Abstract. Given a finite partially ordered set $P$, for subsets or, in other words, coalitions $X$, $Y$ of $P$ let $X\le Y$ mean that there exists an injection $\varphi\colon X \to Y$ such that $x\le\varphi (x)$ for all $x\in X$. The set ${\cal L}(P)$ of all subsets of $P$ equipped with this relation is a partially ordered set. All partially ordered sets $P$ such that ${\cal L}(P)$ is a lattice are determined, and this result is extended to quasiordered set $P$ versus $q$-lattice ${\cal L}(P)$ as well. Some elementary properties of distributive lattices ${\cal L}(P)$ are also given.

AMS Subject Classification (1991): 06B99, 06A99, 90D99

Keyword(s): Lattice, q, -lattice, quasiorder, partially ordered set, coalition

Received July 29, 1994. (Registered under 5626/2009.)