Lie pairs

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 pairs comprise two categories in addition to the universal algebraic definition, one with weak Lie morphisms'' preserving null sums, and the other with $⪯$-morphisms’’ preserving a surpassing relation $⪯$ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories.