Divisors of zero in the Lipschitz semigroup (Q885690)

The Clifford algebra \({\mathcal C}(V,Q)\) is generated by a real vector space \(V\) endowed with a quadratic form \(Q\). The author considers the Lipschitz semigroup \(\Pi\) which is a multiplicative semigroup (with respect to the Clifford product) generated by invertible and non-invertible vectors of \({\mathcal C}(V,Q)\). The complexified Lipschitz semigroup \(\Pi^{*}\) is generated by the complexification \(V^*\) of \(V\). The author discusses the solutions of the following system of equations \[ fg=0, \qquad f {\partial g \over\partial x^{\mu}}=0 \quad (\mu=1,2,\ldots,m), \] where \(f\) and \(g\) are \(\Pi\) (or \(\Pi^{*}\)) valued functions of \(m\) variables. He obtains the general solution of the equation \(xy=0\), where \(x\) and \(y\) are non-zero elements of \(\Pi\).