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{{Algebraic structures}} In [[mathematics]], an '''algebraic structure''' or '''algebraic system'''<ref>{{Cite book |last=F.-V. Kuhlmann (originator) |title=Encyclop ...

...tions of [[group (mathematics)|groups]] that are built on more complicated structures than [[Set (mathematics)|sets]]. A typical example of a group object is a [ ...in the category of [[algebraic variety|algebraic varieties]]. In modern [[algebraic geometry]], one considers the more general [[group scheme]]s, group objects ...

...far more general: they are a common generalisation of ''all'' [[algebraic structures]]. "Subalgebra" can refer to either case. ...algebraic operations are restricted to ''S''. If the axioms of a kind of [[algebraic structure]] is described by [[variety (universal algebra)|equational laws]] ...

==Structures and operations== * [[Linear algebraic group]] ...

...gebraic geometry|geometry]] and [[combinatorics]]. It is named after the [[algebraic structure]] [[magma (algebra)|magma]]. It runs on [[Unix-like]] [[operating * [[Algebraic number theory]] ...

...[vector space]]) that is compatible with the structure in the sense that [[algebraic operation]]s done with equivalent elements will yield equivalent elements.{ ...[equivalence relation]] on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are [[well-defined]] on the [[e ...

<!-- * A ring-like [[List of algebraic structures|algebraic structure]] notability? --> ...

*[[Singular point of an algebraic variety#Nodes|Singular point of an algebraic variety]], a type of singular point of a curve *[[Node (computer science)]], a basic unit used to build data structures ...

...cs)|set]] is the [[Cartesian product]] of the underlying sets of the given structures. The [[direct sum]] of a collection of structures agrees with the direct product in some but not all cases. ...

{{Short description|Theory of algebraic structures in general}} ...udies [[algebraic structure]]s in general, not specific types of algebraic structures. ...

{{Short description|Algebraic structure}} {{Algebraic structures |Ring}} ...

...n be reversed by an [[inverse function|inverse mapping]]. Two mathematical structures are '''isomorphic''' if an isomorphism exists between them, and this is oft ...ommon example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension ...

...polynomials or matrices, one can also define products on many different [[algebraic structure]]s. The [[empty product]] on numbers and most [[algebraic structure]]s has the value of 1 (the [[identity element]] of multiplication ...

{{Short description|Structure-preserving map between two algebraic structures of the same type}} ...[[morphism|structure-preserving]] [[map (mathematics)|map]] between two [[algebraic structure]]s of the same type (such as two [[group (mathematics)|group]]s, ...

...completeness]], cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable [[uniform space]], for example, may have ...]] on an [[algebraic variety]] or on the [[spectrum of a ring]], used in [[algebraic geometry]], ...

...ject into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, [[group homomorphism]], [[ring homomorphism]], ...e {id}_X</math> is the [[identity morphism]] of {{mvar|X}}. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is ...

...-07 }}</ref> This is thus a theorem that they are equivalent for algebraic structures; see ''{{slink|Homomorphism|Monomorphism}}'' for more details. * An injective function which is a homomorphism between two algebraic structures is an [[embedding]]. ...

...ain condition''' ('''DCC''') are finiteness properties satisfied by some [[algebraic structure]]s, most importantly [[Ideal (ring theory)|ideal]]s in certain [[ ...or any [[partially ordered set]]. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler. ...

{{Short description|Algebraic object with an ordered structure}} * any subfield of an ordered field, such as the real [[algebraic numbers]] or the [[computable number]]s, becomes an ordered field by restri ...

{{Algebraic structures|Ring}} ...fg''}} is a power of an [[irreducible polynomial]], which defines the same algebraic set. ...