Relation Theory • 6

Posted on November 4, 2021 by Jon Awbrey

Relation Theory • Species of Dyadic Relations

Returning to 2-adic relations, it is useful to describe several familiar classes of objects in terms of their local and numerical incidence properties.  Let L \subseteq S \times T be an arbitrary 2-adic relation.  The following properties of L can be defined.

Dyadic Relations • Total • Tubular

If L \subseteq S \times T is tubular at S then L is called a partial function or a prefunction from S to T.  This is sometimes indicated by giving L an alternate name, for example, {}^{\backprime\backprime} p {}^{\prime\prime}, and writing L = p : S \rightharpoonup T.  Thus we have the following definition.

\begin{matrix} L & = & p : S \rightharpoonup T & \text{if and only if} & L & \text{is} & \text{tubular} & \text{at}~ S. \end{matrix}

If L is a prefunction p : S \rightharpoonup T which happens to be total at S, then L is called a function from S to T, indicated by writing L = f : S \to T.  To say a relation L \subseteq S \times T is totally tubular at S is to say it is 1-regular at S.  Thus, we may formalize the following definition.

\begin{matrix} L & = & f : S \to T & \text{if and only if} & L & \text{is} & 1\text{-regular} & \text{at}~ S. \end{matrix}

In the case of a function f : S \to T, we have the following additional definitions.

Dyadic Relations • Surjective, Injective, Bijective

Resources

cc: Category Theory • Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation Theory • Laws of Form • Peirce 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.

4 Responses to Relation Theory • 6

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

  2. Pingback: Survey of Relation Theory • 5 | Inquiry Into Inquiry

  3. Pingback: Survey of Relation Theory • 2 | Inquiry Into Inquiry

  4. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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