Related papers: Combinatorial operads from monoids
We construct certain monoids, called tied monoids. These monoids result to be semidirect products finitely presented and commonly built from braid groups and their relatives acting on monoids of set partitions. The nature of our monoids…
This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant…
We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. As all previously known examples, our examples are contracted semigroup algebras and the underlying…
We study Hamiltonian paths and cycles in undirected graphs from an operadic viewpoint. We show that the graphical collection $\mathsf{Ham}$ encoding directed Hamiltonian paths in connected graphs admits an operad-like structure, called a…
We show how to "interleave" the monad for operads and the monad for contractions on the category \coll of collections, to construct the monad for the operads-with-contraction of Leinster. We first decompose the adjunction for operads and…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of…
A certain analysis of all possible associative binary operations on N is presented. This is equivalent with an analysis of all possible monoid structures on N. Several results and a conjecture in this regard are given.
Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the…
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
We develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We build explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the…
Usually a name of the category is inherited from the name of objects. However more relevant for a category of objects and morphisms is an algebra of morphisms. Therefore we prefer to say a category of graphs if every morphism is a graph. In…
It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint $Sym_1$ given by product with the symmetric group operad. It is also well known that this functor does not affect the category of…
In this paper we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular we construct a braided primitive functor…
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs,…
The whole enterprise of spin compositions can be recast as simple enumerative combinatoric problems. We show here that enumerative combinatorics (EC)\citep{book:Stanley-2011} is a natural setting for spin composition, and easily leads to…
In a recent paper, we described a lifting of coordinate rings of groups, loops, quantum groups, etc. to the categoric setup of operads. In most examples of that paper, these rings are non--commutative. Quantum physics of the XX--th century…
Operads were originally defined by May to have right actions of the symmetric groups, but later formulations have also used no groups actions at all or group actions by such families as the braid groups. We call such families action…
For a directed polytope, we construct a colored operad whose Poincare-Hilbert series encodes certain operations on the cellular complex of the polytope. We conjecture that for a class of short polytopes the constructed operads are Koszul…
We construct linear operators factorizing the three bases of symmetric polynomials: monomial symmetric functions m(x), elementary symmetric polynomials E(x), and Schur functions s(x), into products of univariate polynomials.