This paper gives some new characterizations of completeness for trellises by introducing the notion of a cycle-complete trellis. One of our results yields, in particular, a characterization of completeness for trellises of finite length due to K. Gladstien (see K. Gladstien: Characterization of completeness for trellises of finite length, Algebra Universalis 3 (1973), 341–344).
In this paper the notion of weak chain-completeness is introduced for pseudo-ordered sets as an extension of the notion of chain-completeness of posets (see [3]) and it is shown that every isotone map of a weakly chain-complete pseudo-ordered set into itself has a least fixed point.