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Difference between revisions of "Local and residual properties"

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Certain abstract properties (that is, properties preserved under isomorphism) of algebraic systems or universal algebras. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600701.png" /> is an abstract property of algebras, one says that an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600702.png" /> locally has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600703.png" /> if there is a local system of subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600704.png" /> each of which has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600705.png" />. A local system of subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600706.png" /> is a system of non-empty subalgebras, directed by inclusion, whose union coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600707.png" />. If every algebra of some class that locally has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600708.png" /> actually has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l0600709.png" /> itself, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007010.png" /> is called a local property of the algebras of this class. For example, the property of being an Abelian group is a local property in the class of all groups, but the property of being a finite group is not local. For more details about the local nature of properties, see [[Mal'tsev local theorems|Mal'tsev local theorems]].
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Certain abstract properties (that is, properties preserved under isomorphism) of algebraic systems or universal algebras. If $P$ is an abstract property of algebras, one says that an algebra $A$ locally has the property $P$ if there is a local system of subalgebras of $A$ each of which has the property $P$. A local system of subalgebras of $A$ is a system of non-empty subalgebras, directed by inclusion, whose union coincides with $A$. If every algebra of some class that locally has the property $P$ actually has the property $P$ itself, then $P$ is called a local property of the algebras of this class. For example, the property of being an Abelian group is a local property in the class of all groups, but the property of being a finite group is not local. For more details about the local nature of properties, see [[Mal'tsev local theorems|Mal'tsev local theorems]].
  
One says that an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007011.png" /> residually has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007012.png" /> if there is a separating family of congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007014.png" /> such that every quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007015.png" /> has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007016.png" />. A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007017.png" /> is called a separating family of congruences if the intersection of all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007018.png" /> is the diagonal congruence (the equality relation) on the given algebra. An algebra residually has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007019.png" /> if and only if it can be represented as a subdirect product of algebras of the appropriate type having the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007020.png" />. A property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007021.png" /> is said to be residual in a class of algebras if every algebra of this class that residually has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007022.png" /> actually has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007023.png" /> itself. In the class of all groups the property of being Abelian is residual, but finiteness is not residual. Every residual property of algebras that is preserved under transition to homomorphic images is local.
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One says that an algebra $A$ residually has the property $P$ if there is a separating family of congruences $(q_\lambda)_{\lambda\in\Lambda}$ on $A$ such that every quotient algebra $A/q_\lambda$ has the property $P$. A family $(q_\lambda)_{\lambda\in\Lambda}$ is called a separating family of congruences if the intersection of all the $q_\lambda$ is the diagonal congruence (the equality relation) on the given algebra. An algebra residually has the property $P$ if and only if it can be represented as a subdirect product of algebras of the appropriate type having the property $P$. A property $P$ is said to be residual in a class of algebras if every algebra of this class that residually has the property $P$ actually has the property $P$ itself. In the class of all groups the property of being Abelian is residual, but finiteness is not residual. Every residual property of algebras that is preserved under transition to homomorphic images is local.
  
 
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Of course, the precise meaning of the phrase  "local property"  depends on the local system of algebras that is used to define it.
 
Of course, the precise meaning of the phrase  "local property"  depends on the local system of algebras that is used to define it.
  
A most important example of a local system of subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007024.png" /> is the system of all finitely-generated subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007025.png" />; see also e.g. [[Locally free group|Locally free group]]; [[Locally solvable group|Locally solvable group]]; [[Locally solvable algebra|Locally solvable algebra]]; [[Local property|Local property]]; [[Locally nilpotent group|Locally nilpotent group]]; [[Locally nilpotent algebra|Locally nilpotent algebra]]; [[Locally finite group|Locally finite group]]; [[Locally finite algebra|Locally finite algebra]].
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A most important example of a local system of subalgebras of $A$ is the system of all finitely-generated subalgebras of $A$; see also e.g. [[Locally free group|Locally free group]]; [[Locally solvable group|Locally solvable group]]; [[Locally solvable algebra|Locally solvable algebra]]; [[Local property|Local property]]; [[Locally nilpotent group|Locally nilpotent group]]; [[Locally nilpotent algebra|Locally nilpotent algebra]]; [[Locally finite group|Locally finite group]]; [[Locally finite algebra|Locally finite algebra]].
  
The term  "local property"  is used in topology as well as algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007026.png" /> is an abstract property of topological spaces, one says that a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007027.png" /> locally has the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007028.png" /> if every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007029.png" /> possesses a base of neighbourhoods having the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007030.png" />. (For example, the properties  "locally compact"  and  "locally connected"  may be defined in this way.) One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007031.png" /> is a local property if the spaces which locally have the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007032.png" /> are exactly the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007033.png" />-spaces: for example, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060070/l06007034.png" /> [[Separation axiom|separation axiom]] is a local property, but the [[Hausdorff axiom|Hausdorff axiom]] is not.
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The term  "local property"  is used in topology as well as algebra. If $P$ is an abstract property of topological spaces, one says that a space $X$ locally has the property $P$ if every point of $X$ possesses a base of neighbourhoods having the property $P$. (For example, the properties  "locally compact"  and  "locally connected"  may be defined in this way.) One says that $P$ is a local property if the spaces which locally have the property $P$ are exactly the $P$-spaces: for example, the $T_1$ [[Separation axiom|separation axiom]] is a local property, but the [[Hausdorff axiom|Hausdorff axiom]] is not.
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[[Category:General algebraic systems]]

Latest revision as of 22:24, 26 October 2014

Certain abstract properties (that is, properties preserved under isomorphism) of algebraic systems or universal algebras. If $P$ is an abstract property of algebras, one says that an algebra $A$ locally has the property $P$ if there is a local system of subalgebras of $A$ each of which has the property $P$. A local system of subalgebras of $A$ is a system of non-empty subalgebras, directed by inclusion, whose union coincides with $A$. If every algebra of some class that locally has the property $P$ actually has the property $P$ itself, then $P$ is called a local property of the algebras of this class. For example, the property of being an Abelian group is a local property in the class of all groups, but the property of being a finite group is not local. For more details about the local nature of properties, see Mal'tsev local theorems.

One says that an algebra $A$ residually has the property $P$ if there is a separating family of congruences $(q_\lambda)_{\lambda\in\Lambda}$ on $A$ such that every quotient algebra $A/q_\lambda$ has the property $P$. A family $(q_\lambda)_{\lambda\in\Lambda}$ is called a separating family of congruences if the intersection of all the $q_\lambda$ is the diagonal congruence (the equality relation) on the given algebra. An algebra residually has the property $P$ if and only if it can be represented as a subdirect product of algebras of the appropriate type having the property $P$. A property $P$ is said to be residual in a class of algebras if every algebra of this class that residually has the property $P$ actually has the property $P$ itself. In the class of all groups the property of being Abelian is residual, but finiteness is not residual. Every residual property of algebras that is preserved under transition to homomorphic images is local.

References

[1] P.M. Cohn, "Universal algebra" , Reidel (1981)


Comments

Of course, the precise meaning of the phrase "local property" depends on the local system of algebras that is used to define it.

A most important example of a local system of subalgebras of $A$ is the system of all finitely-generated subalgebras of $A$; see also e.g. Locally free group; Locally solvable group; Locally solvable algebra; Local property; Locally nilpotent group; Locally nilpotent algebra; Locally finite group; Locally finite algebra.

The term "local property" is used in topology as well as algebra. If $P$ is an abstract property of topological spaces, one says that a space $X$ locally has the property $P$ if every point of $X$ possesses a base of neighbourhoods having the property $P$. (For example, the properties "locally compact" and "locally connected" may be defined in this way.) One says that $P$ is a local property if the spaces which locally have the property $P$ are exactly the $P$-spaces: for example, the $T_1$ separation axiom is a local property, but the Hausdorff axiom is not.

How to Cite This Entry:
Local and residual properties. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_and_residual_properties&oldid=12157
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article