Core Universal Algebra
Algebras, Types and Signatures
Fix the type before arguing about the theorem. Almost every apparent counterexample in elementary universal algebra is a disagreement about which operations were in the signature.
- Define a type and an algebra of that type precisely.
- Explain how nullary operations change the subalgebra lattice.
- Present groups, rings and lattices as algebras in explicit types.
- Predict how a change of type alters homomorphisms and subalgebras.
- Recognise when two structures are not comparable because their types differ.
01Type, signature, arity
A type (or signature) is a set F of operation symbols together with an arity function assigning a natural number to each. An algebra of type F is a non-empty set A together with, for each n-ary symbol f, a concrete operation fA mapping An to A.
Note what is absent: axioms. An algebra of type (2) is any set with any binary operation whatsoever. Semigroups, groupoids and quasigroups all live in that type, distinguished by the equations they satisfy, not by their type.
02Standard structures as algebras
| Structure | Type | Operations |
|---|---|---|
| Groupoid | (2) | one binary |
| Semigroup | (2) | one binary, associative |
| Monoid | (2, 0) | binary and identity constant |
| Group | (2, 1, 0) | product, inverse, identity |
| Ring with unit | (2, 2, 1, 0, 0) | +, ·, −, 0, 1 |
| Lattice | (2, 2) | join and meet |
| Bounded lattice | (2, 2, 0, 0) | join, meet, 0, 1 |
| Boolean algebra | (2, 2, 1, 0, 0) | ∨, ∧, ′, 0, 1 |
| R-module | (2, 1, 0) + unary for each r ∈ R | addition, negation, zero, scalars |
The module case is worth noting: scalar multiplication by a fixed ring element is a unary operation, so a module over a ring with infinitely many elements has infinitely many operations in its type. Types may be infinite; only individual operations must be finitary.
03Why nullary operations change everything
A nullary operation is a constant, and constants must be preserved by homomorphisms and contained in subalgebras. Whether a distinguished element sits in the type or is merely guaranteed by an axiom is therefore a substantive choice.
When a claim about subalgebras or homomorphisms seems to fail, the first check is the type. A great deal of confusion about whether rings form a variety, whether the empty set is a subalgebra, and whether monoid homomorphisms preserve the identity dissolves once the signature is written down explicitly.
04The empty algebra question
The source requires the underlying set of an algebra to be non-empty, and the convention has consequences worth being explicit about.
- With at least one constantThe question does not arise: any subuniverse contains the constants, so no subuniverse is empty and non-emptiness is automatic.
- With no constantsThe empty set is closed under the operations vacuously, so it would be a subuniverse if empty algebras were permitted. Excluding it keeps statements uniform but costs some closure properties.
- Practical ruleFollow the source's convention and state it. Results about subuniverse lattices differ between conventions in the bottom element only, but that is enough to make cross-text comparisons go wrong.
05Term operations and polynomial operations
Two derived notions appear immediately and are easy to conflate. A term operation is one built from the basic operations and variables alone. A polynomial operation additionally allows elements of the algebra to be substituted as constants.
| Built from | Preserved by | |
|---|---|---|
| Term operation | basic operations, variables | all homomorphisms and subalgebras |
| Polynomial operation | basic operations, variables, elements of A | congruences, but not homomorphisms in general |
The distinction is load-bearing later. Congruences are exactly the equivalence relations compatible with all polynomial operations, and the theory of functional completeness and primality is stated in terms of which functions are polynomial or term operations. Getting the two confused invalidates those arguments.
Frequently asked
Can an algebra have infinitely many operations?
Yes. The type may be of any cardinality; only each individual operation must have finite arity. Modules over an infinite ring are the standard example, with one unary scalar operation per ring element. What is not permitted is an infinitary operation — a map from Aω to A — because the whole theory of finitary closure and algebraic lattices depends on finite arity.
Is a field an algebra in this sense?
Not conveniently. Multiplicative inverse is undefined at zero, so it is not a total operation, and fields are not closed under direct products — a product of two fields has zero divisors. The class of fields is therefore not a variety and is handled by model-theoretic rather than equational methods. This is a genuine limitation of the framework, not an oversight.
Why require finite arity?
Because finitariness is what makes generated subuniverses depend on finitely many generators, which makes the subuniverse closure operator finitary, which makes Sub(A) algebraic. Nearly every structural theorem in the subject traces back to this. Infinitary algebras exist as a study but form a different and much less tractable theory.
- S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
- G. Grätzer, Universal Algebra, 2nd edition, Springer.
- R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.
Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.