Varieties, Free Algebras and Equational Logic
Identities, Satisfaction and Equational Classes
Identities as pairs of terms, what it means for an algebra to satisfy one, and the Galois connection between classes of algebras and sets of identities.
Learning objectives
- Define satisfaction of an identity
- Describe the operators Id and M and the Galois connection they form
- Define equational class and relate it to variety
Identities and satisfaction
A formal expression p ≈ q where p and q are terms over a variable set X.
Satisfaction is implicitly universally quantified. The identity x · y ≈ y · x asserts commutativity for all elements, not the existence of a commuting pair.
In Chapter II the double turnstile relates an algebra to an identity. In Chapter V it relates a structure to an arbitrary first-order sentence. The second generalises the first, but the notation is reused across a large gap in the text.
The Galois connection
- Id<sub><em>K</em></sub>(<em>X</em>)
- the set of identities over X satisfied by every member of K
- <em>M</em>(Σ)
- the class of all algebras satisfying every identity in Σ
These two operators reverse inclusion and form a Galois connection between classes of algebras and sets of identities:
A class of the form M(Σ) for some set of identities Σ. Equivalently, a class closed under the operator MId.
A set of identities of the form Id(K) — equivalently, a set closed under the operator IdM.
The three closure properties
If every member of K satisfies p ≈ q, then so does every homomorphic image, every subalgebra and every direct product of members of K.
| Operator | Reason |
|---|---|
| H | Homomorphisms preserve term operations, so an identity holding upstairs holds in the image |
| S | A subalgebra's term operations are restrictions of the ambient ones; an identity holding on all of A holds on any subset |
| P | Operations act coordinatewise, so an identity holding in every factor holds in the product |
These three facts show every equational class is a variety. The converse — every variety is equational — is the substantial half, and it requires free algebras.
Examples of equational definitions
| Class | Defining identities |
|---|---|
| Commutative semigroups | associativity, xy ≈ yx |
| Bands | associativity, xx ≈ x |
| Abelian groups of exponent n | group axioms, commutativity, xn ≈ e |
| Boolean algebras | lattice axioms, distributivity, complement laws, bounds |
| Nilpotent groups of class 2 | group axioms plus [[x,y],z] ≈ e |
| Rings satisfying x2 ≈ x | ring axioms plus idempotence — the Boolean rings |
Frequently asked questions
Can an equational class be defined by infinitely many identities?
Yes, and sometimes it must be. Whether a finitely axiomatisable class exists is the finite basis problem, and Chapter V devotes a section to when the answer is yes.
Is satisfaction decidable?
For a fixed finite algebra and a given identity, yes — check all assignments. For a variety and an arbitrary identity, it depends: the equational theory of a variety may be undecidable, which is one theme of Chapter V §5.
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.11, book pages 77-79.
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.