"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation.
The 2×2-matrices over the reals R form a four-dimensional unital algebra M(2,R) in the obvious way.
These more general algebras are usually called structures, and they are studied in model theory and in theoretical computer science.
For structures with relations there are notions of weak and of induced substructures.
(Inversion and unit are needed to get the right notions of homomorphism and so that the group laws can be expressed as equations.)