## Thursday, June 7, 2012

### The regularity of relations

The regularity of relations first leads us to functions which are relations with a out degree regularity of one. Functions that also have an in degree regularity are place forms, and if the in degree regularity of a function is one then the function is one-to-one. Since bijections are just special cases of place forms you can apply all place form functions to them:
(setf [reverse coll] '(3 2 1))
;=> (1 2 3)

Functions that cannot be expressed as place forms are in degree irregular. Count is an example of such a function, because there is only one type of empty collection, and there are many other possible collections for the other sizes. Monoids are also irregular functions because there is a different number of partitions for different objects in most monoids, unless you consider the identity.