Related papers: Binary Tree Arithmetic with Generalized Constructo…
A Bialgebra is a module over a ring that is both an associative algebra and a co-associative coalgebra with the product and coproduct additionally satisfying an appropriate commutative relationship. One application of Bialgebras is in the…
The arithmetic of the natural numbers can be extended to arithmetic operations on planar binary trees. This gives rise to a non-commutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and…
We study algebraic synchronization trees, i.e., initial solutions of algebraic recursion schemes over the continuous categorical algebra of synchronization trees. In particular, we investigate the relative expressive power of algebraic…
Following our previous work, we suggest here a large class of algebras of scalars in which simultaneous and correlated computations can be performed owing to the existence of surjective algebra homomorphisms. This may replace the currently…
We introduce a generalization of tridendriform algebras, where each of the three products are replaced by a family of products indexed by a set $\Omega$. We study the needed structure on $\Omega$ for free $\Omega$-tridendriform algebras to…
Alphabetic codes and binary search trees are combinatorial structures that abstract search procedures in ordered sets endowed with probability distributions. In this paper, we design new linear-time algorithms to construct alphabetic codes,…
Ordinals can help prove termination for dependently typed programs. Brouwer trees are a particular ordinal notation that make it very easy to assign sizes to higher order data structures. They extend natural numbers with a limit…
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…
Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally…
First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous…
We introduce the notion of "binary" positive and complex geometries, giving a completely rigid geometric realization of the combinatorics of generalized associahedra attached to any Dynkin diagram. We also define open and closed "cluster…
This paper argues that mathematical objects are constructions and that constructions introduce a flexibility in the ways that mathematical objects are represented (as sets of binary sequences for example) and presented (in a particular…
For $\mathcal{O}$ a reduced operad, a generalized divergence from the derivations of a free $\mathcal{O}$-algebra to a suitable trace space is constructed. In the case of the Lie operad, this corresponds to Satoh's trace map and, for the…
We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the…
We extend the results we obtained in an earlier work. The cocommutative case of rooted ladder trees is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the…
We provide a unified framework for the treatment of special integrable systems which we propose to call "generalized mean field systems". Thereby previous results on integrable classical and quantum systems are generalized. Following…
As a fundamental and ubiquitous combinatorial notion, species has attracted sustained interest, generalizing from set-theoretical combinatorial to algebraic combinatorial and beyond. The Rota-Baxter algebra is one of the algebraic…
Terms are a concise representation of tree structures. Since they can be naturally defined by an inductive type, they offer data structures in functional programming and mechanised reasoning with useful principles such as structural…
In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch…
We study varieties that contain unranked tree languages over all alphabets. Trees are labeled with symbols from two alphabets, an unranked operator alphabet and an alphabet used for leaves only. Syntactic algebras of unranked tree languages…