Abelian Kernels of Some Monoids of Injective Partial Transformations and an Application

In this paper we compute the abelian kernels of the monoids POI_n and POPI_n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI_n (n epsilon N) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI_n (n epsilon N) with the pseudovariety Ab of all finite abelian groups.     

The Monoid of All Injective Order Preserving Partial Transformations on a Finite Chain

In this paper we study several structural properties of the monoids \poi _n of all injective order preserving partial transformations on a chain with n elements. Our main aim is to give a presentation for these monoids. January 27, 1999     

n-Dimensional algebras over a field with a cyclic extension of degree n

We give a geometric method of classifying algebras A _n,K , n-dimensional over a field K, with a cyclic extension of degree n. Algebras A _n,K without zero divisors satisfying some conditions are classified. In particular, we determine all n-dimensional division algebras over a finite field F _q when n is prime and q is large enough.This research was supported in part by a grant from the M U R S T (40 % funds).     

Non-Unique Factorization of Polynomials Over Residue Class Rings of the Integers

We investigate non-unique factorization of polynomials in ? _{p ⁿ} [x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, ? _{p ⁿ} [x] is atomic. We reduce the question of factoring arbitrary nonzero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of ? _{p ⁿ} [x] is a direct sum of monoids corresponding to irreducible polynomials in ? _p [x], and we show that each of these monoids has infinite elasticity. Moreover, for every m ∈ ?, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.     

Bilateral semidirect product decompositions of transformation monoids

In this paper we consider the monoid OR_n\mathcal {OR}_{n} of all full transformations on a chain with n elements that preserve or reverse the orientation, as well as its submonoids OD_n\mathcal {OD}_{n} of all order-preserving or order-reversing elements, OP_n\mathcal {OP}_{n} of all orientation-preserving elements and O_n\mathcal {O}_{n} of all order-preserving elements. By making use of some well known presentations, we show that each of these four monoids is a quotient of a bilateral semidirect product of two of its remarkable submonoids.     

Monoids and Groups of <Emphasis Type="Italic">I</Emphasis>-Type

A monoid S generated by {x₁,. . .,x_n} is said to be of (left) I-type if there exists a map v from the free Abelian monoid FaM_n of rank n generated by {u₁,. . .,u_n} to S so that for all a∈FaM_n one has {v(u₁a),. . .,v(u_na)}={x₁v(a),. . .,x_nv(a)}. Then S has a group of fractions, which is called a group of (left) I-type. These monoids first appeared in the work of Gateva-Ivanova and Van den Bergh, inspired by earlier work of Tate and Van den Bergh. In this paper we show that monoids and groups of left I-type can be characterized as natural submonoids and groups of semidirect products of the free Abelian group Fa_n and the symmetric group of degree n. It follows that these notions are left–right symmetric. As a consequence we determine many aspects of the algebraic structure of such monoids and groups. In particular, they can often be decomposed as products of monoids and groups of the same type but on less generators and many such groups are poly-infinite cyclic. We also prove that the minimal prime ideals of a monoid S of I-type, and of the corresponding monoid algebra, are principal and generated by a normal element. Further, via left–right divisibility, we show that all semiprime ideals of S can be described. The latter yields an ideal chain of S with factors that are semigroups of matrix type over cancellative semigroups. In memory of Paul Wauters Mathematics Subject Classifications (2000) 20F05, 20M05; 16S34, 16S36, 20F16. The authors were supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish–Polish bilateral agreement BIL 01/31, and KBN research grant 2P03A 033 25 (Poland).     

PRESENTATIONS FOR SOME MONOIDS OF PARTIAL TRANSFORMATIONS ON A FINITE CHAIN

ABSTRACT

In this paper we calculate presentations for some natural monoids of transformations on a chain X _n  = {1 < 2 <?s < n}. First we consider 𝒪𝒟 _n [𝒫𝒪𝒟 _n ], the monoid of all full [partial] transformations on X _n that preserve or reverse the order. Two other monoids of partial transformations on X _n we look at are 𝒫𝒪𝒫 _n and 𝒫𝒪? _n –-the elements of the first preserve the orientation and the elements of the second preserve or reverse the orientation.     

The classification of (J, σ)-irreducible monoids of type A 4

In this paper we develop a very basic method to classify (J, σ)-irreducible monoids of type A ₄. As a typical example, we list all the types for the monoids corresponding to the strongly dominant weights. This example also shows that there is no general theorem to determine the cross-section lattices for reductive monoids according to their Dynkin diagrams as Putcha and Renner’s recipe for J-irreducible monoids.     

Partitions of bipartite numbers

Letp _j(m, n) be the number of partitions of (m, n) into at mostj parts. We prove Landman et al.'s conjecture: for allj andn, p _j(x, 2n–x) is a maximum whenx-n. More generally we prove that for all positive integersm, n andj, p _j(n, m)=p_j(m, n)p_j(m–1, n+1) ifmn.     

On sets of integers with prescribed gaps

For a fixed setI of positive integers we consider the set of paths (p _o,...,p _k ) of arbitrary length satisfyingp _l –p _l–1I forl=2,...,k andp ₀=1,p _k =n. Equipping it with the uniform distribution, the random path lengthT _n is studied. Asymptotic expansions of the moments ofT _n are derived and its asymptotic normality is proved. The step lengthsp _l –p _l–1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values ofp ₀ andp _k are not specified. In the special caseI={m+1,m+2,...} (for some fixed m) we derive the exact distribution of a random m-gap subset of {1,...,n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a randomm-gap subset is also presented.     

Schensted-Type Correspondences and Plactic Monoids for Types B n and D n

We use Kashiwara's theory of crystal bases to study plactic monoids for U _q(so _2n+1) and U _q(so _2n ). Simultaneously we describe a Schensted type correspondence in the crystal graphs of tensor powers of vector and spin representations and we derive a Jeu de Taquin for type B from the Sheats sliding algorithm.     

Asymptotic convergence of degree‐raising

It is well known that the degree‐raised Bernstein–Bézier coefficients of degree n of a polynomial g converge to g at the rate 1/n. In this paper we consider the polynomial A _n(g) of degree ⩼ n interpolating the coefficients. We show how A _n can be viewed as an inverse to the Bernstein polynomial operator and that the derivatives A _n(g)(^r) converge uniformly to g(^r) at the rate 1/n for all r. We also give an asymptotic expansion of Voronovskaya type for A _n(g) and discuss some shape preserving properties of this polynomial. This revised version was published online in June 2006 with corrections to the Cover Date.     

Betti Numbers of Toric Varieties and Eulerian Polynomials

It is well known that the Eulerian polynomials, which count permutations in S _n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the S _n -orbit for a generic weight in the weight lattice of S _n . Therefore, the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra.

In this article we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a nongeneric weight, namely, a weight fixed by only the simple reflections J = {s _n , s _n?1, s _n?2,…, s _n?k+2, s _n?k+1} for some k with respect to the A _n root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincaré polynomial of X(J) in terms of the Eulerian polynomials.     

On Singular Artin Monoids and Contributions to Birman's Conjecture

ABSTRACT

Baez and Birman introduced the singular braid monoid on n + 1 strings, 𝒮B _n+1, which Birman uses in understanding knot invariants. 𝒮? _n+1 is the type A _n case of an infinite class of monoids known as singular Artin monoids and denoted by 𝒮G _M for a Coxeter matrix M. Birman conjectured, and Paris proved, that 𝒮B _n+1 embeds in the complex algebra of the braid group under the desingularisation map or Vassiliev homomorphism, η. In effect, Birman's conjecture generalizes to arbitrary types since, as noted by Corran, the Vassiliev homomorphism from 𝒮G _M to the algebra of the corresponding Artin group is well defined. We deduce general combinatorial results regarding divisibility in positive singular Artin monoids, and when M is of finite type, a well-defined positive form for 𝒮G _M is produced. These facts are then invoked to infer that, when M is of finite type, η is injective on pairs of words such that a common multiple exists for their positive form.     

On Genuine Bernstein–Durrmeyer Operators

We continue the studies on the so–called genuine Bernstein–Durrmeyer operators U _n by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by U _n . Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of U _n , compute the moments of T(t) and establish asymptotic formulas. Received: January 26, 2007.     

On the automorphism conjecture for products of ordered sets

I. Rival and A. Rutkowski conjectured that the ratio of the number of automorphisms of an arbitrary poset to the number of order-preserving maps tends to zero as the size of the poset tends to infinity. We prove this hypothesis for direct products of arbitrary posets P=S ₁××S _n under the condition that max_i|S_i|=0(n/logn).     

Presentations for Two Extensions of the Monoid of Order-Preserving Mappings on a Finite Chain

The purpose of this paper is to give presentations for the monoids of orientation-preserving mappings on a finite chain of order n, and orientation-preserving or reversing mappings on such a chain. Both these monoids are natural extensions of the monoid of order-preserving mappings. The obtained presentations are on two and three generators, respectively, and have n + 2 and n + 6 relations, respectively.AMS Subject Classification (1991): 20M05 20M20 06A05