subalgebra

A subalgebra is a piece of an algebra that is self-contained: you can apply the operations inside it and never fall out. Just as a subgroup is a subset closed under the group product and inverses, and a subring is closed under addition and multiplication, a subalgebra is a subset closed under every operation of the parent — a smaller world running on the same rules.

Formally, let A be a Σ-algebra. A non-empty subset B of A is a subalgebra if for every operation symbol f of arity n and all b_1, ..., b_n in B, the element f^A(b_1, ..., b_n) again lies in B; in particular B must contain every constant. Then B, with the restricted operations, is itself a Σ-algebra and the inclusion B -> A is a homomorphism. The subalgebra generated by a subset X is the smallest subalgebra containing X, equivalently the set of all values of terms with arguments in X.

A subtle but important point: closure must be checked against the operations actually in the signature. A subset of a group closed under multiplication but not under inverse is a sub-semigroup, not a subgroup — so whether something is a subalgebra depends on which operations you declared. The subalgebras of a fixed algebra, ordered by inclusion, form a complete lattice, a basic example linking universal algebra to lattice theory.