### 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}$