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ON THE VARIETY OF POSITIVE IMPLICATIVE BCK-ALGEBRAS

ON THE VARIETY OF POSITIVE IMPLICATIVE BCK-ALGEBRAS DEMONSTRATIO MATHEMATICAVol. X V I INo 419J»4Helena bednarczykON THE VARIETY OF POSITIVE IMPLICATIVE BCK-ALGEBRASThe notion of a BCK-algebra was introduced by Y.Imai andK.Iseki [3}. K.Iseki and S.Tanaka ( [5] , [6]) gave muoh information about the general theory and ideal theory in BCK-algebra. H.Yutani [8] has shown that the class at all positive implicative BCK-algebras forms a variety. V.K.Cornish [1] observed that the notion of positive implicative BCK-algebruis exactly order-dual to the notion of positive implicationalgebra in sense-of H.Rasiowa [7].An algebra X a (X,*,0) of type (2 t O) is called a BCK-algebra if it satisfies the following axioms:I.[(x*y) * (x *z)] * (« * y ) = 0,II.[x * (x *y)] * y • 0,III.x * x = 0,IV.0 « x « 0,V.x*y • 0andy *x • 0implyx • y.X is partially ordered set with 0 as the smallest element,where the order relation is defined byVI.x$yiffx * y • 0.If (X,$) is a chain (tree), then the BCK-algebra (X,*,0) iecalled a BCK-chain (BCK-tree).A BCK-algebra (X,«,0) is positive implicative, if X satisfies one of equivalent oonditionB- 869 -2(t)H. Bednarozyk( x * y ) * z » ( x * z ) « ( y « z )»(x*y) * y http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ON THE VARIETY OF POSITIVE IMPLICATIVE BCK-ALGEBRAS

Bednarczyk, Helena
Demonstratio Mathematica , Volume 17 (4): 6 – Oct 1, 1984
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Publisher
de Gruyter
Copyright
© by Helena Bednarczyk
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1984-0408
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. X V I INo 419J»4Helena bednarczykON THE VARIETY OF POSITIVE IMPLICATIVE BCK-ALGEBRASThe notion of a BCK-algebra was introduced by Y.Imai andK.Iseki [3}. K.Iseki and S.Tanaka ( [5] , [6]) gave muoh information about the general theory and ideal theory in BCK-algebra. H.Yutani [8] has shown that the class at all positive implicative BCK-algebras forms a variety. V.K.Cornish [1] observed that the notion of positive implicative BCK-algebruis exactly order-dual to the notion of positive implicationalgebra in sense-of H.Rasiowa [7].An algebra X a (X,*,0) of type (2 t O) is called a BCK-algebra if it satisfies the following axioms:I.[(x*y) * (x *z)] * (« * y ) = 0,II.[x * (x *y)] * y • 0,III.x * x = 0,IV.0 « x « 0,V.x*y • 0andy *x • 0implyx • y.X is partially ordered set with 0 as the smallest element,where the order relation is defined byVI.x$yiffx * y • 0.If (X,$) is a chain (tree), then the BCK-algebra (X,*,0) iecalled a BCK-chain (BCK-tree).A BCK-algebra (X,«,0) is positive implicative, if X satisfies one of equivalent oonditionB- 869 -2(t)H. Bednarozyk( x * y ) * z » ( x * z ) « ( y « z )»(x*y) * y

Journal

Demonstratio Mathematica – de Gruyter

Published: Oct 1, 1984

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