---
tag: 81XXW834SS
kind: definition
name: Identity
msc: [08B05, 03C05]
defines: PKD440G2XW
uses: [6D3WA7N2W4, DH30WTA26T]
sources:
  - url: https://en.wikipedia.org/wiki/Identity_(mathematics)
---

## Definition

Let $\sigma$ be a signature with only function symbols and let $X$ be a set of variables.
An *identity*, or *equation*, over $\sigma$ and $X$ is an ordered pair $(s, t)$ of [[terms|6D3WA7N2W4]] over $\sigma$ and $X$, written $s \approx t$.
An algebra $A$ of signature $\sigma$ *satisfies* $s \approx t$, written $A \models s \approx t$, if $s^A[\alpha] = t^A[\alpha]$ for every map $\alpha \colon X \to A$, where $t^A[\alpha]$ is the [[value|DH30WTA26T]] of $t$.
A class $\mathcal{K}$ of algebras satisfies $s \approx t$ if every member of $\mathcal{K}$ does.

## Remarks

The symbol $\approx$ separates the formal pair from the equality $=$ of elements.
As a first-order sentence, $s \approx t$ stands for the universal closure $\forall x_1 \cdots \forall x_n \, (s = t)$, where $x_1, \ldots, x_n$ are the variables of $s$ and $t$.

Example: the group $(\Z, +, -, 0)$ satisfies $x + y \approx y + x$, since $a + b = b + a$ for all integers $a, b$.

Non-example: the same group does not satisfy $x + x \approx x$.
At $\alpha(x) = 1$ the two sides take the values $2$ and $1$.