Algorithms and computations for (m,n)-fold p-ideals in BCI-algebras

In [C. Lele, S. Moutari, M.L.N. Mbah, Algorithms and computations for foldedness of p-ideals in BCI-algebras, J. Appl. Logic 6 (4) (2008) 580-588], the notion of the n-fold p-ideals in BCI-algebras as a generalization of p-ideals in BCI-algebras, is introduced, but we show that an ideal is an n-fold p-ideal if and only if it is a p-ideal, and that the results of the mentioned paper is the same as those in [Y.B. Jun, J. Meng, Fuzzy P-ideals in BCI-algebra, Math. Japon. 2 (1994) 271-282, X.H. Zhang, J. Hao, S.A. Bhatti, On p-ideals of a BCI-algebra, Punjab Univ. J. Math. 27 (1994) 121-128]. In this paper we observe that, the notions of (m,n)-fold p-ideals and fuzzy (m,n)-fold p-ideals, for each positive integers m,n, are indeed the natural generalization of p-ideals and fuzzy p-ideals, respectively. A characterization of (m,n)-fold p-ideals and fuzzy (m,n)-fold p-ideals is given, and conditions for which an ideal (respectively fuzzy ideal) is an (m,n)-fold p-ideal (respectively fuzzy (m,n)-fold p-ideal) are studied. We also establish extension properties for (m,n)-fold p-ideals and fuzzy (m,n)-fold p-ideals. Furthermore, we construct some algorithms to determine whether certain finite sets provided with a well defined operation, are BCI-algebras, (m,n)-fold p-ideals, fuzzy subsets or fuzzy (m,n)-fold p-ideals.