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Register a new userNonsymmetric operads in combinatorics
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Authors
Samuele Giraudo
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Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operads, together with the algebraic setting and the tools accompanying it, promises advances in these two areas. On the one hand, operads provide a useful abstraction of formal expressions, and also, provide connections with the theory of rewrite systems. On the other hand, a lot of operads involving combinatorial objects highlight some of their properties and allow to discover new ones. This book presents the theory of nonsymmetric operads under a combinatorial point of view. It portrays the main elements of this theory and the links it maintains with several areas of computer science and combinatorics. A lot of examples of operads appearing in combinatorics are studied and some constructions relating operads with known algebraic structures are presented. The modern treatment of operads consisting in considering the space of formal power series associated with an operad is developed. Enrichments of nonsymmetric operads as colored, cyclic, and symmetric operads are reviewed. This text is addressed to any computer scientist or combinatorist who looks a complete and a modern description of the theory of nonsymmetric operads. Evenly, this book is intended to an audience of algebraists who are looking for an original point of view fitting in the context of combinatorics.
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The paper concerns the Gelfand-Kirillov dimension and the generating series of nonsymmetric operads. An analogue of Bergmans gap theorem is proved, namely, no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov dimension strictly between $1$ and $2$. For every $rin {0}cup {1}cup [2,infty)$ or $r=infty$, we construct a single-element generated nonsymmetric operad with Gelfand-Kirillov dimension $r$. We also provide counterexamples to two expectations of Khoroshkin and Piontkovski about the generating series of operads.
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Bore
A new hierarchy of operads over the linear spans of $delta$-cliffs, which are some words of integers, is introduced. These operads are intended to be analogues of the operad of permutations, also known as the associative symmetric operad. We obtain operads whose partial compositions can be described in terms of intervals of the lattice of $delta$-cliffs. These operads are very peculiar in the world of the combinatorial operads since, despite to the relative simplicity for their construction, they are infinitely generated and they have nonquadratic and nonhomogeneous nontrivial relations. We provide a general construction for some of their quotients. We use it to endow the spaces of permutations, $m$-increasing trees, $c$-rectangular paths, and $m$-Dyck paths with operad structures. The operads on $c$-rectangular paths admit, as Koszul duals, operads generalizing the duplicial and triplicial operads.
Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking an uniform approach, this paper gives an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly strict restraints among the replicated copies. The linear compatibility is in Koszul dual to the total compatibility, while the matching compatibility is self dual. Further, each compatibility can be expressed in terms of either one or both of the two Manin square products.
Using the combinatorial species setting, we propose two new operad structures on multigraphs and on pointed oriented multigraphs. The former can be considered as a canonical operad on multigraphs, directly generalizing the Kontsevich-Willwacher operad, and has many interesting suboperads. The latter is a natural extension of the pre-Lie operad in a sense developed here and related to the multigraph operad. We study some of the finitely generated suboperads of the multigraph operad and establish links between them and the commutative operad and the commutative magmatic operad.
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