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Abstract:
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The thesis consists of three chapters. Heyting algebras are studied as an equality category in Chapter 1. The properties of filters and ideals of Heyting algebras are presented together with corresponding properties in distributive nets and Boolean algebras. Free, injective and projective Heyting algebras are presented and a theorem about the representation in algebras with closing is proved. Some properties of Heyting algebras, which are important for study of formal logics closely to intuitionistic logic, are also presented. Complete Heyting algebras are studied in Chapter 2. The family of complete Heyting algebras is obtained by repeating of the construction of the algebra of J-operators. Some properties of this family when the initial Heyting algebras is linear order, are studied. Moreover, the characterization of complete Heyting algebras which can be approximated by complete Boolean algebras is given. Duality of categories of topological spaces and complete Heyting algebras are studied in Chapter 3. Some adjunctions are defined, and for those adjunctions the actions of monad and comonad are studied. It is shown that the category of complete Heyting algebras is reflective in the categories of sets, distributive bounded nets and complete Heyting algebras. It is shown that complete Heyting algebras correspond to "deposited" spaces, and distributive bounded nets correspond to a restriction of Ston’s spaces. |