Core Structure Theory
The Definition of an Algebra and its Type
The central definition of the subject: an algebra is a set with a family of finitary operations indexed by a type. Everything that follows is an elaboration of this one idea.
Learning objectives
- State the definition of an algebra and of a type
- Explain the role of arity and why operations must be finitary
- Distinguish an algebra from its underlying universe
The definition
An algebra A is a pair ⟨A, F⟩ where A is a non-empty set, the universe, and F is a family of finitary operations on A.
A type is a set of operation symbols together with an assignment of a non-negative integer arity to each. Two algebras have the same type when they interpret the same symbols with the same arities.
For each operation symbol f of arity n in the type, an algebra A of that type provides an interpretation fA: An → A. Nullary operations, with n = 0, pick out distinguished constants.
What is deliberately absent
The definition imposes no axioms whatever. There is no requirement of associativity, of identity elements, of inverses, or of any relation among the operations. An algebra is raw structure.
- Results proved at this level hold for every algebraic structure simultaneously — the homomorphism theorems, the correspondence theorem, the subdirect representation theorem.
- Axioms are added back as identities, and the study of which classes arise this way is Birkhoff's theorem. Axiom-adding becomes itself a mathematical subject rather than a preliminary.
The empty algebra causes trouble with the class operators and with free algebras on empty generating sets. Excluding it is a convention, not a deep fact; some authors allow it and pay for the choice in extra case analysis.
Finitary arity is not a technicality
Every operation must take finitely many arguments. This single restriction is responsible for a large share of the subject's structure theory:
Infinitary algebras exist and are studied, but they lose algebraicity of the subuniverse and congruence lattices, and with it most of the machinery developed in Chapters II and IV.
Similarity types in practice
| Structure | Operation symbols with arity | Type |
|---|---|---|
| Semigroup | · (2) | ⟨2⟩ |
| Monoid | · (2), e (0) | ⟨2, 0⟩ |
| Group | · (2), −1 (1), e (0) | ⟨2, 1, 0⟩ |
| Ring with unit | + (2), · (2), − (1), 0 (0), 1 (0) | ⟨2, 2, 1, 0, 0⟩ |
| Lattice | ∨ (2), ∧ (2) | ⟨2, 2⟩ |
| Boolean algebra | ∨ (2), ∧ (2), ′ (1), 0 (0), 1 (0) | ⟨2, 2, 1, 0, 0⟩ |
| R-module | + (2), − (1), 0 (0), r· (1) for each r | One unary symbol per ring element |
Groups can be presented with only the binary operation, with inverse and identity existential rather than operational. Universal algebra insists on including them as operations, because only then is the class of groups a variety — closed under subalgebras. With multiplication alone, a subsemigroup of a group need not be a group.
Frequently asked questions
Can two algebras of different types be compared?
Not directly. Homomorphisms, subalgebras and products are all defined only between algebras of the same type. Comparing across types requires either a reduct — forgetting some operations — or a term-based interpretation of one type in another.
Why include inverse as an operation for groups?
So that the class of groups is closed under subalgebras. With only multiplication, the positive integers form a subsemigroup of the integers under addition but not a subgroup. Including inverse as an operation forces subalgebras to be subgroups.
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.1, book pages 25-26.
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.