Mathematical Structures: History of Relation algebras

History of Relation algebras

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 Revision 15 . . August 4, 2004 12:33 pm by Jipsen Revision 14 . . July 10, 2004 10:53 am by Jipsen Revision 13 . . July 10, 2004 10:52 am by Jipsen Revision 12 . . May 18, 2003 10:33 am by 68.5.251.xxx

Difference (from prior major revision) (author diff)

Changed: 32,33c32
 A relation algebra is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\circ,^{\smile},e\rangle$ such that
 A 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 \href{Boolean_algebras.pdf}{Boolean algebra}

Changed: 36,37c36
 $\langle A,\vee,0, \wedge,1,\neg\rangle$ is a \href{Boolean_algebras.pdf}{Boolean algebras}
 $\langle A,\circ,e\rangle$ is a \href{Monoids.pdf}{monoid}

 $\circ$ is join-preserving: $(x\vee y)\circ z=(x\circ z)\vee (y\circ z)$

Changed: 40c40
 $\langle A,\circ,e\rangle$ is a \href{Monoids.pdf}{monoids}
 $^{\smile}$ is an involution: $x^{\smile\smile}=x$, $(x\circ y)^{\smile}=y^{\smile}\circ x^{\smile}$

 $^{\smile}$ is join-preserving: $(x\vee y)^{\smile}=x^{\smile}\vee y^{\smile}$

Changed: 43,56c44
 $\circ$ is join-preserving: $(x\vee y)\circ z=(x\circ z)\vee (y\circ z)$ $^{\smile}$ is an involution: $x^{\smile\smile}=x$, $(x\circ y)^{\smile}=y^{\smile}\circ x^{\smile}$ $^{\smile}$ is join-preserving: $(x\vee y)^{\smile}=x^{\smile}\vee y^{\smile}$ $\circ$ is residuated: $x^{\smile}\circ(\neg (x\circ y))\le\neg y$
 $\circ$ is residuated: $x^{\smile}\circ(\neg (x\circ y))\le\neg y$