Brauer and Jones tied monoids

We introduce a ramified monoid, attached to each Brauer–type monoid, that is, to the symmetric group, to the Jones and Brauer monoids among others. Ramified monoids correspond to a class of tied monoids arising from knot theory and are interesting in themselves. The ramified monoid attached to the symmetric group is the Coxeter-like version of the so–called tied braid monoid. We give a presentation of the ramified monoid attached to the Brauer monoid. Also, we introduce and study two tied-like monoids that cannot be described as ramified monoids. However, these monoids can also be regarded as tied versions of the Jones and Brauer monoids.     

A special class of tensor categories initiated by inverse braid monoids

In this paper, we introduce the concept of a wide tensor category which is a special class of a tensor category initiated by the inverse braid monoids recently investigated by Easdown and Lavers [The Inverse Braid Monoid, Adv. in Math. 186 (2004) 438-455].The inverse braid monoidsIB_n is an inverse monoid which behaves as the symmetric inverse semigroup so that the braid group B_n can be regarded as the braids acting in the symmetric group. In this paper, the structure of inverse braid monoids is explained by using the language of categories. A partial algebra category, which is a subcategory of the representative category of a bialgebra, is given as an example of wide tensor categories. In addition, some elementary properties of wide tensor categories are given. The main result is to show that for every strongly wide tensor category C, a strict wide tensor category C^str can be constructed and is wide tensor equivalent to C with a wide tensor equivalence F.As a generalization of the universality property of the braid category B, we also illustrate a wide tensor category through the discussion on the universality of the inverse braid category IB (see Theorem 3.3, 3.6 and Proposition 3.7).     

Factorization theory of root closed monoids of small rank

In this note, we investigate ideal and factorization-theoretic properties of some root closed cancellative commutative monoids of rank at most two.     

Finiteness and Dehn functions of automatic monoids having directed fellow traveller property

A left-cancellative automatic monoid having directed fellow traveller property is finitely presented, and the first order Dehn functions of such automatic monoids are bounded above by a quadratic function. These results coincide with those of automatic groups. The research of X. Wang was partially supported by China National Science Funds (No:10771077 and 10671114).     

Inverse monoids of graphs

. IntroductionGraph endomorphism and its regularity property have been investigated in some literatures (of. [1--41 for examples). The invertibility is a stronger algebraic property thanregUlarity in semigroup theory. It is commonly agreed that inverse semigroups are the mostpromising class of semigroups for study. In this paper we first present a combinatorial characterization of an inverse monoid of a graph (Theorem 2.3). Then using this we prove thata bipartite graph with an inverse monoi…     

Brauer Characters of Finite Monoids

We introduce the concept of a Brauer character of a representation of a finite monoid M in characteristic p>0. When p does not divide the order of any subgroup of M, we develop a theory of p-monoid quivers. We apply our results to the full transformation semigroup.     

Resolutions for free partially commutative monoids

We construct a free resolution for a free partially commutative monoid and, using this resolution, estimate the homological dimension of the monoid.     

On the endomorphism monoids of (uniquely) complemented lattices

Let be a lattice with and . An endomorphism of is a -endomorphism, if it satisfies and . The -endomorphisms of form a monoid. In 1970, the authors proved that every monoid can be represented as the -endomorphism monoid of a suitable lattice with and . In this paper, we prove the stronger result that the lattice with a given -endomorphism monoid can be constructed as a uniquely complemented lattice; moreover, if is finite, then can be chosen as a finite complemented lattice.

     

Rationally smooth algebraic monoids

A reductive monoid M is called rationally smooth if it has sufficiently mild singularities as a topological space. We characterize this class of monoids in combinatorial terms. We then use our results to calculate the Betti numbers of certain projective, rationally smooth group embeddings using the “monoid BB-decomposition”.     

Principal ideals of finitely generated commutative monoids

We study the semigroups isomorphic to principal ideals of finitely generated commutative monoids. We define the concept of finite presentation for this kind of semigroups. Furthermore, we show how to obtain information on these semigroups from their presentations.     

Special ideals in partial abelian monoids

Under some conditions, the special congruences of partial abelian monoid are those induced by the special ideals, and a class of special ideals of partial abelian monoid has some upper and lower bound properties.     

Catalan monoids,monoids of local endomorphisms,and their presentations

The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of finite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Aîzen?tat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a finite symmetric chain.     

半群上Rees矩阵半群的半格的结构

推广了Ｍ．Ｐｅｔｒｉｃｈ在文［１］中所用的方法，得到了幺半群上Ｒｅｅｓ矩阵半群的半格的一个结构定理．研究了单幂幺半群上Ｒｅｅｓ矩阵半群的半格的性质并给出了矩形单幂幺半群的半格的若干等价刻划．     

On amenability of two-generator groups

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that if the group is generated by two elements and is amenable then there are words being relators whose most of the consecutive pairs of the letters belong to a certain four-element set of pairs. This fact is applied to reproving non-amenability of a braid group. The same group provides an example showing that such type of condition is not su?cient for amenabilty.     

On minimum delta set values in block monoids over cyclic groups

The difference in length between two distinct factorizations of an element in a Dedekind domain or in the corresponding block monoid is an object of study in the theory of non-unique factorizations. It provides an alternate way, distinct from what the elasticity provides, of measuring the degree of non-uniqueness of factorizations. In this paper, we discuss the difference in consecutive lengths of irreducible factorizations in block monoids of the form where . We will show that the greatest integer r, denoted by , which divides every difference in lengths of factorizations in can be immediately determined by considering the continued fraction of . We then consider the set including necessary and sufficient conditions (which depend on p) for a value to be an element of . 2000 Mathematics Subject Classification Primary—20M14, 11A55, 20D60, 11A51 Parts of this work are contained in the first author’s Doctoral Dissertation written at the University of North Carolina at Chapel Hill under the direction of the third author.     

Matrix presentations of braids and applications

We show that there exists a one-to-one correspondence between the class of certain block tridiagonal matrices with the entries or and the free monoid generated by generators and relation and give some applications for braids. In particular, we give new formulation of the reduced Alexander matrices for closed braids.

     

Posets whose monoids of order-preserving maps are regular

The purpose of this paper is to determine all posetsP such that the monoid of all order-preserving maps ofP intoP is regular in the semigroup-theoretic sense. AMS subject classifications (1980). 06A10, 20M20.     

The catenary degrees of elements in numerical monoids generated by arithmetic sequences

We compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This allows us to define and compute the dissonance number- the largest element with a catenary degree different from the fixed value. We determine the dissonance number in terms of the arithmetic sequence’s starting point and its number of generators.     

左中心rpp半群的半直积

讨论了幂等元都是左中心元的rpp半群的半直积,给出了这种半群半直积的充要条件,推广了一些已知的结果.     

Elasticity in certain block monoids via the Euclidean table

This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤ _n , S) where S = (hereafter denoted ℬ _a (n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬ _a (n). In Section 3 we use the Euclidean table to compute the elasticity of ℬ _a (n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬ _a (n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬ _a (p). Part of this work was completed while the second author was on an Academic Leave granted by the Trinity University Faculty Development Committee.