Note on " Fuzzy universal algebras on L-sets"

Document Type : Research Paper


Department of Mathematics, Ocean University of China, Qingdao, 266100, P. R. China


This note is the continuation of the paper entitled ``Fuzzy universal algebras on $L$-sets"(IJFS, Volume 16, Number 4, (2019), pp. 175--187) 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.