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.

• 14T10 - Foundations of tropical geometry and relations with algebra
• 15A80 - Max-plus and related algebras
• 16S30 - Universal enveloping algebras of Lie algebras
• 16Y60 - Semirings
• 17B99 - Lie algebras and Lie superalgebras

• 1 zbMATH Open