Addition and multiplication of real numbers are the prototypical examples of operations that combine two elements of a set to produce a third element of the set. These operations obey several algebraic laws. For example, a + (b +

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One set with operations
Simple structures: no binary operation:
• Set: a degenerate algebraic structure S having no operations.
• Pointed set: S has one or more distinguished

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Algebraic structures can also coexist with added structure of non-algebraic nature, such as partial order or a topology. The added structure must be compatible, in some sense, with the algebraic structure.
• Topological group: a group with a topology compatible with

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Algebraic structures are defined through different configurations of axioms. Universal algebraabstractly studies such objects. One major

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Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of objects with associated morphisms. Every algebraic structure has its own notion of homomorphism, namely any functioncompatible

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In a slight abuse of notation, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring structure on the set ," means that we have defined ring

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• Free object
• List of algebraic structures
• Mathematical structure
• Outline of algebraic structures

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