The word problem for one-relation monoids: a survey

This survey is intended to provide an overview of one of the oldest and most celebrated open problems in combinatorial algebra: the word problem for one-relation monoids. We provide a history of the problem starting in 1914, and give a detailed overview of the proofs of central results, especially those due to Adian and his student Oganesian. After showing how to reduce the problem to the left cancellative case, the second half of the survey focuses on various methods for solving partial cases in this family. We finish with some modern and very recent results pertaining to this problem, including a link to the Collatz conjecture. Along the way, we emphasise and address a number of incorrect and inaccurate statements that have appeared in the literature over the years. We also fill a gap in the proof of a theorem linking special inverse monoids to one-relation monoids, and slightly strengthen the statement of this theorem.

Special arrangements of lines: Codimension 2 ACM varieties in ℙ1 × ℙ1 × ℙ1

In this paper, we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimension 2 arithmetically Cohen–Macaulay (ACM) varieties in [Formula: see text], called varieties of lines. We also describe their ACM property from a combinatorial algebra point of view.

Combinatorial Algebra and Its Antisymmetrized Algebra I

The simple non-associative algebra N(eAS, q, n, t)k and its simple subalgebras are defined in [1, 3, 5–7, 13]. In this work, we define the combinatorial algebra N(e𝔄p, n, t)k and its antisymmetrized algebra [Formula: see text] and their subalgebras. We prove that these algebras are simple. Some authors [2, 5–7, 10, 13, 14, 16, 17] found all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra. We find all the derivations of the subalgebra N(e±x1x2 ⋯xn, 0, n)[1] of N(e𝔄p, n, t)k and the Lie subalgebra [Formula: see text] of [Formula: see text].

Structure Coefficients of the Hecke Algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$

The Hecke algebra of the pair $(\mathcal{S}_{2n},\mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial algebra which projects onto the Hecke algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$ for every $n$. To build it, by using partial bijections we introduce and study a new class of finite dimensional algebras.