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nonassociative_relation_algebras [2010/07/29 15:46] (current)
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 +=====Nonassociative relation algebras=====
 +
 +Abbreviation: **NA**
 +====Definition====
 +A \emph{nonassociative 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]]
 +
 +
 +$e$ is an \emph{identity} for $\circ$:  $x\circ e=x$, $e\circ x=x$
 +
 +
 +$\circ$ is \emph{join-preserving}:  
 +$(x\vee y)\circ z=(x\circ z)\vee (y\circ z)$
 +
 +
 +$^{\smile}$ is an \emph{involution}:  
 +${x^\smile}^\smile=x$, $(x\circ y)^{\smile} z=y^{\smile}\circ x^{\smile}$
 +
 +
 +$^{\smile}$ is \emph{join-preserving}:  
 +$(x\vee y)^{\smile} z=x^{\smile}\vee y^{\smile}$
 +
 +
 +$\circ$ is residuated:  $x^{\smile}\circ(\neg (x\circ y))\le\neg y$
 +
 +
 +Remark:
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be relation algebras.
 +A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\circ$, $^{\smile}$, $e$:
 +
 +$h(x\circ y)=h(x)\circ h(y)$, $h(x^{\smile})=h(x)^{\smile}$, $h(e)=e$
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  |decidable |
 +^[[Quasiequational theory]]  |undecidable |
 +^[[First-order theory]]  |undecidable |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[Congruence distributive]]  |yes |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  |yes, $n=2$ |
 +^[[Congruence regular]]  |yes |
 +^[[Congruence uniform]]  |yes |
 +^[[Congruence extension property]]  |yes |
 +^[[Definable principal congruences]]  | |
 +^[[Equationally def. pr. cong.]]  | |
 +^[[Discriminator variety]]  |no |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &\\
 +f(3)= &\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Weakly associative relation algebras]]
 +
 +====Superclasses====
 +[[Nonassociative sequential algebras]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]
 +
 +
 +
 +
 +