Beschreibung:
Extending the theory of systems, we introduce a theory of Lie semialgebra``pairs'' which parallels the classical theory of Lie algebras, but with a``null set'' replacing $0$. A selection of examples is given. These Lie pairscomprise two categories in addition to the universal algebraic definition, onewith ``weak Lie morphisms'' preserving null sums, and the other with``$\preceq$-morphisms'' preserving a surpassing relation $\preceq$ thatreplaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt)Theorem in these three categories.