal languages introduced by M. Fitting. Each modal language in this family is built on an underlying space of truth values, a Heyting algebra H. All the truth values are directly represented in the language, which is interpreted on relational frames with an H-valued accessibility relati...

In this paper we define and examine frame constructions for the family of many-valued modal logics introduced by M. Fitting in the ’90s. Every language of this family is built on an underlying space of truth values, a Heyting algebra H. We generalize Fitting’s original work by considering complete Heyting algebras as truth spaces and proceed to define a suitable notion of H-indexed families of ...

A review is given of recent work aimed at constructing a quantum theory of cosmology in which all observables refer to information measurable by observers inside the universe. At the classical level the algebra of observables should be modified to take into account the fact that observers can only give truth values to observables that have to do with their backwards light cone. The resulting al...

In Coecke (2002a) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic o...

In this paper it is shown that Heyting and Co-Heyting mereological systems provide a convenient conceptual framework for spatial reasoning, in which spatial concepts such as connectedness, interior parts, (exterior) contact, and boundary can be defined in a natural and intuitively appealing way. This fact refutes the wide-spread contention that mereology cannot deal with the more advanced aspec...