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{{about|the algebraic identity|relation between chemical material and attenuation of light|Beer–L ...by the absorption law is called a [[Lattice (order)#Lattices as algebraic structures|lattice]]; in this case, both operations are necessarily [[Idempotence|idem ...

...thematics)|field]]s, etc.), [[Topology|topologies]], [[Metric space|metric structures]] ([[Geometry|geometries]]), [[Order theory|orders]], [[graph theory|graphs ...which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topol ...

{{redirect|Ordered group|groups with a total or linear order|Linearly ordered group}} In [[abstract algebra]], a '''partially ordered group''' is a [[group (mathematics)|group]] (''G'', +) equipped with a [[pa ...

...and the [[Zariski closure]] of a set {{mvar|V}} of points is the smallest algebraic set that contains {{mvar|V}}. ===In algebraic structures=== ...

...t the advent of domain theory. Scott domains are very closely related to [[algebraic lattice]]s, being different only in possibly lacking a [[greatest element]] ..."Scott domains". Additionally, Scott domains appear with other names like "algebraic semilattice" in some publications. ...

{{Short description|1=Overview of and topical guide to algebraic structures}} {{Algebraic structures}} ...

...peration (mathematics)|operations]], but their precise definitions lead to structures such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [ == Algebraic equations == ...

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

...substructure, especially when the context suggests a theory of which both structures are models. ...al property that every substructure of an ordered group which is itself an ordered group, is an induced substructure. ...

...(cons) cells, conses, non-atomic [[s-expressions]] ("NATSes"), or (cons) [[ordered pair|pairs]]. In Lisp jargon, the expression "to cons ''x'' onto ''y''" mea ...en arguments, and more closely related to the constructor function of an [[algebraic data type]] system. ...

...th>X</math> to the [[carrier set]] of <math>A</math> can be turned into an algebraic structure of the same type in an analogous way. ...partially ordered sets|posets]], the set of functions ''A'' → ''B'' can be ordered by defining ''f'' ≤ ''g'' if {{math|(∀''x'' ∈ A) ''f''(''x'') ≤ ''g''(''x'' ...

...of a [[theory (logic)|theory]], generalizing the notion of dimension in [[algebraic geometry]]. ...'&nbsp;=&nbsp;''x'' is strongly minimal. Morley rank and strongly minimal structures are key tools in the proof of [[Morley's categoricity theorem]] and in the ...

...ly]], a '''meet-semilattice''' (or '''lower semilattice''') is a partially ordered set which has a [[meet (mathematics)|meet]] (or [[greatest lower bound]]) f A [[lattice (order)|lattice]] is a partially ordered set that is both a meet- and join-semilattice with respect to the same part ...

...far more general: they are a common generalisation of ''all'' [[algebraic structures]]. "Subalgebra" can refer to either case. ...mon_associative_algebras_over_the_reals.svg|thumb|right|300px|Algebras are ordered by inclusion as in this [[lattice (order)|lattice]] of associative algebras ...

{{Short description|Algebraic object with an ordered structure}} ...elements that is compatible with the field operations. Basic examples of ordered fields are the [[Rational number|rational numbers]] and the [[real numbers] ...

...in condition]] on its [[submodule]]s, where the submodules are [[partially ordered]] by [[inclusion (set theory)|inclusion]].<ref>{{harvnb|Roman|2008|loc=p. 1 ==Use in other structures== ...

...(or equivalences) between mathematical objects or [[Mathematical structure|structures]]. Roughly speaking, a collection ''Y'' of mathematical objects may be said ...ion theory]]''', which studies the representing of elements of [[algebraic structures]] by [[linear transformations]] of [[vector spaces]].<ref name=":0" /> ...

{{Short description|Special ordered sets}} [[Category:Algebraic structures]] ...

...ategory of ''non-empty finite ordinals'' as objects, thought of as totally ordered sets, and ''(non-strictly) order-preserving functions'' as [[morphisms]]. T [[Category:Algebraic topology]] ...

{{Short description|Mathematical property of algebraic structures}} ...y|Syracuse]], is a property held by some [[algebraic structure]]s, such as ordered or normed [[group (algebra)|groups]], and [[field (mathematics)|fields]]. ...