What to call substructures in universal algebra in which we restrict the signature?

2012-05-03T22:15:09Z

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 by Joel David Hamkins for What to call substructures in universal algebra in which we restrict the signature?

2012-05-03T22:19:27Z

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.

What to call substructures in universal algebra in which we restrict the signature? - MathOverflow

What to call substructures in universal algebra in which we restrict the signature?

Answer by Joel David Hamkins for What to call substructures in universal algebra in which we restrict the signature?