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In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to pairs of algebras. The notion of B-\((\ell , r)\)-cumulants is defined via a bi-Boolean moment-cumulant formula over the lattice of bi-interval partitions, and it is demonstrated that bi-Boolean independence is equivalent to the vanishing of mixed B-\((\ell , r)\)-cumulants. Furthermore, some of the simplest bi-Boolean convolutions are considered, and a bi-Boolean partial \(\eta \)-transform is constructed for the study of limit theorems and infinite divisibility with respect to the additive bi-Boolean convolution. In particular, a bi-Boolean Lévy–Hinčin formula is derived in perfect analogy with the bi-free case, and some Bercovici–Pata type bijections are provided. Additional topics considered include the additive bi-Fermi convolution, some relations between the \((\ell , r)\)- and B-\((\ell , r)\)-cumulants, and bi-Boolean independence in an amalgamated setting.

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We would like to express our gratitude to the anonymous referee for a careful reading and detailed comments that improved the quality of the paper.

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Correspondence to Yinzheng Gu.

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Communicated by Hari Bercovici.

The work of Yinzheng Gu was partially supported by CIMI (Centre International de Mathématiques et d’Informatique) Excellence Program, ANR-11-LABX-0040-CIMI within the Program ANR-11-IDEX-0002-02, while visiting the Institute of Mathematics of Toulouse. He would like to thank the institute for the generous hospitality and Serban Belinschi for his constant support and valuable advice when this research was conducted.

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Gu, Y., Skoufranis, P. Bi-Boolean Independence for Pairs of Algebras. Complex Anal. Oper. Theory 13, 3023–3089 (2019).

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