On lattices from combinatorial game theory: infinite case

Given a set of combinatorial games, the children are all those games that can be generated using as options the games of the original set. It is known that the partial order of the children of all games whose birthday is less than a fixed ordinal is a distributive lattice and also that the children of any set of games form a complete lattice. We are interested in the converse. In a previous paper, we showed that for any finite lattice there exists a finite set of games such that the partial order of the children, minus the top and bottom elements, is isomorphic to the original lattice. Here, the main part of the paper is to extend the result to infinite complete lattices. An original motivating question was to characterize those sets whose children generate distributive lattices. While we do not solve it, we show that if the process of taking children is iterated, eventually the corresponding lattice is distributive