We characterize the associative, idempotent, symmetric, and order-preserving binary operations on (finite) chains in terms of properties of (the Hasse diagram of) their associated semilattice order. In particular, we prove that the number of associative, idempotent, symmetric, and order-preserving operations on an n -element chain is the n th Catalan number.

中文翻译：

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链上的关联、幂等、对称和保序操作

我们根据（有限）链上的关联、幂等、对称和保序二元运算，根据其相关半格阶（的哈斯图）的性质来描述它们的特征。特别地，我们证明了 n 元素链上的结合、幂等、对称和保序操作的数量是第 n 个加泰罗尼亚数。