Fuzzy pogroupoids

In this paper we show that, for given a pogroupoid (X, ·), the associated poset (X, ⩽) is (C2 + 1)-free if and only if the relation ▷μ is transitive for any fuzzy subset μ of X. Also we determine the set C(X, ·) of fuzzy subsets μ such that μ(x · y) = μ(y · x) for all x, y ∈ X. Furthermore, with a given finite poset (X, ⩽) or the associated pogroupoid (X, ·) we may then associate a polynomial whose coefficients count the number of congruence classes mod(X,  ·) of fuzzy subsets of X along with another polynomial invariant of interest.