Ian Hodkinson

A shorter proof that wRRA is a variety

The class wRRA of "weakly representable" relation algebras was introduced by Jónsson in 1959. It is the class of all relation algebras that have a representation (an embedding into an algebra of binary relations on some base set) that respects all the relation algebra operations (including boolean meet) except perhaps boolean join and complement - even though these operations are present in the algebra.

In Problem 1 of his 1959 paper, Jónsson asked implicitly whether wRRA is a variety - axiomatisable by equations. (His explicit question concerned algebras without join and complement.) For wRRA, this was answered positively by Pécsi in 2009. His proof used reduced products, and was short. I will give a similar proof, using compactness, that seems to be even shorter.