## Sequential algebras

Abbreviation: SeA

### Definition

A sequential algebra is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\circ,e,\triangleright,\triangleleft\rangle$ such that

$\langle A,\vee,0, \wedge,1,\neg\rangle$ is a Boolean algebra

$\langle A,\circ,e\rangle$ is a monoid

$\triangleright$ is the right-conjugate of $\circ$: $(x\circ y)\wedge z=0 \iff (x\triangleright z)\wedge y=0$

$\triangleleft$ is the left-conjugate of $\circ$: $(x\circ y)\wedge z=0 \iff (z\triangleleft y)\wedge x=0$

$\triangleright,\triangleleft$ are balanced: $x\triangleright e=e\triangleleft x$

$\circ$ is euclidean: $x\cdot(y\triangleleft z)\leq (x\cdot y)\triangleleft z$

Remark:

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be sequential algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\circ$, $\triangleright$, $\triangleleft$, $e$:

$h(x\circ y)=h(x)\circ h(y)$, $h(x\triangleright y)=h(x)\triangleright h(y)$, $h(x\triangleleft y)=h(x)\triangleleft h(y)$, $h(e)=e$

Example 1:

### Properties

Classtype variety undecidable undecidable undecidable no unbounded yes yes yes, $n=2$ yes yes yes yes yes no no no no

### Finite members

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