For a given fixed poset $\mathcal P$ we say that family of subsets $[n]$ is P$-saturated if it does not contain an induced copy P$, but whenever add to new set, formed. The size the smallest such denoted by $\text{sat}^*(n, \mathcal P)$. diamond D_2$ (the two-dimensional Boolean lattice), Martin, Smith and Walker proved $\sqrt n\leq\text{sat}^*(n, D_2)\leq n+1$. In this paper prove D_2)\geq (2\...

In this paper, a Krein-Milman  type theorem in $T_0$ semitopological cone is proved,  in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.