Table of Contents
Sequential algebras
Abbreviation: SeA
Definition
A \emph{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 \emph{right-conjugate} of $\circ$: $(x\circ y)\wedge z=0 \iff (x\triangleright z)\wedge y=0$
$\triangleleft$ is the \emph{left-conjugate} of $\circ$: $(x\circ y)\wedge z=0 \iff (z\triangleleft y)\wedge x=0$
$\triangleright,\triangleleft$ are \emph{balanced}: $x\triangleright e=e\triangleleft x$
$\circ$ is \emph{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 $\circ$, $\triangleright$, $\triangleleft$, $e$:
$h(x\circ y)=h(x)\circ h(y)$, $h(x\triangleright y)=h(x)\triangleright h(y)$, $h(x\triangleleft y)=h(x)\triangleleft h(y)$, $h(e)=e$
Examples
Example 1:
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | undecidable |
| 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 | yes |
| Equationally def. pr. cong. | yes |
| Discriminator variety | no |
| Amalgamation property | no |
| Strong amalgamation property | no |
| Epimorphisms are surjective | no |
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$
