P-closure ideals in BCI-algebras

In this paper, for any non-empty subset A of a BCI-algebra X, we introduce the concept of p-closure of A, denoted by Apc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{pc}$$\end{document}, and investigate some related properties. Applying this concept, we characterize the minimal elements of BCI-algebras. We also give a characterization of the p-closure of subalgebras of X by some branches of X. We state a necessary and sufficient condition for a BCI-algebra (1) to be p-semisimple; (2) to be a BCK-algebra. Moreover, we show that the p-closure can be used to define a closure operator. We investigate the relationship between f(Apc)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(A^{pc})$$\end{document} and (f(A))pc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(f(A))^{pc}$$\end{document} for a BCI-homomorphism f. Finally, we prove that Apc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{pc}$$\end{document} is the least closed p-ideal containing A for any ideal A of X.