The Interaction between Fuzzy Subsets and Groupoids

We discuss properties of a class of real-valued functions on a set X (2) constructed as finite (real) linear combinations of functions denoted as [(X, ∗); μ], where (X, ∗) is a groupoid (binary system) and μ is a fuzzy subset of X and where [(X, ∗); μ](x, y): = μ(x∗y) − min⁡{μ(x), μ(y)}. Many properties, for example, μ being a fuzzy subgroupoid of X, ∗), can be restated as some properties of [(X, ∗); μ]. Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin(X), □) for example.