Abbreviation: SeA

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:

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:

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

Relation algebras

Representable sequential algebras

Distributive residuated lattices

Semiassociative sequential algebras