Symmetries and functional dependiencies

We can combine the functional dependencies of a set of elements with a group of symmetries of the set in order to construct an advanced system of reasoning about relations. One immediate application of the combination of these two notions is that we can combine symmetry and dependency to find involutions amongst the elements.

The set of relations with certain functional dependencies and symmetries forms a meet subsemilattice of the set of all relations. This allows us to use the combination of symmetries and functional dependencies to reason logically about classes of relations such as functions, bijections, constants, symmetric relations, binary operators, cancellative operations, and commutative operations.