CLIFFORD GROUPS AND MONOIDS ASSOCIATED WITH ARBITRARY QUADRATIC MODULES OVER COMMUTATIVE RINGS

For a quadratic module, degenerate or not, over a commutative ring with 1, we determine some of the important subalgebras of its Clifford algebra under some conditions, and consider the Clifford group and related ones with the maps to the orthogonal group of the module. By allowing elements that are not necessarily invertible, but only have a property that we call pseudo-invertibility, we extend these groups to larger monoids, which still carry natural maps to the orthogonal group, with the image sometimes containing orthogonal transformations that are not in the image of any group from the classical theory. Our objects may contain elements that are not locally homogeneous, which allows us obtain an analogous theory in the paravector setting.