Study Of Bck Algebra And De Morgan Algebra

The BCK algebra is the algebra system that was in 1966 proposed by the Japanese mathematician Y.Ismai and the K.Iseki , this theory contact to be suffused with universal algebra, group theory, ring theory, lattice theory,Boolean algebra,set point topology, toplogical algebra etc. we have already obtained the large quantity result to the research of the BCK algebra.In the meantime, the BCK algebra , Boole algebra,MV algebra,BL algebra,Fuzzy imply algebra,the Ro- algebra and lattice implicative algebra, etc all have a close relation, the associated BCK - algebra is the Boole algebra, bounded commutative fuzzy implication algebra and MV algebra are equivalent, the BCK algebra and fuzzy implication algebra are equivalent, lattice implication algebra and MV algebra are equivalent, the MV algebra is the BL algebra. with Wang guo-jun and other fomous mathematicians' efforts, relations of these algebras become clearer and clearer.In fact, the BCK algebra and Heyting algebra,Demorgan algebra,Kleene algebra,the Stone algebra...etc. all have close relation, and the research paper of the relation between the BCK algebra and these algebras are less.Li Zhi-Wei,Zheng Chong-You proved the Heyting algebra to be a Fuzzy implication algebra, Then the Heyting algebra is the BCK algebra, but didn't give what condition under the BCK algebra become the Heyting algebra.This text studied the relations between the BCK algebra and the Heyting algebra etc, proving the associated BCK algebra to be the Heyting algebra, and the Fuzzy implication algebra becomes the Heyting algebra to" → "should satisfy of condition, prove to have the bounded commutative BCK algebra is a Demorgan algebra and isn't the Kleene algebra.The bounded associated BCK algebra is the Stone algebra, is also a pair of Stone algebras, this text maininnovation point have:(l)Proving the bounded associated BCK algebra is the Heyting algebra, give a sufficient and necessary condition of BCK algebra constitute Heyting algebra. (2)Proving bounded commutative BCK algebra is a Demorgan algebra, and it needn't a Kleene algebra.(3)Proving bounded commutative BCK algebra is Nelson algebra.(4)Proving bounded commutative BCK algebra is Pseudocomplemented Lattice, is a Stone algebra, and it is a pair of Stong algebras.