It is elementary and well known that if an element x of a bounded modular lattice L $ \mathbf L $ has a complement in L $ \mathbf L $ then x has a relative complement in every interval [a, b] containing x. We show that the relatively strong assumption of modularity of L $ \mathbf L $ can be replaced by a weaker one formulated in the language of so-called modular triples. We further show that, in general, we need not suppose that x has a complement in L $ \mathbf L $ . By introducing the concept of modular triples in posets, we extend our results obtained for lattices to posets. It should be remarked that the notion of a complement can be introduced also in posets that are not bounded.

中文翻译：

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迫使格和偏序集中存在相对补集的条件

这是基本的和众所周知的，如果一个元素X有界模格的 大号 $ \mathbf L $ 有补语 大号 $ \mathbf L $ 然后X在每个区间都有一个相对补集[A,b] 含X. 我们表明，相对强的模块化假设 大号 $ \mathbf L $ 可以用所谓的模块化三元组语言表述的较弱的替换。我们进一步表明，一般来说，我们不需要假设X有补语 大号 $ \mathbf L $ . 通过在偏序集中引入模三元组的概念，我们将对格获得的结果扩展到偏序集中。应该注意的是，补集的概念也可以引入到无界偏序集中。