---
tag: HJ1159KYP3
kind: definition
name: Taylor term
msc: [08B05, 08A70]
defines: QEAMRNRJJD
uses: [AJNZJD8442, N9M2N27END, PKD440G2XW, 9FS1JFZSX4, 9879ZMW5AA, Y203PPTS61]
sources:
  - url: https://en.wikipedia.org/wiki/Constraint_satisfaction_problem
---

## Definition

Let $\mathcal{V}$ be a [[variety of algebras|AJNZJD8442]] and let $n \ge 2$.
A *Taylor term* for $\mathcal{V}$ is an $n$-ary [[term|N9M2N27END]] $t(x_1, \dots, x_n)$ such that:

1. $t$ is [[idempotent|9FS1JFZSX4]], that is, $\mathcal{V}$ satisfies $t(x, x, \dots, x) \approx x$.
2. For each $i \in \set{1, \dots, n}$, $\mathcal{V}$ satisfies an [[identity|PKD440G2XW]] $t(u_1, \dots, u_n) \approx t(v_1, \dots, v_n)$ with $u_i = x$ and $v_i = y$.
   Here every $u_j$ and every $v_j$ is one of the two variables $x$ and $y$.

An algebra has a Taylor term if the variety it generates has one.

## Remarks

Example: a semilattice term $x \wedge y$ is a Taylor term of the variety of semilattices.
Check: $x \wedge x \approx x$, and the identity $x \wedge y \approx y \wedge x$ has $x$ on the left and $y$ on the right in place $1$.
Swapping the names of $x$ and $y$ in it gives the required identity for place $2$.
Non-example: in the variety of sets with no operations, the term $x_1$ is idempotent but fails clause 2 for $i = 1$.
An identity with $x_1$ on both sides reads $u_1 \approx v_1$, which is false for $u_1 = x$ and $v_1 = y$.
Every [[majority term|9879ZMW5AA]] and every [[Mal'cev term|Y203PPTS61]] is a Taylor term.
For a majority term $m$, the identity $m(x, x, y) \approx m(y, x, x)$ serves places $1$ and $3$, and $m(x, y, x) \approx m(x, x, y)$ handles place $2$ after renaming.