Graphical algebras — a new approach to congruence lattices

In 1970, H. Werner considered the question of which sublattices of partition lattices are congruence lattices for an algebra on the underlying set of the partition lattices. He showed that a complete sublattice of a partition lattice is a congruence lattice if and only if it is closed under a new operation called graphical composition. We study the properties of this new operation, viewed as an operation on an abstract lattice. We obtain some necessary properties, and we also obtain some sufficient conditions for an operation on an abstract lattice L to be this operation on a congruence lattice isomorphic to L. We use this result to give a new proof of Grätzer and Schmidt’s result that any algebraic lattice occurs as a congruence lattice.     

The General Theory of Diads

A diad is a generalisation of a monad and a comonad. The idea is that we ignore the unit or counit, and consider only the natural transformations between T and T ². It turns out that almost all the constructions that we form for a monad or comonad can also be constructed from a related diad. Diads were introduced in Kenney (Appl. Categ. Structures, 2008), where they give a generalisation of the results that the category of coalgebras for a finite-limit preserving comonad on a topos is another topos, and that the category of algebras for a finite-limit preserving idempotent monad on a topos is another topos. In that paper, we were only interested in a special class of diads called codistributive diads, and we considered only the part of the theory of diads necessary to prove the result about finite-limit preserving diads in topoi. Here, we will study general diads in greater detail. We will develop the general theory with constructions that extend the standard constructions for monads and comonads.