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Abstract:
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The thesis consists of three chapters. In Chapter 1 non-strict deductive implicative algebras are studied. Weak deductive, n-deductive and ω-deductive implicative algebras are introduced. Two kinds of complements, pseudo-complement and contraposition complement in deductive implicative algebras are defined, and the connection between these algebras and deductive implicative algebras with complement are presented. Certain properties of several implicative filters in implicative algebras and their connections with homomorphsms and congruences of these algebras are studied. In the last part of Chapter 1 the representation theorems for implicative algebras mentioned in the previous parts of the chapter are proved. The strict deductive implicative algebras and their properties, which are analogous to the properties algebras from the first chapter, are studied in Chapter 2. In the last part of that chapter the representation theorems for the strict implicative algebras are proved. In Chapter 3 deductive implicative algebras in the context of deductive (sub)nets are studied. Important notions of different forms of limited distribution and many interesting connections between these distributions and the properties of deductive nets are presented. It is shown that an implicative algebra can be drowned isomorphicaly into finite deductive subnet of sets or a net such that the implication are preserved. |