Subalgebra of an F-algebra

Question

In the literature for F-algebras there appears frequently the notion of F-subalgebra. I'm trying to interpret this as some kind of subobject in the category of all F-algebras and F-algebras homomorphisms.

Is my interpretation right? What would be the right way to encode it?

Answer 1

Here is a reference taken from Universal Algebra and Coalgebra from Denecke and Wismath point 5.3:

Let $\mathcal{A} = (A,\beta_A)$ be an $F$-algebra and $S \subseteq A$.

The set $S$ is said to be closed if htere is a mapping $\beta_S:F(S) \to S$ such that the embedding (injection) mapping $i: S \to A$ is a homomorphism.

In this case, $\mathcal{S} = (S,\beta_S)$ is called a subalgebra of $\mathcal{A}$.