×

Essential ideals and homomorphisms in BCI-algebras. (English) Zbl 0779.06013

We consider essential closed ideals of BCI-algebras and show that if \(X\neq \{0\}\) is a BCK-algebra and \(Y\) is a \(p\)-semisimple BCI-algebra, then \(X\) is an essential closed ideal of \((Y-\{0\})\cup X\) where \((Y- \{0\})\cup X\) is the BCI-algebra with \(X\) as its \(p\)-radical and \(Y\) as its \(p\)-semisimple part. It is shown that a closed ideal \(I\) of a BCI- algebra \(X\) is an essential closed ideal if and only if the inclusion map \(I\to X\) is an essential homomorphism. Finally, it is shown that a BCI- algebra has an injective envelope if and only if it is \(p\)-semisimple and its natural abelian group structure has an injective envelope.
Reviewer: C.S.Hoo (Edmonton)

MSC:

06F35 BCK-algebras, BCI-algebras
PDFBibTeX XMLCite