Exponential series without denominators

For a commutative algebra which comes from a Zinbiel algebra the exponential series can be written without denominators. When lifted to dendriform algebras this new series satisfies a functional equation analogous to the Baker-Campbell-Hausdorff formula. We make it explicit by showing that the obstruction series is the sum of the brace products. In the multilinear case we show that the role the Eulerian idempotent is played by the iterated pre-Lie product.Comment: 13

The YY game

We introduce a new one-person game similar to the Sudoku game. It is based on combinatorial objects called planar binary rooted trees. It is related to the four color conjecture. Its mathematical analysis makes use of the Tamari poset, hence the Stasheff associahedron.Comment: 7 page

On Products and Duality of Binary, Quadratic, Regular Operads

Since its introduction by Loday in 1995 with motivation from algebraic K-theory, dendriform dialgebras have been studied quite extensively with connections to several areas in mathematics and physics. A few more similar structures have been found recently, such as the tri-, quadri-, ennea- and octo-algebras, with increasing complexity in their constructions and properties. We consider these constructions as operads and their products and duals, in terms of generators and relations, with the goal to clarify and simplify the process of obtaining new algebra structures from known structures and from linear operators.Comment: 22 page