## 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 $h(1)=1$.

Example 1:

### Properties

Classtype variety 1) undecidable undecidable no unbounded yes 2) yes yes, $n=4$ 3)

### Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &3 f(4)= &20 f(5)= &149 f(6)= &1488 \end{array}$

### References

1) Vaughan Pratt, \emph{Action logic and pure induction}, Logics in AI (Amsterdam, 1990)'', Lecture Notes in Comput. Sci., 478, 1991, 97–120, 92d:03016
2), 3) C.J. van Alten and J.G. Raftery, \emph{Embedding Theorems and Rule Separation in Logics without Weakening}, Studia Logica, 2004, …–…, preprint