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 $η$-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.
The last section of this paper also includes an errata that will be published with this copy of the paper.
Bigraph independence : a mixture of the five natural independences
CNRS, MAP5 Universit \'e Paris Cit \'e 45 rue des Saints-P \`e res, F-75006, Paris, France；Department of Mathematical Sciences University of Copenhagen Universitetsparken 5, 2100 K benhavn , Denmark
An asymptotic property of large matrices with identically distributed Boolean independent entries
受近年来关于非交换元随机矩阵渐近独立关系的研究成果的启发，本文研究了布尔独立元同分布大型矩阵的极限分布和渐近独立关系.更准确地说，我们表明，在某些时刻的条件下，这样的随机矩阵是渐近$ B $ -对角和布尔相互独立。本文还给出了一个组合条件，在此条件下，这些矩阵是渐近布尔独立的矩阵置换的项目（从而推广了最近的结果在布尔概率）。特别是，我们认为随机矩阵是渐近布尔独立于他们的部分转置。本文的主要结果是基于组合技术。
