A multiple conjugation quandle is an algebraic structure whose axioms are motivated from handlebody-knot theory. By using an MCQ Alexander pair f, which is a pair of maps corresponding to a linear extension of a multiple conjugation quandle, we can construct the f-twisted Alexander matrices, which produce invariants for handlebody-knots. The purpose of this paper is to show the sufficiency of the definition of MCQ Alexander pairs in constructing the f-twisted Alexander matrices from the aspect of linear extensions of multiple conjugation quandles. Furthermore, we introduce the notion of cohomologous for MCQ Alexander pairs, which induces the same invariant for handlebody-knots.
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