---
tag: DJFQWTQG9V
kind: concept
name: McKenzie–Hobby tame congruence theory
msc: [08A05, 08B05]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_algebra
---

## Summary

Tame congruence theory is a structure theory for finite algebras and [[locally finite varieties|8QPW65PZVW]], developed by David Hobby and Ralph McKenzie.
It studies each covering pair $\alpha \prec \beta$ in the [[congruence lattice|REE07CSXHR]] of a finite algebra through the polynomial operations restricted to small subsets, the minimal sets.
On these subsets the algebra behaves in one of five ways, which gives each covering pair a type: unary, affine, Boolean, lattice or semilattice.
The types that occur in a locally finite variety control its congruence lattices and its Mal'cev conditions.
For example, a locally finite variety omits the unary type exactly when it has a [[Taylor term|QEAMRNRJJD]].

## Explanations

### L1 human

A finite algebra can be very complicated, but tame congruence theory looks at it one small step at a time.
Each step goes from one congruence to the next larger one, and the theory finds a small piece of the algebra where this step is visible.
On that piece, the algebra looks like one of five simple kinds of structure: a set with permutations, a vector space, a Boolean algebra, a lattice, or a semilattice.
Knowing which kinds appear tells a great deal about the whole algebra and the variety it generates.