grounded relation

A grounded relation over a sequence of sets is a mathematical object consisting of two components. The first component is a subset of the cartesian product taken over the given sequence of sets, which sets are called the domains of the relation. The second component is just the cartesian product itself.

For example, if $L$ is a grounded relation over a finite sequence of sets, $X_1,\mathrm{\dots },X_k$, then $L$ has the form $L=(F(L),G(L))$, where $F(L)\subseteq G(L)=X_1\times \mathrm{\dots }\times X_k$.

1 Remarks

• In various language that is used, $F(L)$ may be called the figure or the graph of $L$, while $G(L)$ may be called the ground of $L$.

• The default assumption in almost all applications is that the domains of the grounded relation are nonempty sets, hence departures from this assumption need to be noted explicitly.

• In many applications all relations are considered relative to explicitly specified grounds. In these settings it is conventional to refer to grounded relations somewhat more simply as “relations”.

• One often hears or reads the usage when the speaker or writer really means . Be charitable in your interpretations.

• The cardinality of $G(L)$ is referred to as the adicity or the arity of the relation. For example, in the finite case, $L$ may be described as $k$-adic or $k$-ary.

• The set ${\mathrm{dom}}_j(L):=X_j$ is referred to as the $j^{\text{th}}$ domain of the relation.

• In the special case where $k=2$, the set $X_1$ is called “the domain” and the set $X_2$ is called “the codomain” of the relation.