Note on " Fuzzy universal algebras on L-sets"

This note is the continuation of the paper entitled ``Fuzzy universal algebras on L-sets"(IJFS, Volume ۱۶, Number ۴, (۲۰۱۹), pp. ۱۷۵--۱۸۷) and it focuses on \mathcal{Q}-valued (universal) algebras on \mathcal{Q}-typed sets. When \mathcal{Q} is an involutive quantaloid, some basic related notions in \mathcal{Q}-valued algebra such as subalgebra, quotient algebra, homomorphism, congruence, direct product and variety etc are given and the properties of them are studied. When \mathcal{Q} is still a symmetric quantaloid, the \mathcal{Q}-valued algebra is lifted to \mathcal{Q}-valued power algebra, and the power isomorphism theorem is given.  As a special case of \mathcal{Q}-valued algebra, the fuzzy universal algebra on L-set is presented naturally when L is a commutative and divisible quantale.

Keywords:

mathcal{Q}-typed set , universal algebra , mathcal{Q}-valued algebra , congruence , power algebra