International Journal of Algebra and Computation

A semigroup S is of the type in the class of the title if S has a congruence ρ such that S/ρ is a normal band (i.e. satisfies the identities x² = x and axya = ayxa) and all ρ-classes are commutative cancellative semigroups. We consider semigroups S with such a congruence first for completely regular semigroups, then characterize the general case in several ways, including some special cases. When S is an order in a normal band of abelian groups Q, we study the restrictions of Green's relations on Q to S. The paper concludes with the discussion of a free semigroup in the title on two generators.

In [Bardakov and Gongopadhyay, Palindromic width of free nilpotent groups, J. Algebra 402 (2014) 379–391] the authors provided a bound for the palindromic widths of free abelian-by-nilpotent group AN_n of rank n and free nilpotent group N_n,r of rank n and step r. In the present paper, we study palindromic widths of groups and . We denote by the quotient of the group G_n = 〈x₁, …, x_n〉, which is free in some variety by the normal subgroup generated by . We prove that the palindromic width of the quotient is finite and bounded by 3n. We also prove that the palindromic width of the quotient is precisely 2(n - 1). As a corollary to this result, we improve the lower bound of the palindromic width of N_n,r. We also improve the bound of the palindromic width of a free metabelian group. We prove that the palindromic width of a free metabelian group of rank n is at most 4n - 1.

We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier–Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata).

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.

Let S be a numerical monoid with minimal generating set 〈n₁, …, n_t〉. For m ∈ S, if , then is called a factorization length of m. We denote by ℒ(m) = {m₁, …, m_k} (where m_i < m_i+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by Δ(m) = {m_i+1 - m_i | 1 ≤ i < k} and the Delta set of S by Δ(S) = ⋃_m∈SΔ(m). In this paper, we expand on the study of Δ(S) begun in [C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On delta sets of numerical monoids, J. Algebra Appl. 5 (2006) 1–24] in the following manner. Let r₁, r₂, …, r_t be an increasing sequence of positive integers and M_n = 〈n, n + r₁, n + r₂, …, n + r_t〉 a numerical monoid where n is some positive integer. We prove that there exists a positive integer N such that if n > N then |Δ(M_n)| = 1. If t = 2 and r₁ and r₂ are relatively prime, then we determine a value for N which is sharp.

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faà di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss Möbius inversion in certain Hopf algebras built from Bell polynomials.

We introduce ring theoretic constructions that are similar to the construction of wreath product of groups [M. Kargapolov and Y. Merzlyakov, Fundamentals of the Theory of Groups (Springer-Verlag, New York, 1979)]. In particular, for a given graph Γ = (V, E) and an associate algebra A, we construct an algebra B = A wr L(Γ) with the following property: B has an ideal I, which consists of (possibly infinite) matrices over A, B/I ≅ L(Γ), the Leavitt path algebra of the graph Γ. Let W ⊂ V be a hereditary saturated subset of the set of vertices [G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334], Γ(W) = (W, E(W, W)) is the restriction of the graph Γ to W, Γ/W is the quotient graph [G. Abrams and G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293(2) (2005) 319–334]. Then L(Γ) ≅ L(W) wr L(Γ/W). As an application we use wreath products to construct new examples of (i) affine algebras with non-nil Jacobson radicals, (ii) affine algebras with non-nilpotent locally nilpotent radicals.

We investigate Borel ideals on the Hilbert scheme components of arithmetically Cohen–Macaulay (ACM) codimension two schemes in ℙⁿ. We give a basic necessary criterion for a Borel ideal to be on such a component. Then considering ACM curves in ℙ³ on a quadric we compute in several examples all the Borel ideals on their Hilbert scheme component. Based on this we conjecture which Borel ideals are on such a component, and for a range of Borel ideals we prove that they are on the component.

Let G be the free product of groups A and B with commuting subgroups H ≤ A and K ≤ B, and let 𝒞 be the class of all finite groups or the class of all finite p-groups. We derive the description of all 𝒞-separable cyclic subgroups of G provided this group is residually a 𝒞-group. We prove, in particular, that if A, B are finitely generated nilpotent groups and H, K are p′-isolated in the free factors, then all p′-isolated cyclic subgroups of G are separable in the class of all finite p-groups. The same statement is true provided A, B are free and H, K are p′-isolated and cyclic.

Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red(C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.