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  2. Algebraic structure - Wikipedia

    en.wikipedia.org/wiki/Algebraic_structure

    An axiom of an algebraic structure often has the form of an identity, that is, an equation such that the two sides of the equals sign are expressions that involve operations of the algebraic structure and variables. If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true.

  3. Outline of algebraic structures - Wikipedia

    en.wikipedia.org/.../Outline_of_algebraic_structures

    In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a ...

  4. Mathematical structure - Wikipedia

    en.wikipedia.org/wiki/Mathematical_structure

    Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field. A measure: intervals of the real line have a specific length , which can be extended to the Lebesgue measure on many of its subsets .

  5. Abelian variety - Wikipedia

    en.wikipedia.org/wiki/Abelian_variety

    In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry ...

  6. Category:Algebraic structures - Wikipedia

    en.wikipedia.org/wiki/Category:Algebraic_structures

    In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra

  7. Abstract algebra - Wikipedia

    en.wikipedia.org/wiki/Abstract_algebra

    In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. [1] Algebraic structures include groups , rings , fields , modules , vector spaces , lattices , and algebras over a field .

  8. Semigroup - Wikipedia

    en.wikipedia.org/wiki/Semigroup

    Algebraic structures between magmas and groups: A semigroup is a magma with associativity. A monoid is a semigroup with an identity element. In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.

  9. Klein four-group - Wikipedia

    en.wikipedia.org/wiki/Klein_four-group

    V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.

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