What to call substructures in universal algebra in which we restrict the signature? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:02:17Z http://mathoverflow.net/feeds/question/95925 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95925/what-to-call-substructures-in-universal-algebra-in-which-we-restrict-the-signatur What to call substructures in universal algebra in which we restrict the signature? Andrej Bauer 2012-05-03T22:15:09Z 2012-05-03T22:28:45Z <p>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.</p> <p>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.</p> <p>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\$.</p> http://mathoverflow.net/questions/95925/what-to-call-substructures-in-universal-algebra-in-which-we-restrict-the-signatur/95927#95927 Answer by Joel David Hamkins for What to call substructures in universal algebra in which we restrict the signature? Joel David Hamkins 2012-05-03T22:19:27Z 2012-05-03T22:28:45Z <p>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 <em>reduct</em> 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 <em>reduct substructure</em> in exactly this situation, but I haven't seen this terminology elsewhere and I don't think there is an established terminology.</p>