The geometry of Boolean space and its elementary figures. I (Q1336670)

The authors give a characterization of Boolean algebras \((B,+,\cdot, ', \circ, 1)\) in terms of so-called Boolean spaces \((B, \leq)\) [cf. the authors, ibid. 90, 75-84 (1991; Zbl 0734.51012)] by a geometrically inspired system of axioms which refers to a given partial ordering \(\leq\) on the pointset \(B\) and its intervals (termed linear subspaces by the authors). ``Symmetric difference'' \(A\oplus B\) of \(A, B\in B\) is considered as a distance function with values in \(B\) [cf. Chapter XV in \textit{L. M. Blumenthal's} book: Theory and applications of distance geometry (1953; Zbl 0050.385)]. This function gives rise to a notion of perpendicularity of intervals \([AB, A+ B]\) and \([CD, C+ D]\) by requiring \((A\oplus B)(C\oplus D)= 0\), which then is examined in some detail: Among others, perpendicular projections, triangles, heights, circumspheres and inspheres of triangles are studied; many peculiarities are shown to arise in the Boolean setting, and a Boolean analogue of the Pythagorean theorem is given.