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, $(\mathbb{B}, m_0,m_1)$, 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

$x \leq l_0 m_1 x \cdot l_1 m_0 x$

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