Identity

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 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 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 $1$ .

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Term concept
Term evaluation concept
Identity definition

Tag	`81XXW834SS`
URI	`m2://81XXW834SS`
Kind	definition
Writing	W0 stub
MSC	08B05, 03C05
Defines	Identity
Uses	Term, Term evaluation
Changed	2 November 2026

Identity

Definition

Remarks

Reading path

Further reading

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