signature

Before you can even write down what a group or a ring is, you need to declare the vocabulary: which symbols will name operations, and how many inputs each one takes. The signature is exactly that declaration — the grammar of an algebraic language, fixed in advance and shared by every structure of the same kind. Two algebras can only be compared, or be homomorphic, if they speak the same signature.

Formally, a signature (also called a type or similarity type) is a set Σ of operation symbols together with an arity function ar: Σ -> N assigning each symbol a non-negative integer. A Σ-algebra is then a carrier A together with, for each symbol f of arity n, an actual operation f^A: A^n -> A interpreting that symbol. Constants are simply the nullary symbols. One often writes a signature as a list like (·, ^{-1}, e) with arities (2, 1, 0) for groups, or (+, ·, -, 0, 1) for rings.

The signature governs everything downstream: terms are built only from these symbols and variables, identities are equations between such terms, and homomorphisms are the maps that commute with every operation in the signature. Choosing the signature is a genuine design decision — recording the group inverse as an operation (rather than asserting its mere existence) is what makes groups an equationally axiomatizable variety rather than merely an elementary class.