J. Lawson and Y. Lim
Geometric Means and Reflection Quasigroup

In this paper we show how the category of reflection quasigroups forms a natural and suitable context for the development of an abstract theory of the geometric mean, as it appears in matrix and operator theory. We provide a new characterization of those quasigroups that arise when reflection quasigroups are endowed with the mean operation. We also show how the notion of the geometric mean can be enlarged to that of weighted means, develop basic properties of the latter, and illustrate their usefulness in solving equations involving the mean operation.

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