8.2.1 Foundations

Let $A$ be a set.

Definition 8.2.1.1.1 ▶ Symmetric Relations

A relation $R$ on $A$ is symmetric if we have $R^{\dagger }=R$.

Remark 8.2.1.1.2 ▶ Unwinding Definition 8.2.1.1.1

In detail, a relation $R$ is symmetric if it satisfies the following condition:

  • For each $a,b\in A$, if $a\sim _{R}b$, then $b\sim _{R}a$.

Definition 8.2.1.1.3 ▶ The Po/Set of Symmetric Relations on a Set

Let $A$ be a set.

  1. The set of symmetric relations on $A$ is the subset $\smash {\mathrm{Rel}^{\mathrm{symm}}\webleft (A,A\webright )}$ of $\mathrm{Rel}\webleft (A,A\webright )$ spanned by the symmetric relations.
  2. The poset of relations on $A$ is is the subposet $\smash {\mathbf{Rel}^{\mathsf{symm}}\webleft (A,A\webright )}$ of $\mathbf{Rel}\webleft (A,A\webright )$ spanned by the symmetric relations.

Proposition 8.2.1.1.4 ▶ Properties of Symmetric Relations

Let $R$ and $S$ be relations on $A$.

  1. Interaction With Inverses. If $R$ is symmetric, then so is $R^{\dagger }$.
  2. Interaction With Composition. If $R$ and $S$ are symmetric, then so is $S\mathbin {\diamond }R$.

Proof of Proposition 8.2.1.1.4.
Item 1: Interaction With Inverses
Clear.
Item 2: Interaction With Composition
Clear.

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