Help understanding possible world formalism

Question

I am a total novice to philosophy trying to make sense of a computer science article which makes reference to possible world theory. I don't really understand the formalization in the last sentence below (see attached image for proper formatting). This is because I don't understand the word "arity" or the relation suggested by the superscript. It's be great if someone could lend some insight.

Let S be an arbitrary system, D an arbitrary set of distinguished elements of S, and W the set of world states for S (also called worlds, or possible worlds). The tuple is called a domain space for S, as it intuitively fixes the space of variability of the universe of discourse D with respect to the possible states of S. An intensional relation ρ^n of arity n on n is a total function ρ^n : W → 2^D^n from the set W into the set of all n-ary (extensional) relations on D.

Answer 1

The arity of a relation R is the number of its "argument places".

Thus, the relation "less than" is a binary relation: it has arity 2. We usually write: x < y, but we can write it "more formally" as: <(x,y).

An extensional relation of arity n on D is a subset of D^n (i.e. D x D x ... x D: the set of all n-uples).

The set of all extensional relations of arity n is 2^D^n (i.e. the power set of D^n).