Pseudo Treealgebras

Abstract

A pseudotree $ \langle T, \leq \rangle$ is a partially ordered set for which $ \{ u \in T: u \leq t \}$ is a linear ordering for each $ t \in T$. Define $ \mathcal{B}(T)$, the pseudo treealgebra over T, as the subalgebra of the power set of T generated by $ \{ b_{t} : t \in T \}$ where $ b_{t} = \{ u \in T : t \leq u \}$. It is shown that every pseudo treealgebra is embeddable into an interval algebra; thus it is a retractive Boolean algebra. Moreover, superatomicity of $ \mathcal{B}(T)$ is described using conditions on $ \langle T, \leq \rangle$.