Core Structure Theory
Subalgebras and Algebra Isomorphism
Subalgebras as subsets closed under the operations, the notion of embedding, and isomorphism as the equivalence under which algebras are classified.
Learning objectives
- Define subuniverse, subalgebra and embedding
- Test whether a subset is a subuniverse
- Distinguish isomorphism from equality of algebras
Subuniverses and subalgebras
A subset B of the universe of an algebra A is a subuniverse if it is closed under every basic operation: for each n-ary f in the type and all b1,…,bn in B, the element fA(b1,…,bn) lies in B.
If B is a non-empty subuniverse of A, then B = ⟨B, FB⟩ with operations restricted from A is a subalgebra of A.
If the type contains any constants, every subuniverse must contain them, so the empty set is not a subuniverse. If the type has no constants, the empty set is a subuniverse but does not yield an algebra, since algebras have non-empty universes. This is the standard reason for the subuniverse/subalgebra terminological split.
The closure test
Checking whether a subset is a subuniverse is mechanical: apply each basic operation to every tuple from the subset and verify membership. Two shortcuts are worth knowing.
Basic operations suffice
Closure under the basic operations automatically gives closure under all term operations, by induction on term structure. There is no need to check derived operations.
Intersections are free
Any intersection of subuniverses is a subuniverse, so subuniverses form a closure system and Sub(A) is a complete lattice.
| Algebra | Subuniverses are |
|---|---|
| Group ⟨G, ·, −1, e⟩ | Subgroups |
| Ring with unit | Subrings containing 1 |
| R-module | Submodules |
| Lattice | Sublattices |
| Semigroup | Subsemigroups |
| Boolean algebra | Subalgebras containing 0 and 1 |
Embeddings and isomorphisms
An injective homomorphism. Its image is a subuniverse, and the map is an isomorphism onto the corresponding subalgebra.
A bijective homomorphism. Algebras A and B are isomorphic, written A ≅ B, if such a map exists.
The inverse of an isomorphism is automatically a homomorphism, so isomorphism is a genuine equivalence relation on any set of algebras of a fixed type.
Everything expressible in terms of the operations: the subalgebra lattice, the congruence lattice, satisfaction of every identity and indeed every first-order sentence. Isomorphic algebras are indistinguishable by the methods of the subject, which is why classification is always up to isomorphism.
The operator I
The class operator I takes a class K to the class of all algebras isomorphic to a member of K. It is the least interesting of the class operators but is included because it makes statements about the others precise.
A class K is said to be abstract if I(K) = K — closed under isomorphism. Every class arising naturally in the subject is abstract, and I is often absorbed silently into the other operators. It appears explicitly in identities such as SP ≤ PS, where keeping track of isomorphic copies matters.
Frequently asked questions
Is every subset closed under the operations a subalgebra?
It is a subuniverse; it is a subalgebra provided it is non-empty. When the type contains constants the distinction evaporates, since every subuniverse then contains those constants.
Can two non-isomorphic algebras have isomorphic congruence lattices?
Easily. The congruence lattice is a coarse invariant — every simple algebra has the two-element congruence lattice, and simple algebras exist in enormous variety.
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.2, book pages 31-32.
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.