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relation_algebras [2010/09/17 20:42]
jipsen
relation_algebras [2010/09/17 20:45] (current)
jipsen
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Abbreviation: **RA** Abbreviation: **RA**
====Definition==== ====Definition====
-A \emph{relation algebra} is a structure $\mathbf{A}=\langle A,\vee,0,\wedge,1,\neg,\circ,^{\smallsmile},e\rangle$ such that+A \emph{relation algebra} is a structure $\mathbf{A}=\langle A,\vee,0,\wedge,1,\neg,\circ,^{\smile},e\rangle$ such that
$\langle A,\vee,0,\wedge,1,\neg\rangle$ is a [[Boolean algebra]] $\langle A,\vee,0,\wedge,1,\neg\rangle$ is a [[Boolean algebra]]
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====Examples==== ====Examples====
-Example 1: $\langle \mathcal P(U^2), \cup, \emptyset, \cap, U^2, -, \circ, ^\smallsmile, id_U \rangle$ the full relation algebra of binary relations on a set $U$.+Example 1: $\langle \mathcal P(U^2), \cup, \emptyset, \cap, U^2, -, \circ, ^\smile, id_U \rangle$ the full relation algebra of binary relations on a set $U$.
-Example 2: $\langle \mathcal P(G), \cup, \emptyset, \cap, G, -, \circ, ^\smallsmile, \{e\} \rangle$ the group relation algebra of a [[group]] $\langle G, *, ^{-1}, e \rangle$, where $X\circ Y=\{x*y : x\in X, y\in Y\}$ and $X^\smallsmile=\{x^{-1} : x\in X\}$.+Example 2: $\langle \mathcal P(G), \cup, \emptyset, \cap, G, -, \circ, ^\smile, \{e\} \rangle$ the group relation algebra of a [[group]] $\langle G, *, ^{-1}, e \rangle$, where $X\circ Y=\{x*y : x\in X, y\in Y\}$ and $X^\smile=\{x^{-1} : x\in X\}$.
====Basic results==== ====Basic results====