# Terminology for generalized relations

I have a simple terminology request: recall that given sets $A$ and $B$, a relation $R$ 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 $R$.

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.

Is there a standard terminology for such a situation?

I thought of using ordered relation, but that is dangerous because it causes immediate confusion with order relation. 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.

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Technically it looks like a valuation, or a weight function or labelling of the directed edges of an A-B bipartite graph. However, your application might be specific enough to deserve its own name. Gerhard "Boy Or Girl Name? Hmmm." Paseman, 2012.09.21 –  Gerhard Paseman Sep 21 '12 at 21:06
In the case that the range is contained in $[0,1]$ it would be called a fuzzy relation, I think. –  Trevor Wilson Sep 21 '12 at 21:17
Gerhard: Valuation has at least three different meanings (at least one each in logic, measure theory and algebra according to wikipedia), none of which appear to be what the OP wants. In light of your comment, I'd recommend "weighted relation". For bonus points, a map $A \times B \to W$ for any $W$, ordered or otherwise, could be called a $W$-weighted relation. –  Vidit Nanda Sep 21 '12 at 21:30
I believe that the kind of a relation you are looking for is called a labelled transition in computer science literature. –  Eugene Lerman Sep 22 '12 at 12:36

This is called an $L$-valued relation, when $L$ is the target of the function, which can be viewed as the collection of possible truth values.

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.

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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.

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.

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But internally, $\Omega$ is just the analogue of $\{0,1\}$. So I don't think your second paragraph implies your first. In my experience, when speaking externally, Heyting-valued functions aren't usually called "relations" without any qualification. –  Mike Shulman Sep 21 '12 at 22:16
I like Joel's answer better than mine, but I would like to point out that one can sensibly interchange "$L$-valued relation" and "relation in the category of $L$-valued sets". If it's clear that I'm talking about the category of $L$-valued sets, I'm going to say "let $R\rightarrow A\times B$ be a relation", not "let $R$ be an $L$-valued relation". –  Shawn Henry Sep 21 '12 at 22:28