PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] subalgebra of an algebraic system (Definition)

Let $ (A,O)$ be an algebraic system ( $ A\ne \varnothing$ is the underlying set and $ O$ is the set of operators on $ A$).

Subalgebras of an Algebra

Let $ B$ be a non-empty subset of $ A$. $ B$ is closed under operators of $ A$ if for each $ n$-ary operator $ \omega_A$ on $ A$, and any $ b_1,\ldots,b_n\in B$, we have $ \omega_A(b_1,\ldots,b_n)\in B$.

Suppose $ B$ is closed under operators of $ A$. For each $ n$-ary operator $ \omega_A$ on $ A$, we define $ \omega_B:B^n\to B$ by $ \omega_B(b_1,\ldots,b_n):= \omega_A(b_1,\ldots,b_n)$. Each of these operators is well-defined and is called a restriction (of the corresponding $ \omega_A$). Furthermore, $ (B,O)$ is a well-defined algebraic system, and is called the subalgebra of $ (A,O)$. When $ (B,O)$ is a subalgebra of $ (A,O)$, we also say that $ (A,O)$ is an extension of $ (B,O)$.

$ (A,O)$ is clearly a subalgebra of itself. Any other subalgebra of $ (A,O)$ is called a proper subalgebra.

Remark. If $ (A,O)$ contains constants, then any subalgebra of $ (A,O)$ must contain the exact same constants. For example, the ring $ \mathbb{Z}$ of integers is an algebraic system with no proper subalgebras. Indeed, if $ R$ is a subring of $ \mathbb{Z}$, $ 1\in R$, so $ R=\mathbb{Z}$.

Since we are operating under the same operator set, we can, for convenience, drop $ O$ and simply call $ A$ an algebra, $ B$ a subalgebra of $ A$, etc... If $ B_1,B_2$ are subalgebras of $ A$, then $ B_1\cap B_2$ is also a subalgebra. In fact, given any set of subalgebras $ B_i$ of $ A$, their intersection $ \bigcap B_i$ is also a subalgebra.

Generating Set of an Algebra

Let $ C$ be any subset of an algebra $ A$. Consider the collection $ [C]$ of all subalgebras of $ A$ containing $ C$. This collection is non-empty because $ A\in [C]$. The intersection of all these subalgebras is again a subalgebra containing the set $ C$. Denote this subalgebra by $ \langle C\rangle$. $ \langle C\rangle$ is called the subalgebra spanned by $ C$, and $ C$ is called the spanning set of $ \langle C\rangle$. Conversely, any subalgebra $ B$ of $ A$ has a spanning set, namely itself: $ B=\langle B\rangle$.

Given a subalgebra $ B$ of $ A$, a minimal spanning set $ X$ of $ B$ is called a generating set of $ B$. By minimal we mean that the set obtained by deleting any element from $ X$ no longer spans $ B$. When $ B$ has a generating set $ X$, we also say that $ X$ generates $ B$. If $ B$ can be generated by a finite set, we say that $ B$ is finitely generated. If $ B$ can be generated by a single element, we say that $ B$ is cyclic.

Remark. $ \langle \varnothing\rangle =$ the subalgebra generated by the constants of $ A$. If no such constants exist, $ \langle \varnothing \rangle :=\varnothing$.

From the discussion above, the set of subalgebras of an algebraic system forms a complete lattice. Given subalgebras $ A_i$, $ \bigvee A_i$ is the intersection of all $ A_i$, and $ \bigvee A_i$ is the subalgebra $ \langle \bigcup A_i\rangle$. The lattice of all subalgebras of $ A$ is called the subalgebra latttice of $ A$, and is denoted by $ \operatorname{Sub}(A)$.



"subalgebra of an algebraic system" is owned by CWoo.
(view preamble)

View style:

Other names:  subalgebra lattice
Also defines:  subalgebra, generating set, subalgebra generated by, extension of an algebraic system, restriction, proper subalgebra, lattice of subalgebras, spanning set, finitely generated, cyclic

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: lattice, complete lattice, finite set, generated by, generates, spans, mean, minimal, collection, intersection, operator set, subring, integers, ring, contains, extension, well-defined, closed under, closed, subset, algebra, operators, algebraic system
There are 165 references to this entry.

This is version 6 of subalgebra of an algebraic system, born on 2007-02-23, modified 2007-03-21.
Object id is 8961, canonical name is SubalgebraOfAnAlgebraicSystem.
Accessed 3907 times total.

Classification:
AMS MSC: 08A62 (General algebraic systems :: Algebraic structures :: Finitary algebras)
 08A05 (General algebraic systems :: Algebraic structures :: Structure theory)
 08A30 (General algebraic systems :: Algebraic structures :: Subalgebras, congruence relations)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)