Question

Suppose $\Sigma$ is a signature in the sense of universal algebra and $\Sigma' \subseteq \Sigma$ a sub-signature. Every $\Sigma$-algebra is also a $\Sigma'$-algebra in a forgetful way. Suppose $A$ is a $\Sigma$-algebra and $B \subseteq A$ is a $\Sigma'$-subalgebra of $A$ viewed as a $\Sigma'$-algebra. Is there an accepted phrase which describes the relationship of $B$ to $A$? For example, we might say that $B$ is a $\Sigma'$-restriction of $A$, or something like that? It seems wrong to use the word "subalgebra" in this context.

Here is an example: the semiring of natural numbers $\mathbb{N}$ is contained in the ring of real numbers $\mathbb{R}$. This makes $\mathbb{N}$ a what of $\mathbb{R}$? A subsemiring? What is the general phrase? A $\Sigma'$-subalgebra? I would prefer a word which does not refer to the signature explicitly.

The concrete example which I need this for is when $A$ is the $\Sigma$-algebra freely generated by a set of generators $X$ and $B$ is the free $\Sigma'$-aglebr freely generated by the same set of generators $X$.

Answer 1

If the domain of $B$ is the same as $A$, but you only forget the interpretation of the extra language elements, then $B$ is called a reduct of $A$ to signature $\Sigma'$. But you don't merely have a reduct, since you are taking a substructure in the smaller language. Thus, what you have is that $B$ is a substructure of the reduct of $A$ to $\Sigma'$. Having needed this concept in a recent article, I used the term reduct substructure in exactly this situation, but I haven't seen this terminology elsewhere and I don't think there is an established terminology.