Local and residual properties

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.

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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.

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.