ExtensionΒΆ

Definition

Given a set, or algebra \(S\), we say that \(T\) is an extension of \(S\), or extends \(S\), if one or more of the following holds:

  • \(S \subseteq T\) and \(T\) has operations that \(S\) does not.
  • \(S \subset T\) and any operation \(op\) on \(S\) can be obtained from an operation \(OP\) on \(T\) by restricting \(OP\) to elements only in \(S\).

Examples of extensions

  • Any set \(S\) with no pre-defined operations on it extends to its set algebra: \(\bigg[P(S), \big \{ [ \cup, \varnothing ] , [ \cap, M ] \big\} , \big\{\ '\ \big\} , \big\{ \subset \big\} \bigg]\).

  • Let \(\mathbb{Z}\) represent the set of integers, and \(\mathbb{Q}\) represent the set of rational numbers. The set \(\mathbb{Z}\) has a natural algebra under addition, and multiplication, and negation with signature: \(\bigg[ \mathbb{Z}, \big \{[+,0],[\cdot ,1]\}, \big \{-\}, \bigg]\). This algebra naturally extends to an algebra on the rational numbers with signature: \(\bigg[ \mathbb{Q}, \big \{[+,0],[\cdot ,1]\}, \big \{-\}, \bigg]\).

  • Let us take the algebra of integers with signature \(\bigg[ \mathbb{Z}, \big \{[+,0],[\cdot ,1]\}, \big \{-\}, \bigg]\) as in the previous example. Since \(\mathbb{Z}\) is a set, it also posesses the set algebra \(\bigg[P(\mathbb{Z}), \big \{ [ \cup, \varnothing ] , [ \cap, M ] \big\} , \big\{\ '\ \big\} , \big\{ \subset \big\} \bigg]\). We can extend \(\bigg[ \mathbb{Z}, \big \{[+,0],[\cdot ,1]\}, \big \{-\}, \bigg]\) to the set algebra by defining addition, multiplication, and negation on subsets of \(\mathbb{Z}\) as follows: Given subsets \(A,B\subset\mathbb{Z}\)

    \[A+B:=\{c\in \mathbb{Z}:c=a+b\text{ for some }a\in A\text{\ and for some }%b\in B\}\]
    \[A\cdot B:=\{c\in \mathbb{Z}:c=a\cdot b\text{ for some }a\in A\text{\ and for some }b\in B\}\]
    \[-A:=\{c\in \mathbb{Z}:c=-a\text{ for some }a\in A\text{\ }\}.\]

    In words, the above equations say that over the integers: - The sum of sets is the set of sums. - The product of sets the set of products. - The negative of a set is the set of negatives.

    So for example if \(A=\{3,-5,9\}\) and \(B=\{4,12\}\), then

    \[\begin{split}\begin{eqnarray*} \{3,-5,9\}+\{4,12\} &=&\left\{ \begin{array}{c} 3+4,3+12, \\ -5+4,-5+12, \\ 9+4,9+12% \end{array}% \right\} \\ &=&\left\{ \begin{array}{c} 7,15, \\ -1,7, \\ 13,21% \end{array}% \right\} \\ &=&\{7,15,-1,13,21\}. \end{eqnarray*}\end{split}\]

    and,

    \[\begin{split}\begin{eqnarray*} \{3,-5,9\}\cdot \{4,12\} &=&\left\{ \begin{array}{c} 3\cdot 4,3\cdot 12, \\ -5\cdot 4,-5\cdot 12, \\ 9\cdot 4,9\cdot 12% \end{array}% \right\} \\ &=&\left\{ \begin{array}{c} 12,36, \\ -20,-60, \\ 36,108% \end{array}% \right\} \\ &=&\{12,36,-20,-60,36,108\}. \end{eqnarray*}\end{split}\]

    and \(-A=-\{3,-5,9\}=\{-3,5,-9\}\).

    In conclusion, this shows that the algebra \(\bigg[ \mathbb{Z}, \big \{[+,0],[\cdot ,1]\}, \big \{-\}, \bigg]\) extendss to the algebra

    \[\begin{equation*} \bigg[ P(\mathbb{Z}),\{[\cup ,\varnothing ],[\mathbb{\cap },% \mathbb{Z}],[+,\{0\}],[\cdot ,\{1\}]\},\{-,^{\prime }\},\{\subset \}\bigg] , \end{equation*}\]

  • The algebra of couplets on a set \(M\) with signature \(\bigg[ M \times M , \big\{\ \circ\ \big\} , \big\{ \leftrightarrow \big\} \bigg]\) extends to the algebra of relations \(\bigg[P(M \times M),\big\{[ \circ, D_M ] \big\} , \big\{ \leftrightarrow \big\}\bigg]\), where \(\circ\) is composition, \(\leftrightarrow\) is transposition, and \(D_M\) is the diagonal of \(M\).

  • The power set of a power set has an algebra given by the algebra of sets. Hence, if \(U\) is a set, we have an algebra with signature \(\bigg[ P^{2}(U),[\cup ,\varnothing ],[\mathbb{\cap },P(U)],\subset ,\prime \bigg]\). This algebra extends to the algebra \(\bigg[ P^{2}(U),[\cup ,\varnothing ],[\mathbb{\cap },P(U)], [\blacktriangledown ,\{\varnothing\}],[\mathbb{\blacktriangle },\{U\}],\subset ,\prime \bigg]\), where \(\blacktriangledown\) is the cross-intersection and \(\blacktriangle\) is the cross-union.