# Differences

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

nonassociative_relation_algebras [2010/07/29 15:46] (current)
Line 1: Line 1:
+=====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>
+)]
+
+
+
+
+

##### Toolbox 