Terminology for generalized relations - MathOverflow

Terminology for generalized relations

2012-09-21T20:48:26Z

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.

Answer by Shawn Henry for Terminology for generalized relations

2012-09-21T21:35:33Z

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.

Answer by Joel David Hamkins for Terminology for generalized relations

2012-09-21T22:16:43Z

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.