echanism and the low-energy spectrum of $^9$He. This supports the idea that $^8$He (having closed $p_{3/2}$ subshell) presents a “good” core in the $^{9}$He structure.

*Acknowledgments.* — This work was supported by the Russian Foundation for Basic Research grants 02-02-16550, 02-02-16174, 05-02-16404, and 05-02-17535 by the INTAS grantS 03-51-4496 and 03-54-6545. LVG acknowledge the financial support from the Royal Swedish Academy of Science and Russian Ministry of Industry and Science grant NS-1885.2003.2.

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---
abstract: 'In this article we study left I-orders in the bicyclic monoid $\mathcal{B}$. We give necessary and sufficient conditions for a subsemigroup of $\mathcal{B}$ to be a left I-oreder in $\mathcal{B}$. We then prove that any left I-order in $\mathcal{B}$ is straight.'
address: |
    Department of Mathematics\
    University of York\
    Heslington\
    York YO10 5DD\
    UK
author:
- 'N. Ghroda'
title: 'Bicyclic semigroups of left I-quotients'
---

Introduction
============

The first published description of the bicyclic semigroup was given by Evgenii Lyapin in 1953 [@Lyp]. A description of the subsemigroups of the bicyclic monoid was given in 2005 [@ruskuc]. In this article, we use this description to study left I-orders in the bicyclic monoid.

Many definitions of semigroups of quotients have been proposed and studied. The first, that