# Characteristic subalgebra

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.

View more such properties

## Contents

## 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

Variety of interest | Characteristic subalgebra in that variety | Facts peculiar to characteristic subalgebras in that variety |
---|---|---|

variety of groups, with the usual universal algebra axiomatization of groups | characteristic subgroup | characteristic subalgebras are ideals in the variety of groups |

variety of Lie rings, with the usual universal algebra axiomatization of Lie rings | characteristic Lie subring | |

variety of powered groups for a set of primes, i.e., fix a set of primes and look at the variety of groups powered over those primes | characteristic subgroup that is in addition powered over those primes (note that any powering-invariant characteristic subgroup would satisfy this condition but the converse is not necessarily true, since "powering-invariant" means invariant under powering by all primes the overgroup is powered by, not just the primes for which we are considering the variety. |

## 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

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

fully invariant subalgebra | invariant under all endomorphisms | |FULL LIST, MORE INFO | ||

verbal subalgebra | union of the images of word maps | |FULL LIST, MORE INFO | ||

homomorph-containing subalgebra | contains every homomorphism image | |FULL LIST, MORE INFO | ||

isomorph-containing subalgebra | contains every subalgebra of the whole algebra that is isomorphic to it | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

I-automorphism-invariant subalgebra | |FULL LIST, MORE INFO |