Pseudo-BCK-algebras And Rough Pseudo-BCK-algebras And Their Filter Theories

Logic algebras are the algebraic foundations of reasoning mechanism of many fields such ascomputer sciences, information sciences, cybernetics, artificial intelligence and so on. Pseudo-BCK-algebras are non-commutative extension of well-known logical algebras named BCK-algebras. Inorder to characterize furtherly the structure of pseudo-BCK-algebras, many scholars introduced thenotions of pseudo-filters of different types, and investigated some related properties. In recent years,the study on applying the rough set and hyper-structures theory to algebras structions have becomeone of focus paid close attention by scholars in the field of logic. Based on these new results, thisthesis mainly researched on pseudo-BCK-algebra and its filter theory, and investigated somerelated properties of introduced rough pseudo-BCK-algebras and hyper pseudo-BCK-algebras.The innovation and main results in this thesis are as following:(1) By introducing the notion of NP-type poset, we analysed its characteristic in construc-tion and obtained all NP-type posets of order n6from {1}. On this basis, we calculated all non-isomorphic proper pseudo-BCK algebras of order n6.(2) By introducing the notions of pseudo-R0implication, strong pseudo-De Morgan-algebras, we investigated the relations between pseudo-De Morgan-algebras and pseudo-involutivepseudo-BCK-algebras, pseudo-regular residuated lattices.(3) We introduced the notions of the positive implicative pseudo-BCK-algebra oftype1and positive implicative pseudo-filter of type1, and proved the theorem that the structures of1-type positive implicative pseudo-BCK-algebras are characterized.(4) We introduced the notions of rough pseudo-BCK-algebras and rough pseudo-filters,and investigated their related properties. Proper rough pseudo-BCK-algebras inducedby set are proposed and discussed.(5) The notion of hyper pseudo-BCK-algebras is introdued and its properties isinvestigated. By introducing the notion of hyper pseudo-filter, we gived thecharacterized of hyper pseudo-BCK-algebras. Finally, a negative answer to the openproblem of R. A. Borzooei is given.