Weakly partitive families on infinite sets

Given a finite or infinite set S and a positive integer k, a binary structure B of base S and of rank k is a function (S × S) \ {(x, x) : x ∈ S} −→ {0, . . . , k− 1}. A subset X of S is an interval of B if for a, b ∈ X and x ∈ S \X, B(a, x) = B(b, x) and B(x, a) = B(x, b). The family of intervals of B satisfies the following: ∅, B and {x}, where x ∈ B, are intervals of B; for every family F of intervals of B, the intersection of all the elements of F is an interval of B; given intervals X and Y of B, if X ∩ Y 6= ∅, then X ∪ Y is an interval of B; given intervals X and Y of B, if X \ Y 6= ∅, then Y \ X is an interval of B; for every up-directed family F of intervals of B, the union of all the elements of F is an interval of B. Given a set S, a family of subsets of S is weakly partitive if it satisfies the properties above. After suitably characterizing the elements of a weakly partitive family, we propose a new approach to establish the following [6]: given a weakly partitive family I on a set S, there is a binary structure of base S and of rank ≤ 3 whose intervals are exactly the elements of I.