Initial algebra semantics

Summary

Initial algebra semantics gives meaning to syntax through the universal property of an initial algebra. The terms of a signature form the term algebra, which is an initial algebra: it has exactly one homomorphism to every algebra of the signature. Choosing a semantic algebra then fixes the meaning of every term, and the meaning is compositional because the map is a homomorphism. The same idea, with equations added, gives the standard meaning of an algebraic specification and of an abstract data type.