# Definition:Relational Loop

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This page is about Relational Loop. For other uses, see Loop.

## Definition

Let $\RR$ be a relation on a set $S$.

Let $a_1, a_2, \ldots a_n$ be elements of $S$.

A relational loop on $S$ takes the form:

$\tuple {a_1 \mathrel \RR a_2 \land a_2 \mathrel \RR a_3 \dots \land a_{n - 1} \mathrel \RR a_n \land a_n \mathrel \RR a_1}$

That is, it is a subset of $\RR$ of the form:

$\set {\tuple {a_1, a_2}, \tuple {a_2, a_3}, \ldots, \tuple {a_{n - 1}, a_n}, \tuple {a_n, a_1} }$