---
tag: 5W801R7ZF5
kind: definition
name: Free spectrum of a variety
msc: [08B20]
defines: W0RYRVJY5Z
uses: [GTS4DRRMZR, AJNZJD8442, 8QPW65PZVW]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Definition

Let $\mathcal{V}$ be a [[variety|AJNZJD8442]], and for each natural number $n$ let $F_{\mathcal{V}}(n)$ be the [[free algebra|GTS4DRRMZR]] of $\mathcal{V}$ on a set of $n$ generators.
The *free spectrum* of $\mathcal{V}$ is the function
$$f_{\mathcal{V}}(n) = \abs{F_{\mathcal{V}}(n)}, \qquad n \in \N,$$
with values in $\N \cup \set{\infty}$.

## Remarks

The values are all finite exactly when $\mathcal{V}$ is a [[locally finite variety|8QPW65PZVW]].
When $\mathcal{V}$ is generated by a finite algebra $A$, the free algebra $F_{\mathcal{V}}(n)$ embeds in $A^{A^n}$, so $f_{\mathcal{V}}(n) \le \abs{A}^{\abs{A}^n}$.
Example: for Boolean algebras, $f(n) = 2^{2^n}$, since $F(n)$ is the algebra of all Boolean functions $\set{0, 1}^n \to \set{0, 1}$.
Example: for semilattices, $f(n) = 2^n - 1$, since $F(n)$ is the set of nonempty subsets of the generators under union.
Non-example: the variety of all groups has no finite spectrum value for $n \ge 1$, since the free group on one generator is $\Z$.
Some authors start the spectrum at $n = 1$, or use the related function $\log_2 f_{\mathcal{V}}(n)$.