Fuzzy Sets in UP-algebras with Respect to A Triangular Norm

Abstract

In this paper, we apply the notion of fuzzy sets with respect to a triangular norm to UP-algebras. We introduce the notions of $T$-fuzzy UP-subalgebras, $T$-fuzzy near UP-filters, $T$-fuzzy UP-filters, $T$-fuzzy UP-ideals, and $T$-fuzzy strongly UP-ideals, their properties are investigated and some useful examples are discussed. We discuss the relations between $T$-fuzzy UP-subalgebras (resp., $T$-fuzzy near UP-filters, $T$-fuzzy UP-filters, $T$-fuzzy UP-ideals, and $T$-fuzzy strongly UP-ideals) and a notion of UP-subalgebras (resp., near UP-filters, UP-filters, UP-ideals, strongly UP-ideals), and their level subsets and UP-homomorphisms are studied. Moreover, we have introduced the notion of fuzzy sets with respect to a triangular norm of anti-type in UP-algebras, and studied the properties as well as previous notions.