Partial function systems

Families of partial functions can be used to represent many of the most important algebraic structures: given a standard of keys (like 0,1,2) that are reused partial function systems can be used to represent the n-ary relations of database tables. Given a single output value (like true) a partial function system can be used to represent a set system by associating each partial function with its set of keys.

Besides relations partial functions can be used to represent all transformation systems including systems of partial symmetries, transformation monoids, and transformation groups by associating the system of partial functions with some compositional properties such as closure and associativity. Partial function systems are therefore one of the most generally applicable structures available to use in modern mathematics.