Differences

This shows you the differences between two versions of the page.

relation_algebras [2010/09/17 20:38]
jipsen
relation_algebras [2010/09/17 20:45] (current)
jipsen
Line 3: Line 3:
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]]
Line 24: Line 24:
====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====
Line 50: Line 50:
^[[Strong amalgamation property]]  |no | ^[[Strong amalgamation property]]  |no |
^[[Epimorphisms are surjective]]  |no | ^[[Epimorphisms are surjective]]  |no |
+
+
====Finite members==== ====Finite members====
Line 62: Line 64:
+
====Subclasses==== ====Subclasses====
Line 71: Line 74:
[[Square-increasing relation algebras]] [[Square-increasing relation algebras]]
+
====Superclasses==== ====Superclasses====
Line 80: Line 84:
====References==== ====References====
-[(Ln19xx> +/*[(Ln19xx> )]*/
-)] +
- +
- +
- +
- +