Abstract
The subject matter of this work is hoping for a new relationship between the Lie algebras and the algebra of logic, which will constitute an important part of our study of "pure'' algebra theory. $BCL$ algebras as a class of logical algebras is can be generated by a Lie algebra. The opposite is also true that when special conditions occur. The aim of this paper is to prove several theorems on Lie algebras with $BCL$ algebras. I introduce the notion of a "pseudo-association'' which I propose as the adjoint notion of $BCL$ algebra in the abelian group.
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