Universal Algebra - Basic Constructions

Basic Constructions

We assume that the type, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.

A homomorphism between two algebras A and B is a function h: A → B from the set A to the set B such that, for every operation f_A of A and corresponding f_B of B (of arity, say, n), h(f_A(x₁,...,x_n)) = f_B(h(x₁),...,h(x_n)). (Sometimes the subscripts on f are taken off when it is clear from context which algebra your function is from) For example, if e is a constant (nullary operation), then h(e_A) = e_B. If ~ is a unary operation, then h(~x) = ~h(x). If * is a binary operation, then h(x * y) = h(x) * h(y). And so on. A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under the entry Homomorphism. In particular, we can take the homomorphic image of an algebra, h(A).

A subalgebra of A is a subset of A that is closed under all the operations of A. A product of some set of algebraic structures is the cartesian product of the sets with the operations defined coordinatewise.