Sam Staton asks about relations with the property: If x R y and x' R y and x' R y' then x R y'. Given an equational theory all relations in its category of models satisfy this property iff there's a Mal'cev operator,(a favorite topic among "universal algebraists"). Years ago I used the phrase Mal'cev property (MP) to mean that all relations in an allegory satisfy the condition. Some easy lemmas: MP implies all reflexive relations are symmetric (RIS). RIS implies equivalence relations commute (ERC). MP implies all reflexive relations are transitive (RIT). RIT implies ERC. ERC implies that the smallest equivalence relation containing a given pair of equivalence relations is their composition and that easily implies that the lattice of equivalence relations on any object is a modular lattice. ERC does not imply MP (there are simple examples for RIS not implying RIT and RIT not implying RIS). But in an allegory in which every relation is spanned by a pair of maps (in particular, in the calculus of relations arising from any regular category) it's easy to see that ERC does implies MP. For the record: txyz is defined to be a Mal'cev operator if it satisfies the two equations txxz = z txzz = x. In any theory that includes the theory of groups xy^{-1}z is such. For Heyting algebras take txyz = ((x -> y) -> z) ^ ((z -> y) -> z). That generalizes to a one-object division allegory: tPQR = (R/(1 ^ (R\Q))) ^ (R/(1 ^ (P\Q))). The provably simplest Mal'cev theory has one binary operation x*y and one equation x*(y*x) = y (e.g. in the presence of a group structure x*y = x^{-1}y^{-1}). Take txyz = (x*x)*(z*(x*y)). There are another 23 Mal'cev terms of the same size. If one weakens the theory of groups to the theory of quasigroups: that is, three binary relations and four equations (x/y)y = x x(x\y) = y (xy)/y = x x\(xy) = y then txyz = (x/x)\((x/y)z) is a Mal'cev operator. If we stick to terms of the same size there are 72 versions. Heavenly. But this one uses only the first and fourth equation (and, consequently, its mirror image uses only the second and third equations). The fact that the existence of a Mal'cev operator implies that congruence lattices are modular was well known by universal algebraists.