nLab
temporal algebra

Idea

A temporal algebra is a modal algebra with two operations that reflect the future and past modal operators of a temporal logic.

Definitions

Definition

A temporal algebra is a Boolean algebra, $(𝔹,m_0,m_1)$, with operators, of type $2$, with the condition that the operators are conjugate:

$m_{0}x\cdot y=0$ if, and only if, $m_{1}y\cdot x=0$.

Equivalently (and equationally) this can be written as

$x\le l_0{m}_{1}x\cdot l_1{m}_{0}x$

where $l_i$ is the dual of $m_1$.