Core Structure Theory
Principal and Generated Congruences
The congruence generated by a set of pairs, the principal congruences generated by a single pair, and the reason principal congruences are the compact building blocks of Con(A).
Learning objectives
- Define Θ(X) and Θ(a, b) and describe their elements
- Explain Mal'cev's description of generated congruences
- Connect principal congruences to compactness in Con(A)
Generated congruences
For a set X of pairs from A × A, Θ(X) is the smallest congruence on A containing X — the intersection of all congruences containing X.
Θ(a, b) is the congruence generated by the single pair ⟨a, b⟩.
Every congruence is the join of the principal congruences it contains: θ = ⋁{Θ(a, b) : ⟨a, b⟩ ∈ θ}.
Mal'cev's description
⟨c, d⟩ lies in Θ(a, b) if and only if there is a finite sequence c = e0, e1, …, en = d and unary polynomial functions p1,…,pn of A such that each consecutive pair {ei−1, ei} equals {pi(a), pi(b)}.
The description is constructive and is the standard tool for computing principal congruences. It says: to relate c to d, chain together translates of the original pair by unary polynomials.
Each chain is finite. That is what makes Θ an algebraic closure operator and hence makes principal congruences compact in Con(A). The compactness of principal congruences is used throughout Chapter V, particularly in the analysis of principal congruence formulas.
Principal congruences as compact elements
The compact elements of Con A are exactly the finite joins Θ(a1, b1) ∨ … ∨ Θ(an, bn) of principal congruences.
In particular each principal congruence is compact. This is the algebraicity of Con(A) stated at the level of generators.
Computing principal congruences
| Variety | Θ(<em>a</em>, <em>b</em>) corresponds to |
|---|---|
| Group | The normal subgroup generated by ab−1 |
| Ring | The two-sided ideal generated by a − b |
| R-module | The submodule generated by a − b |
| Boolean algebra | The filter generated by (a ∧ b) ∨ (a′ ∧ b′) |
| Lattice | No such reduction; computed by Mal'cev chains |
In congruence-permutable varieties the Mal'cev chains collapse to length one, which is why groups, rings and modules admit the closed-form descriptions above. Lattices are not permutable, so their principal congruences genuinely require the chain construction.
Definability and Chapter V
A variety has definable principal congruences when membership in Θ(a, b) is expressible by a single first-order formula, uniformly across the variety — equivalently, when the Mal'cev chains can be bounded in length.
Chapter V §3 develops principal congruence formulas for exactly this purpose, and the bounded-chain condition is what drives Baker's finite basis theorem in §4.
Frequently asked questions
Are principal congruences always small?
No. A principal congruence can be all of ∇ — this happens precisely when the algebra is simple and a ≠ b. 'Principal' refers to being generated by one pair, not to being small.
What is a unary polynomial function?
A function of one variable obtained from a term by substituting fixed elements of the algebra for all but one variable. Polynomials differ from terms exactly in allowing parameters from the algebra.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section II.5, book pages 41-44.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.