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08A30-SubalgebraOfAnAlgebraicSystem.tex
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08A30-SubalgebraOfAnAlgebraicSystem.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{SubalgebraOfAnAlgebraicSystem}
\pmcreated{2013-03-22 16:44:19}
\pmmodified{2013-03-22 16:44:19}
\pmowner{CWoo}{3771}
\pmmodifier{CWoo}{3771}
\pmtitle{subalgebra of an algebraic system}
\pmrecord{9}{38961}
\pmprivacy{1}
\pmauthor{CWoo}{3771}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{08A30}
\pmclassification{msc}{08A05}
\pmclassification{msc}{08A62}
\pmsynonym{subalgebra lattice}{SubalgebraOfAnAlgebraicSystem}
\pmdefines{subalgebra}
\pmdefines{generating set}
\pmdefines{subalgebra generated by}
\pmdefines{extension of an algebraic system}
\pmdefines{restriction}
\pmdefines{proper subalgebra}
\pmdefines{lattice of subalgebras}
\pmdefines{spanning set}
\pmdefines{finitely generated}
\pmdefines{cyclic}
\endmetadata
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\begin{document}
Let $(A,O)$ be an algebraic system ($A\ne \varnothing$ is the underlying set and $O$ is the set of operators on $A$).
\textbf{Subalgebras of an Algebra}
Let $B$ be a non-empty subset of $A$. $B$ is \emph{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 \emph{restriction} (of the corresponding $\omega_A$). Furthermore, $(B,O)$ is a well-defined algebraic system, and is called the \emph{subalgebra} of $(A,O)$. When $(B,O)$ is a subalgebra of $(A,O)$, we also say that $(A,O)$ is an \emph{extension} of $(B,O)$.
$(A,O)$ is clearly a subalgebra of itself. Any other subalgebra of $(A,O)$ is called a \emph{proper subalgebra}.
\textbf{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.
\textbf{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 \emph{spanned} by $C$, and $C$ is called the \emph{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 \emph{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 $X$ \emph{generates} $B$. If $B$ can be generated by a finite set, we say that $B$ is \emph{finitely generated}. If $B$ can be generated by a single element, we say that $B$ is \emph{cyclic}.
\textbf{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 \emph{subalgebra latttice} of $A$, and is denoted by $\operatorname{Sub}(A)$.
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\end{document}