One definition for all of them
In Volume I you proved theorems about groups, rings and fields one structure at a time. Universal algebra is the observation that these proofs often share a skeleton, and that the skeleton can be studied directly. An algebra is a nonempty set A together with a collection of operations, each of a fixed arity n (a function A^n → A). A nullary operation (arity 0) is just a chosen constant. The list of operation symbols together with their arities is called the signature (or type).
A group, in this language, is an algebra of signature (·, ⁻¹, e) with arities (2, 1, 0): one binary multiplication, one unary inverse, one nullary identity. A lattice has signature (∨, ∧), both binary. A ring has (+, ·, −, 0) with arities (2, 2, 1, 0). The point is not novelty — it is that once you fix the signature, the notions of substructure, homomorphism and quotient are forced on you, identically, for every structure of that type.
Terms and identities
Given a signature and a set of variables X, a term is any well-formed expression built from variables and operation symbols respecting arities. Terms are syntax; they form the free object we will call the term algebra. An identity is a formal equation s ≈ t between two terms, and an algebra satisfies it when the equation holds for every assignment of elements to the variables. Associativity x·(y·z) ≈ (x·y)·z is an identity; so is the absorption law x ∨ (x ∧ y) ≈ x of lattices.
Why insist that axioms be identities rather than arbitrary first-order sentences? Because identities are preserved by all the constructions we care about — subalgebras, products, homomorphic images. The class of fields is *not* an equational class (x ≠ 0 → x·x⁻¹ = 1 uses an implication and inequality), which is exactly why a subring of a field need not be a field, and why a product of fields is not a field. Equationally axiomatised classes behave far better, and characterising them is Birkhoff's theorem (Guide 4).
Signature of a group: F = { · (arity 2), inv (arity 1), e (arity 0) }
Three terms in variables x, y, z:
t1 = ·(x, ·(y, z)) -- usually written x(yz)
t2 = ·(·(x, y), z) -- usually written (xy)z
t3 = ·(x, inv(x)) -- usually written x x^{-1}
Group axioms as identities (s approx t):
associativity: ·(x, ·(y,z)) approx ·(·(x,y), z)
right identity: ·(x, e) approx x
right inverse: ·(x, inv(x)) approx e
Check in A = Z/6Z under addition (so '·' is +, 'inv' is negation, e = 0):
assign x = 5. Then ·(x, inv(x)) = 5 + (-5) = 5 + 1 = 6 = 0 = e. OK
The identity x x^{-1} approx e holds for ALL x in Z/6Z, hence Z/6Z
satisfies it. A single failing assignment would refute the identity.What carries over from Volume I
- A subalgebra is a subset closed under all operations (including constants, so it is never empty). For groups this recovers subgroups; for rings, subrings containing 1 when 1 is a nullary operation.
- A homomorphism f: A → B is a map preserving every operation: f(ω(a₁,…,aₙ)) = ω(f(a₁),…,f(aₙ)) for each operation symbol ω. This single line subsumes group, ring and lattice homomorphisms.
- The image of a homomorphism is a subalgebra of B, and the composite of homomorphisms is a homomorphism — both proved once, in the abstract, never again.