Fuzzy universal algebras on $L$-sets

Abstract

This paper attempts to generalize universal algebras on classical sets to $L$-sets when $L$ is a GL-quantale. Some basic notions of fuzzy universal algebra on an $L$-set are introduced, such as subalgebra, quotient algebra, homomorphism, congruence, and direct product etc. The properties of them are studied. $L$-valued power algebra is also introduced and it is shown there is an onto homomorphism from $P(A)/R^{+}$ to $P(A/R)$ for any congruence $R$ on $L$-set $A$.