Generalized Boolean algebra

Summary

A generalized Boolean algebra is a distributive lattice with a least element $0$ in which any two elements $a$ and $b$ have a difference $a \setminus b$. This difference is the complement of $a \wedge b$ in the interval $[0, a]$. It is a Boolean algebra that may lack a greatest element. Each interval $[0, a]$ in it is a Boolean algebra. Generalized Boolean algebras correspond to Boolean rings that need not have a unit, in the same way as Boolean algebras correspond to Boolean rings.