具有任意多秩且产生相等牌组的非同构有序集
Nonisomorphic Ordered Sets with Arbitrarily Many Ranks That Produce Equal Decks
AI总结：
证明存在具有任意多秩的非同构有序集对，其极大、极小、邻域及特定秩的移除点牌组均相等，仅在秩1和2处牌组不等。
摘要：
我们证明，对于任意 $n$，存在一对非同构有序集 $(P_1^n, P_2^n)$，使得 $P_1^n$ 和 $P_2^n$ 具有相等的极大和极小牌组、相等的邻域牌组，并且存在 $n+1$ 个秩 $k_0, \ldots, k_n$，使得对于每个 $i$，通过移除秩为 $k_i$ 的点所得到的牌组是相等的。秩 $k_1, \ldots, k_n$ 不包含极元，而在其余每个秩上，都存在移除后会产生同构卡片的元素。此外，我们证明可以构造这样的集合，使得仅在秩 $1$ 和 $2$（两者均无极元）处，通过移除秩为 $r_i$ 的点所得到的牌组是不相等的。
英文摘要：
We prove that for any $n$ there is a pair $(P_1 ^n , P_2 ^n )$ of nonisomorphic ordered sets such that $P_1 ^n $ and $P_2 ^n $ have equal maximal and minimal decks, equal neighborhood decks, and there are $n+1$ ranks $k_0 , \ldots , k_n $ such that for each $i$ the decks obtained by removing the points of rank $k_i $ are equal. The ranks $k_1 , \ldots , k_n $ do not contain extremal elements and at each of the other ranks there are elements whose removal will produce isomorphic cards. Moreover, we show that such sets can be constructed such that only for ranks $1$ and $2$, both without extremal elements, the decks obtained by removing the points of rank $r_i $ are not equal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
