subalgebra of an algebraic system

Let $(A,O)$ be an algebraic system ($A\ne \mathrm{\varnothing }$ 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 $b_1,\mathrm{\dots },b_n\in B$, we have ${\omega }_A(b_1,\mathrm{\dots },b_n)\in B$.

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 ${\omega }_B(b_1,\mathrm{\dots },b_n):={\omega }_A(b_1,\mathrm{\dots },b_n)$. 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 $⟨C⟩$. $⟨C⟩$ is called the subalgebra spanned by $C$, and $C$ is called the spanning set of $⟨C⟩$. Conversely, any subalgebra $B$ of $A$ has a spanning set, namely itself: $B=⟨B⟩$.

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. $⟨\mathrm{\varnothing }⟩=$ the subalgebra generated by the constants of $A$. If no such constants exist, $⟨\mathrm{\varnothing }⟩:=\mathrm{\varnothing }$.

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 $⟨\bigcup A_i⟩$. The lattice of all subalgebras of $A$ is called the subalgebra latttice of $A$, and is denoted by $\mathrm{Sub}(A)$.

Title	subalgebra of an algebraic system
Canonical name	SubalgebraOfAnAlgebraicSystem
Date of creation	2013-03-22 16:44:19
Last modified on	2013-03-22 16:44:19
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	9
Author	CWoo (3771)
Entry type	Definition
Classification	msc 08A30
Classification	msc 08A05
Classification	msc 08A62
Synonym	subalgebra lattice
Defines	subalgebra
Defines	generating set
Defines	subalgebra generated by
Defines	extension of an algebraic system
Defines	restriction
Defines	proper subalgebra
Defines	lattice of subalgebras
Defines	spanning set
Defines	finitely generated
Defines	cyclic