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Combinatorial operads from monoids. (English) Zbl 1308.05110

Summary: We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operads obtained from usual monoids such as the additive and multiplicative monoids of integers and cyclic monoids. They involve various familiar combinatorial objects: endofunctions, parking functions, packed words, permutations, planar rooted trees, trees with a fixed arity, Schröder trees, Motzkin words, integer compositions, directed animals, and segmented integer compositions. We also recover some already known (symmetric or not) operads: the magmatic operad, the associative commutative operad, the diassociative operad, and the triassociative operad. We provide presentations by generators and relations of all constructed nonsymmetric operads.

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05C05 Trees
55P48 Loop space machines and operads in algebraic topology

Keywords:

operad; monoid; rewriting
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