Abbreviation: NA

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:

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 $\circ$, $^{\smile}$, $e$:

$h(x\circ y)=h(x)\circ h(y)$, $h(x^{\smile})=h(x)^{\smile}$, $h(e)=e$

Example 1:

$\begin{array}{lr} f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$

Weakly associative relation algebras

Nonassociative sequential algebras