Table of Contents
Action algebras
Abbreviation: Act
Definition
An \emph{action algebra} is a structure $\mathbf{A}=\langle A,\vee,\bot,\cdot,1,^*,\backslash,/\rangle$ of type $\langle 2,0,2,0,1,2,2\rangle$ such that
$\langle A,\vee,\bot,\cdot,1,^*\rangle$ is a Kleene algebra
$\backslash $ is the left residual of $\cdot$: $y\leq x\backslash z\Longleftrightarrow xy\leq z$
$/$ is the right residual of $\cdot$: $x\leq z/y\Longleftrightarrow xy\leq z$
Remark: These equivalences can be written equationally.
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be action algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(x^*)=h(x)^*$, $h(\bot)=\bot$ and $h(1)=1$.
Examples
Example 1:
Basic results
Properties
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &3
f(4)= &20
f(5)= &149
f(6)= &1488
\end{array}$
