e $$a^mb^n=(a^{p+m+n}b^m)^{-1}(a^{p+m+n}b^n).$$It is clear that $a^{p+m+n}b^m,a^{p+m+n}b^n \in S$.

\[twostraightl\] If $S=F_{D}\cup \widehat{F}\cup \widehat{\Lambda}_{I,p,d}\cup \Sigma_{p,d,P}$ is a left I-order in $\mathcal{B}$, then it is straight.

From corollaries  \[twostraightr\] and  \[twostraightl\], we have the main result in this section which is the third result in this article.

Let $S$ be a two-sided subsemigroup of $\mathcal{B}$. If $S$ is a left I-order in $\mathcal{B}$, then it is straight.

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J. B. Fountain and Mario Petrich, Completely 0-simple semigroups of quotients, *Journal of Algebra* **101** (1986), 365-402.

N. Ghroda and Victoria Gould, Semigroups of inverse quotients, *Periodica Mathematica Hungarica*, to appear. N. Ghroda, Bisimple inverse $\omega$-semigroups of left I-quotients, [*ArXiv:1008.3241*]{}.

V. Gould, Bisimple inverse  $\omega$-semigroup of left quotients, *Proc. London Math. Soc.* **52** (1986), 95-118. E. S. Lyapin, Canonical form of elements of an associative system given by defining relations, *Leningrad Gos. Ped. Inst. Uch. Zap.* **89** (1953), 45-54 \[Russian\].


---
author:
- 'Xiao Fang$^*$, Shige Peng$^\dagger$, Qi-Man Shao$^*$, Yongsheng Song$^\ddagger$'
date: '*The Chinese University of Hong Kong$^*$, Shandong University$^\dagger$, Chinese Academy of Sciences$^\ddagger$*'
title: Limit theorems with rate of convergence under sublinear expectations
---

[**Abstract:**]{} Under the sublinear expectation $\mathbbm{E}[\cdot]:=\sup_{\theta\in \Theta} E_\theta[\cdot]$ for a given set of linear expectations $\{E_\theta: \theta\in \Theta\}$, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical infere