## Relation Theory • 3

### Relation Theory • Definition

It is convenient to begin with the definition of a $k$-place relation, where $k$ is a positive integer.

Definition.  A $k$-place relation $L \subseteq X_1 \times \ldots \times X_k$ over the nonempty sets $X_1, \ldots, X_k$ is
a $(k+1)$-tuple $(X_1, \ldots, X_k, L)$ where $L$ is a subset of the cartesian product $X_1 \times \ldots \times X_k.$

Several items of terminology are useful in discussing relations.

• The sets $X_1, \ldots, X_k$ are called the domains of the relation $L \subseteq X_1 \times \ldots \times X_k,$ with ${X_j}$ being the $j^\text{th}$ domain.
• If all the ${X_j}$ are the same set $X$ then $L \subseteq X_1 \times \ldots \times X_k$ is more simply described as a
$k$-place relation over $X.$
• The set $L$ is called the graph of the relation $L \subseteq X_1 \times \ldots \times X_k,$ on analogy with the graph of a function.
• If the sequence of sets $X_1, \ldots, X_k$ is constant throughout a given discussion or is otherwise determinate in context then the relation $L \subseteq X_1 \times \ldots \times X_k$ is determined by its graph $L,$ making it acceptable to denote the relation by referring to its graph.
• Other synonyms for the adjective $k$-place are $k$-adic and $k$-ary, all of which leads to the integer $k$ being called the dimension, adicity, or arity of the relation $L.$

### Resources

cc: Category Theory • Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation TheoryLaws of FormPeirce List

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