BCI-代数与BCK-代数的粘合(Ⅰ)

构造了BCI-代数范畴中一种自然的粘合,先前许多作者定义的粘合是这种构造的特殊情况,这种构造的自然性表现在:任一BCI-代数与BCK-代数能以此法粘合;导出同态的粘合;保留两个代数的许多性质.

Pasting a BCI-algebra to a BCK-algebra(Ⅰ)

In this note we′ll make a structure to paste any BCI-algebra and any BCK-algebra naturally.Let(Y,*,0) be a BCI-algebra with its BCK-part K and (X,*,0) a BCK-algebra satisfying X∩Y={0}.Let ΓX be a root set of X.Denote A(α)={x∈X|x≤α},T=TΓ=∪α∈ΓA(α).Set Z=X∪Y.Define a binary operation on Z to be:x1x2=x1*x2,if x1,x2∈X;xy=0,if x∈T,y∈K-{0};xy=x*x′,if x∈X-T,y∈K-{0},xy=0*y,if x∈X,y∈Y-K;yx=y,if x∈X,y∈Y;y1y2=y1*y2,if y1,y2∈Y,where x′∈Γ s.t.x*x′≤x*a for all a∈Γ,is called a representative of x.Then (Z,,0) is a BCI-algebra and will be denoted by X∨ΓY.