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'subalgebra of a partial algebra'
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| Title of object: |
subalgebra of a partial algebra |
| Canonical Name: |
SubalgebraOfAPartialAlgebra |
| Type: |
Definition |
| Created on: |
2009-01-10 02:35:31 |
| Modified on: |
2009-01-12 15:20:11 |
| Classification: |
msc:08A62, msc:03E99, msc:08A55 |
| Defines: |
weak subalgebra, relative subalgebra, subalgebra |
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Content:
Unlike an algebraic system, where there is only one way to define a subalgebra, there are several ways to define a subalgebra of a partial algebra.
Suppose $\boldsymbol{A}$ and $\boldsymbol{B}$ are partial algebras of type $\tau$:
\begin{enumerate}
\item $\boldsymbol{B}$ is a \emph{weak subalgebra} of $\boldsymbol{A}$ if $B\subseteq A$, and $f_{\boldsymbol{B}}$ is a subfunction of $f_{\boldsymbol{A}}$ for every operator symbol $f\in \tau$.
In words, $\boldsymbol{B}$ is a weak subalgebra of $\boldsymbol{A}$ iff $B\subseteq A$, and for each $n$-ary symbol $f\in \tau$, if $b_1,\ldots, b_n \in B$ such that $f_B(b_1,\ldots, b_n)$ is defined, then $f_A(b_1,\ldots, b_n)$ is also defined, and is equal to $f_B(b_1, \ldots, b_n)$.
\item $\boldsymbol{B}$ is a \emph{relative subalgebra} of $\boldsymbol{A}$ if $B\subseteq A$, and $f_{\boldsymbol{B}}$ is a \PMlinkname{restriction of $f_{\boldsymbol{A}}$ relative to $B$}{Subfunction} for every operator symbol $f\in \tau$.
In words, $\boldsymbol{B}$ is a relative subalgebra of $\boldsymbol{A}$ iff $B\subseteq A$, and for each $n$-ary symbol $f\in \tau$, given $b_1,\ldots, b_n \in B$, $f_B(b_1,\ldots, b_n)$ is defined iff $f_A(b_1,\ldots, b_n)$ is and belongs to $B$, and they are equal.
\item $\boldsymbol{B}$ is a \emph{subalgebra} of $\boldsymbol{A}$ if $B\subseteq A$, and $f_{\boldsymbol{B}}$ is a \PMlinkname{restriction}{Subfunction} of $f_{\boldsymbol{A}}$ for every operator symbol $f\in \tau$.
In words, $\boldsymbol{B}$ is a subalgebra of $\boldsymbol{A}$ iff $B\subseteq A$, and for each $n$-ary symbol $f\in \tau$, given $b_1,\ldots, b_n \in B$, $f_B(b_1,\ldots, b_n)$ is defined iff $f_A(b_1,\ldots, b_n)$ is, and they are equal.
\end{enumerate}
Every subalgebra is a relative subalgebra, and every relative subalgebra is a weak subalgebra. But the converse is false for both statements. Below are examples.
Let $A$ be the set of all non-negative integers, and $-_A$ the ordinary subtraction on integers. Consider the partial algebra $(A, -_A)$.
\begin{enumerate}
\item Let $B=A$ and $-_B$ the ordinary subtraction on integers, but only defined when the pair of elements in $B$ have the same parity, so that, for example, $7-_B 6$ is not defined, but $7-_B 5$ is. Then $(B, -_B)$ is a weak subalgebra of $(A, -_A)$. However, $(B,-_B)$ is not a relative subalgebra of $(A,-_A)$.
\item Let $C$ be the set of all positive integers, and $-_C$ the ordinary subtraction. Then $(C,-_C)$ is a relative subalgebra of $(A,-_A)$. However, $(C,-_C)$ is not a subalgebra of $(A,-_A)$.
\item Let $D$ be the set $\lbrace 0,1,\ldots, n\rbrace$ and $-_D$ the ordinary subtraction. Then $(D,-_D)$ is a subalgebra of $(A,-_A)$.
\end{enumerate}
A weak subalgebra $\boldsymbol{B}$ of $\boldsymbol{A}$ is a relative subalgebra iff given $b_1,\ldots, b_n\in B$ such that $f_A(b_1,\ldots, b_n)$ is defined and is in $B$, then $f_B(b_1,\ldots, b_n)$ is defined. A relative subalgebra $\boldsymbol{B}$ of $\boldsymbol{A}$ is a subalgebra iff whenever $f_A(b_1,\ldots, b_n)$ is defined for $b_i\in B$, it is in $B$.
\textbf{Remark}. When $\boldsymbol{A}$ is an algebra, all three notions of subalgebras are equivalent (assuming that the partial operations on a weak subalgebra are all total).
\begin{thebibliography}{7}
\bibitem{gg} G. Gr\"{a}tzer: {\em Universal Algebra}, 2nd Edition, Springer, New York (1978).
\end{thebibliography} |
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