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text: 'Roughly speaking, an algebraic structure is a set $X$, known as the [3gz underlying set], paired with a collection of [3h7 operations] that obey a given set of laws. For example, a [3gd group] is a set paired with a single binary operation that satisfies the four group axioms, and a [3gq ring] is a set paired with two binary operations that satisfy the ten ring axioms.\n\nIn fact, algebraic structures can have more than one underlying set. Most have only one (including [3h3 monoids], [3gd groups], [3gq rings], [algebraic_field fields], [algebraic_lattice lattices], and [algebraic_arithmetic arithmetics]), and differ in how their associated operations work. More complex algebraic structures (such as [algebraic_algebra algebras], [algebraic_module modules], and [3w0 vector spaces]) have two underlying sets. For example, vector spaces are defined using both an underlying [algebraic_field field] of scalars and an underlying [3h2 commutative group] of vectors.\n\nFor a map of algebraic structures and how they relate to each other, see the [algebraic_structure_tree tree of algebraic structures].',
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