| The lattice implication algebra and the graph theory are two branches of mathematics which offer the basic surport for the theory and application of artificial intelligence and intelligent information processing. The lattice implication algebra is a theoretical basis for the theory of lattice valued logic and approximate reasoning. In 1987,Alavi and others conjectured that every graph with positive size has an ascending subgraph decomposition. This conjecture is an unsolved problem in graph theory.This thesis contains the author's research work on the lattice implication algebra and the Alavi Conjecture. Following are the main works contained in this thesis.1. Studied the axioms of lattice implication algebra,the axioms which can construct a lattice implication algebra on a set without any algebraic structure were given.2. Pointed out a family of congruence relations induced by a fuzzy filter of a lattice implication algebra and proved that this family of congruence relations is a complete chain,and then discussed the relations between the quotient lattice implication algebras induced by this family of congruence relations.3. Discussed the property of fuzzy LI-ideals,pointed out a family of congruence relations induced by a fuzzy LI-ideal of a lattice implication algebra and proved that this family of congruence relations is a complete chain,and then discussed the relations between the quotient lattice implication algebras induced by this family of congruence relations.4. Determined the relation between fuzzy filter and fuzzy LI-ideal.5. Proposed the concept of associative ideal of a lattice implication algebra,discussed the property of associative ideal,pointed out the relation between associative ideal and LI-ideal and the relation between associative ideal and associative filter.6. Proposed the concept of fuzzy associative ideal of lattice implication algebras,studied the the properties of fuzzy associative ideal,discussed the relations between fuzzy associative ideal and associative ideal,between fuzzy associative ideal and fuzzy associative filter and between fuzzy associative ideal and fuzzy LI-ideal.7. Introduced the concept of L-fuzzylattice implication algebra and discussed it's properties.8. Proposed the concepts of L-fuzzyfilter and L-fuzzyideal of lattice implication algebra and investigated their properties and the relation between them.9. Proposed the concepts of L-fuzzyassociative filter,discussed it's properties andthe relations between L-fuzzyassociative filter and associative filter and between filter and L-fuzzyassociative filter.10. Proposed the concepts of L-fuzzyassociative ideal,discussed it's properties and the relations between L-fuzzyassociative ideal and associative ideal,between L-fuzzyassociative ideal and L/-ideal,and between L-fuzzyassociative filter and L-fuzzyassociative ideal.11. Proved that Kn - H has an ascending star decomposition when the size of H is less than 4p,where H is a subgraph of the complete graph Kn.12. Proposed the concept of comic and proved the following theorems by using comic.(1) Let w0,w1....wn-2 is n-1 different vertices of Kn,if n 2 then Kn has an ascending comic decompositionsuch that w0,w1.....wn-2 is the centers of the comics C0,C1,...,Cn-2 respectively.(2) Let n 3. if the size of Hn-1 is n - 1 then Kn - Hn-1 has an ascending comic decompositionsuch that the centers of any two of these comics are different when choose a convenient center for each one of the comics C0,C1 and C2.(3) Let H is a subgrpah of the complete graph Kn with order n. If n 7 and the size of H is In - 3 then Kn - H can be decomposed intoand the centers of any two of the comics C3,...,Cn-5,Cn-4 are different.(4) Let H is a subgrpah of the complete graph Kn with order n. If n 6 and e(H) <- 4 then Kn - H can be decomposed into subgraphswhere Gi = 5,when i 3 and Gi = Ci~ when i > 3,and the centers of any two comics in the decomposition are different.
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