Terminology for generalized relations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T08:31:45Z http://mathoverflow.net/feeds/question/107803 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107803/terminology-for-generalized-relations Terminology for generalized relations Bern Oay 2012-09-21T20:48:26Z 2012-09-21T23:21:50Z <p>I have a simple terminology request: recall that given sets $A$ and $B$, a <em>relation</em> $R$ from $A$ to $B$ is any subset of the product $A \times B$. Thus, one may view a relation as a function $A \times B \to \lbrace 0,1 \rbrace$ where $(a,b)$ maps to $1$ if and only if it lies in $R$.</p> <p>What I'm looking for is the suitable adjective to describe the situation where $A \times B$ maps into a more general ordered space, like say $\mathbb{R}^+$. The "relation" in this case is not just a yes/no binary affair, but rather a ranking of some sort.</p> <blockquote> <p>Is there a standard terminology for such a situation?</p> </blockquote> <p>I thought of using <em>ordered</em> relation, but that is dangerous because it causes immediate confusion with <em>order relation</em>. Sorry for the possibly silly question, but I have been searching textbooks and internet for a few days with no luck. It seems likely that someone in set theory or combinatorics has named and used this type of relation before. Thank you for the help.</p> http://mathoverflow.net/questions/107803/terminology-for-generalized-relations/107806#107806 Answer by Shawn Henry for Terminology for generalized relations Shawn Henry 2012-09-21T21:35:33Z 2012-09-21T21:35:33Z <p>I'm not sure if there is a name for situations like $\mathbb{R}^{+}$, but if your ordered space is a Heyting algebra, then it's still just called a relation. </p> <p>The notion of a relation makes sense in any category with finite products, even when the objects don't have an underlying set structure. A relation from $A$ to $B$ is just a subobject of $A\times B$. In a topos, the subobjects of $A\times B$ are in one-to-one correspondence with morphisms from $A\times B$ to the subobject classifier $\Omega$, which is an internal (complete) Heyting algebra.</p> http://mathoverflow.net/questions/107803/terminology-for-generalized-relations/107807#107807 Answer by Joel David Hamkins for Terminology for generalized relations Joel David Hamkins 2012-09-21T22:16:43Z 2012-09-21T23:21:50Z <p>This is called an <em>$L$-valued relation</em>, when $L$ is the target of the function, which can be viewed as the collection of possible truth values. </p> <p>Thus, a $2$-valued relation is just an ordinary relation of classical logic, where every instance has truth value either true or false. But for any Boolean algebra $\mathbb{B}$ we have $\mathbb{B}$-valued relations, which arise throughout forcing, or more generally with a Heyting algebra, or an $[0,1]$-valued relation, as in fuzzy logic. </p>