Braids and factorizable inverse monoids

What is the untangling effect on a braid if one is allowed to snip a string, or if two specified strings are allowed to pass through each other, or even allowed to merge and part as newly reconstituted strings? To calculate the effects, one works in an appropriate factorizable inverse monoid, some aspects of a general theory of which are discussed in this paper. The coset monoid of a group arises, and turns out to have a universal property within a certain class of factorizable inverse monoids. This theory is dual to the classical construction of fundamental inverse semigroups from semilattices. In our braid examples, we will focus mainly on the ``merge and part'' alternative, and introduce a monoid which is a natural preimage of the largest factorizable inverse submonoid of the dual symmetric inverse monoid on a finite set, and prove that it embeds in the coset monoid of the braid group

Presentations of factorizable inverse monoids

It is well-known that an inverse monoid is factorizable if and only if it is a homomorphic image of a semidirect product of a semilattice (with identity) by a group. We use this structure to describe a presentation of an arbitrary factorizable inverse monoid in terms of presentations of its group of units and semilattice of idempotents, together with some other data. We apply this theory to quickly deduce a well known presentation of the symmetric inverse monoid on a nite set

Periodic elements of the free idempotent generated semigroup on a biordered set

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup

On regularity and the word problem for free idempotent generated semigroups

The category of all idempotent generated semigroups with a prescribed structure E of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over E, defined by a presentation over alphabet E, and denoted by IG(E). Recently, much effort has been put into investigating the structure of semigroups of the form IG(E), especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for IG(E). We prove two principal results, one positive and one negative. We show that, for a finite biordered set E, it is decidable whether a given word w ∈ E∗represents a regular element; if in addition one assumes that all maximal subgroups of IG(E) have decidable word problems, then the word problem in IG(E) restricted to regular words is decidable. On the other hand, we exhibit a biorder E arising from a finite idempotent semigroup S, such that the word problem for IG(E) is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of IG(E) to the subgroup membership problem in finitely presented groups

Periodic Elements of the Free Idempotent Generated Semigroup on a Biordered Set

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup. © 2010 World Scientific Publishing Company.The second author was supported in part by the NSF grant DMS-0700811 and by a BSF (USA-Israeli) grant. The third author acknowledges support from the Russian Foundation for Basic Research, grant 09-01-12142, and from the Federal Education Agency of Russia, grant 2.1.1/3537

Are Labour Markets Necessarily Local? Spatiality, Segmentation and Scale

This paper draws on recent debates about scale to approach the geography of labour markets from a dynamic perspective sensitive to the spatiality and scale of labour market restructuring. Its exploration of labour market reconfigurations after the collapse of a major firm (Ansett Airlines) raises questions about geography’s faith in the inherently ‘local’ constitution of labour markets. Through an examination of the job reallocation process after redundancy, the paper suggests that multiple labour markets use and articulate scale in different ways. It argues that labour market rescaling processes are enacted at the critical moment of recruitment, where social networks, personal aspirations and employer preferences combine to shape workers’ destinations

Vegetable Soybean: A Crop with Immense Potential to Improve Human Nutrition and Diversify Cropping Systems in Eastern India- A Review

Vegetable soybean was introduced into Jharkhand state in India to provide local communities with an alternative protein source as well as to diversify the cropping system. The paper covers the introduction of the crop, testing of different lines in farmers’ fields, a taste survey, the release of a vegetable soybean cultivar and its popularization, and seed production by the community. The acceptance of the crop by the local community has led to seed production of the cultivar ‘Swarna Vasundhara’ by farmers. A major challenge is to expand beyond household level consumption and local Jharkhand markets to create awareness among other consumers and establish a larger market throughout India for the crop. Infrastructure such as cold storage facilities will be required to support a viable value chain for vegetable soybean. The good progress made in Jharkhand has given greater hope of expanding vegetable soybean production and consumption across India, particularly in the major grain soybean growing states