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Associative, idempotent, symmetric, and order-preserving operations on chains. (English) Zbl 1456.06004
Associativity of binary operations is important because numerous algebraic structures are defined with associative operations as semigroups, groups, rings, lattices, etc. Associativity has been considered in conjunction with other properties such as idempotency or quasitriviality. The authors characterize the associative, idempotent, symmetric, and order-preserving binary operations on finite chains in terms of their associated semilattice order. They prove that the number of associative, idempotent, symmetric and order-preserving operations on an \(n\)-element chain is just the \(n\)-th Catalan number.
MSC:
06A12 Semilattices
06A05 Total orders
06A07 Combinatorics of partially ordered sets
20M14 Commutative semigroups
Software:
OEIS
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