In mathematics, a signature for an algebraic structure A over a set S is a list of the operations that characterize A, along with their arities. Signatures are employed in model, category, and type theory, and most of all, in universal algebra. Euclid, detail from The School of Athens by Raphael. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... The word operation can mean any of several things: The method, act, process, or effect of using a device or system. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ...

Common notation is to list the operations, separated by commas and enclosed between 〈 and 〉. An operation is a mapping Sⁿ→S, where n, a natural number, is the arity of the operation. Distinguished members of S, such as identity and inverse elements, are treated as operations of arity 0. Common practice is to enclose the arities between another pair of 〈 and 〉, listed in the same order as that of the operations. This second bracketed list makes up the type of A. It is customary to list operations in declining order of arity, but nothing compels this. The signature of an algebra captures much of its essential nature apart from its axioms. A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ...

Example: an additive group is an algebra with signature 〈+,-,0〉 of type 〈2,1,0〉. An additive group is a group, and any group can be written as an additive group, so the adjective additive does not describe a class of groups, but rather the notation used to write the group operation. ...

To allow for external operations, it is assumed that there may be various "kinds," so that each operation also has a "type," namely the cartesian products of kinds that are accepted and returned by the operation. In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...