Relation Theory • 3

Relation TheoryDefinition

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

This entry was posted in C.S. Peirce, Category Theory, Dyadic Relations, Logic, Logic of Relatives, Logical Graphs, Mathematics, Nominalism, Peirce, Pragmatism, Realism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization and tagged , , , , , , , , , , , , , , , . Bookmark the permalink.

6 Responses to Relation Theory • 3

  1. Pingback: Survey of Relation Theory • 4 | Inquiry Into Inquiry

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