  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
``$\preceq$-morphisms'' preserving a surpassing relation $\preceq$ that
replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt)
Theorem in these three categories.

Rings and Algebras