Definition:Completely Irreducible

Definition

Let $\struct {S, \preceq}$ be an ordered set.

Let $p \in S$.

An element $p$ is completely irreducible if and only if

$p^\succeq \setminus \set p$ admits a minimum element

where $p^\succeq$ denotes the upper closure of $p$.

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL16:def 3