Related papers: Operated semigroups, Motzkin paths and rooted tree…
In this article we tackle the combinatorics of coloured hard-dimer objects. This is achieved by identifying coloured hard-dimer configurations with a certain class of rooted trees that allow for an algebraic treatment in terms of…
One of the main virtues of trees is to represent formal solutions of various functional equations which can be cast in the form of fixed point problems. Basic examples include differential equations and functional (Lagrange) inversion in…
We introduce partial group algebras with relations in a purely algebraic framework. Given a group and a set of relations, we define an algebraic partial action and prove that the resulting partial skew group ring is isomorphic to the…
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial…
The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a…
For associative commutative algebras $A$ with Rota-Baxter operator $R$ identities of the algebra $AR=(A,\circ)$, where $a\circ b= aR(b),$ are found.
In this paper, we propose the concept of an $\Omega$-Rota-Baxter system, which is a generalization of a Rota-Baxter system and an $\Omega$-Rota-Baxter algebra of weight zero. In the framework of operated algebras, we obtain a linear basis…
We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid to what we call the semi-balanced case, i.e., when each cycle has the same number of edges pointing in the two directions.…
This is a survey of recent progress in several areas of combinatorial algebra. We consider combinatorial problems about free groups, polynomial algebras, free associative and Lie algebras. Our main idea is to study automorphisms and, more…
We give a brief account of a construction called tokens here, which is significant in algebra, analysis, combinatorics, and physics. Tokens allow to express a semigroup on one set via a semigroup convolution on another set. Therefore tokens…
The algebraic study of special integral operators led to the notions of Rota-Baxter operators and shuffle products which have found broad applications. This paper carries out an algebraic study of general integral operators and equations,…
We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows…
The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are…
We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the…
We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…
We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of…
We describe affine monoids whose group of invertible elements is an active semidirect product of a unipotent group and a torus, in terms of comultiplications on the algebra of regular functions. We introduce the notion of a root monoid,…
Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always…
In this paper, we first construct the free Rota-Baxter family algebra generated by some set $X$ in terms of typed angularly $X$-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra…
Programming languages tend to evolve over time to use more and more concepts from theoretical computer science. Still, there is a gap between programming and pure mathematics. Not all theoretical results have realized their promising…