You are best off finding a representation of your algebra (e.g. if it is associative then maybe you can find a matrix representation), but there is something you can do with generators and relations.

Note that it is the relations part that is tricky. If you never cared if two expressions were equal then you can represent multiplication symbolically e.g. f[x,y] being x times y (or use NonCommutativeMultiply instead of f). You can use ordinary + for addition and at least Plus will simplify as expected. You can also give an Output form for f[x,y] so it displays nicely.

One thing you can try with relations is to use the optional second argument in Simplify. This example comes from the documentation under Applications and it shows that f must be commutative if it satisfies the Wolfram Axiom (which is the difficult step in showing that f has to be a Boolean Algebra).

Simplify[f[a, b] == f[b, a], ForAll[{p, q, r}, f[f[f[p, q], r], f[p, f[f[p, r], p]]] == r]]