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.

中文翻译：

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组合博弈论中的格：无限情况

给定一组组合游戏，孩子们是所有可以使用原始游戏集作为选项生成的游戏。已知生日小于固定序数的所有游戏的孩子的偏序是一个分配格，并且任何游戏集的孩子都构成一个完整的格。我们对相反感兴趣。在之前的一篇论文中，我们表明，对于任何有限格，都存在一组有限的博弈，使得子元素的偏序（减去顶部和底部元素）与原始格同构。在这里，论文的主要部分是将结果扩展到无限完全格。一个原始的激励问题是表征那些子集生成分配格的集合。虽然我们不解决它，