Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems

An associative commutative algebra of distributions that contains homogeneous and associated homogeneous distributions is constructed. This algebra is used to analyze generalized solutions to strictly hyperbolic partial differential equations. Possible types of singularities are studied and the necessary (analogues of Hugoniot conditions for shock waves) and sufficient conditions for the existence of such solutions are obtained.