Mal'cev allegories

Mal'cev allegories

[Peter Freyd]

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.