Study On The Structures Of Several Logic Algebra Systems

Logic algebras are the algebraic foundations of reasoning mechanism of many fields such as computer sciences, information sciences, cybernetics, artificial intelligence and so on. BCK/BCI algebras are two classes of logic algebras. Recent researches have shown that, pocrims (partially ordered commutative residuated integral monoids) and BCK algebras with condition (S) are categorically equivalent, and residuated lattices and bounded BCK lattices with condition (S) are categorically equivalent. Hence, most of the algebras related to logic, such as MTL algebras, BL algebras, Heyting algebras, MV algebras, NM algebras, Boolean algebras, are all natural expansions of BCK algebras. Because p-semisimple BCI algebras and abelian groups are categorically equivalent, abelian groups can be seen natural expansion of BCI algebras. These show that BCK/BCI algebras are considerably wide structures. So, it is important significance for investigation of BCK/BCI algebras. Any results on BCK/BCI algebras are also hold in the above-mentioned logic algebras.In recent years, as theory and application motivate, the study on the t-norm based logic system and the corresponding pseudo-logic system have become one of focus paid close attention by scholaxs in the field of logic, in which the t-norm based logical investigations were first to the algebras and in the case of the pseudo-logic system the algebraic development was first to the logic. NM algebras (R₀ algebras) and lattice implication algebras are t-norm based logical algebras.This thesis is devoted to study the structural properties of BCK/BCI algebras and their expansions NM algebras and lattice implication algebras. Details are as follows:1. The notion of BCI implicative ideals is introduced. It is a natural generalization of the notion of implicative ideals in BCK algebras. It is shown that a nonempty subset of a BCI algebra is a BCI implicative ideal if and only if it is both a BCI commutative ideal and a BCI positive implicative ideal. This generalizes the famous result in BCK algebras: a nonempty subset of a BCK algebra is an implicative ideal if and only if it is both a commutative ideal and a positive implicative ideal. The implicative BCI algebras are described completely via BCI implicative ideals. The notions of FSI ideals and FSC ideals are introduced. It is shown that a fuzzy subset of a BCI algebra is an FSI ideal if and only if it is both an FSC ideal and a fuzzy BCI positive implicative ideal.2. A new class of quotient BCK/BCI algebras and a new class of bounded quotient BCK algebra are constructed, respectively. By investigating their applications, it is shown that these quotient structures are more reasonable than ever before and possess good properties. The fact that a fuzzy ideal of a BCI algebra is closed if and only if it is a fuzzy subalgebra is proved. It is ponited that, in some important classes of BCI algebras, any fuzzy ideal must be closed.3. A new definition of fuzzy maximal ideals in BCK/BCI algebras is introduced. It is shown that this new notion is more concise and general than ever before. Its properties are...