Rostock. Math. Kolloq. 63, 55–62 (2008) Subject Classification (AMS) Primary 04A05, 20L13; Secondary 44A35, 46A22 Zoltán Boros, Árpád Száz Reflexivity, Transitivity, Symmetry, and Anti- Symmetry of the Intersection Convolution of Relations ABSTRACT. We give some sufficient conditions in order that the intersection convolution F * G of two relations F and G on a groupoid X be reflexive, transitive, symmetric, and anti-symmetric. Here, F * G is a relation on X such that ( F * G ) (x)= \  F (u)+ G (v): x = u + v, F (u) 6= ∅ , G (v) 6= ∅  for all x ∈ X . KEY WORDS. Groupoids, binary relations, intersection convolution, reflexivity, transitivity, symmetry, and anti-symmetry 1 A few basic facts on relations and groupoids A subset F of a product set X × Y is called a relation on X to Y . If in particular F ⊂ X 2 , then we may simply say that F is a relation on X . Thus, a relation F on X to Y is also a relation on X ∪ Y . If F is a relation on X to Y , then for any x ∈ X and A ⊂ X the sets F (x)= {y ∈ X : (x, y) ∈ F } and F [A]=  a ∈ A F (a) are called the images of x and A under F , respectively. Moreover, the sets D F = {x ∈ X : F (x) 6= ∅} and R F = F [D F ] are called the domain and range of F , respectively. If in particular D F = X , then we say that F is a relation of X to Y , or that F is a total relation on X to Y . Now, a relation F on X is called (1) reflexive if x ∈ F (x) for all x ∈ D F ; (2) symmetric if y ∈ F (x) implies x ∈ F (y) ; (3) transitive if y ∈ F (x) and z ∈ F (y) implies z ∈ F (x) ;