Fuzzy subalgebras of type \((\alpha,\beta)\) in BCK/BCI-algebras. (English) Zbl 1135.06012

Summary: Using the belongs to relation \((\in)\) and quasi-coincident with relation (q) between fuzzy points and fuzzy sets, the concept of \((\alpha,\beta)\)-fuzzy subalgebras, where \(\alpha\) and \(\beta\) are any two relations from the set \(\{\in,\text{q},{\in}\vee\text{q}, {\in}\wedge \text{q}\}\) with \(\alpha\neq{\in}\wedge\text{q}\), was already introduced, and related properties were investigated [Y. B. Jun, Bull. Korean Math. Soc. 42, No. 4, 703–711 (2005; Zbl 1081.06027)]. In this paper, we give a condition for an \((\in,{\in}\vee\text{q})\)-fuzzy subalgebra to be an \((\in,\in)\)-fuzzy subalgebra. We provide characterizations of an \((\in,{\in}\vee\text{q})\)-fuzzy subalgebra. We show that a proper \((\in,\in)\)-fuzzy subalgebra \({\mathcal A}\) of \(X\) with additional conditions can be expressed as the union of two proper non-equivalent \((\in,\in)\)-fuzzy subalgebras of \(X\). We also prove that if \({\mathcal A}\) is a proper \((\in,{\in}\vee\text{q})\)-fuzzy subalgebra of a BCK/BCI-algebra \(X\) such that \(\#\{{\mathcal A}(x)\mid{\mathcal A}(x)< 0.5\}\geq2\), then there exist two proper non-equivalent \((\in,{\in}\vee\text{q})\)-fuzzy subalgebras of \(X\) such that \({\mathcal A}\) can be expressed as the union of them.


06F35 BCK-algebras, BCI-algebras
03E72 Theory of fuzzy sets, etc.
08A72 Fuzzy algebraic structures


Zbl 1081.06027