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Lie pairs
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abstract
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.
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Semirings
Rowen consolidates the pair/surpassing-relation framework that extends classical algebra (roots, matrices, linear algebra, geometry) to semirings without cancellation, adding new root-factor theorems and a map of open...
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