Weakly invariant fuzzy quasi-pseudometrics on semigroups

We obtain conditions on fuzzy quasi-pseudometrics on either semigroups or groups which imply that they are either fuzzy topological semigroups or topological groups. Our main results are: (1) Let (S, M, *) be a fuzzy quasi-pseudometric right topological semigroup (resp., group) such that (M, *) is left weakly invariant; then (S, M, *) is a fuzzy quasi-pseudometric topological semigroup (resp., group); (2) Suppose that (M, *) is a left weakly invariant fuzzy quasi-pseudometric on a monoid G such that each left translation of G is open and every right translation is continuous at the identity e of (G, M, *); then (G, M, *) is a fuzzy quasi-pseudometric topological semigroup. Many results in S & aacute;nchez and Sanchis (Fuzzy Sets Syst 330:79-86, 2018) are improved. We also study complete weakly invariant fuzzy metrics (in the sense of Kramosil and Mich & aacute;lek) on semigroups.