This article is about a property that can be evaluated for a subalgebra in an algebra with specified operations, where algebra is used in the universal algebra sense.
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Definition
Suppose
is an algebra (in the universal algebra sense) with a given collection of algebra operations. A subalgebra
of
is termed a characteristic subalgebra if it satisfies the following equivalent conditions:
- For every automorphism
of
,
.
- For every automorphism
of
,
is a subalgebra of
.
- For every automorphism
of
,
.
Particular cases
Metaproperties
Metaproperty name |
Satisfied? |
Proof |
Statement with symbols
|
transitive algebra property |
Yes |
characteristicity is transitive for any variety of algebras |
Consider a variety , an algebra in , a characteristic subalgebra of , and a characteristic subalgebra of . Then, is characteristic in .
|
strongly intersection-closed algebra property |
Yes |
characteristicity is strongly intersection-closed for any variety of algebras |
Consider a variety , an algebra in , and characteristic subalgebras of , the intersection is also a characteristic subalgebra of .
|
strongly join-closed algebra property |
Yes |
characteristicity is strongly join-closed for any variety of algebras |
Consider a variety , an algebra in , and characteristic subalgebras of , the join is also a characteristic subalgebra of .
|
Facts
Note that the notion of subalgebra depends on precisely what operations we consider part of the algebra structure of
. For instance, if a group is treated as an algebra in the variety of groups with multiplication, identity, and inverse map operations, the subalgebras are subgroups. If, on the other hand, a group is treated as an algebra in the variety of semigroups (so we just remember the multiplication) then the subalgebras are subsemigroups.
Both the notion of "subalgebra" and the notion of "automorphism" are independent of the choice of subvariety that we are looking at, and hence the notion of "characteristic subalgebra" is also independent. For instance, the notion of characteristic subgroup remains the same whether we are working in the variety of groups or the variety of abelian groups.
Relation with other properties
Stronger properties
Weaker properties