---
tag: KEXKKBDXR3
kind: definition
name: Equational basis
msc: [08B05]
defines: 9Q008GMZSW
uses: [PKD440G2XW, F1M51B4MDG, AJNZJD8442]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Definition

Let $\sigma$ be a signature with only function symbols and let $\mathcal{K}$ be a class of algebras of signature $\sigma$, with [[equational theory|F1M51B4MDG]] $\operatorname{Id}(\mathcal{K})$.
A set $\Sigma$ of [[identities|PKD440G2XW]] over $\sigma$ is an *equational basis* of $\mathcal{K}$ if the following hold:

1. $\Sigma \subseteq \operatorname{Id}(\mathcal{K})$;
2. every identity in $\operatorname{Id}(\mathcal{K})$ is satisfied by every algebra that satisfies all identities of $\Sigma$.

An equational basis of a single algebra $A$ is an equational basis of $\set{A}$.

## Remarks

Equivalently, the class of all algebras satisfying $\Sigma$ is the [[variety|AJNZJD8442]] generated by $\mathcal{K}$.
By Birkhoff's completeness theorem, clause 2 can be read syntactically: every identity of $\mathcal{K}$ is derivable from $\Sigma$ in equational logic.
The whole theory $\operatorname{Id}(\mathcal{K})$ is always an equational basis of $\mathcal{K}$, so the interest lies in small bases.

Example: in the signature $(\cdot, {}^{-1}, e)$, the set $\set{x(yz) \approx (xy)z, \ x e \approx x, \ x x^{-1} \approx e}$ is an equational basis of the class of groups.
Clause 1 holds because every group satisfies these laws, and clause 2 holds because every algebra satisfying them is a group.

Non-example: in the same signature, $\set{x(yz) \approx (xy)z}$ is not an equational basis of the class of groups, since clause 2 fails.
Let $A = \set{0, 1}$ with $a \cdot b = 0$ for all $a, b$, with $a^{-1} = a$, and with $e = 1$.
Then $A$ satisfies associativity, since all products are $0$, but $1 \cdot e = 0 \ne 1$, so $A$ does not satisfy $x e \approx x$.