Clifford semialgebras

Lie pairs

“…25. If A is an algebra (i.e., with additive inverses) and c i = −b i and d i = 0, the matrix C as in (8) defines the usual cross product (for A = R).…”

“…The simple but effective idea for remedying the lack of negation is to introduce an endomorphism (−), whose square is the identity, to which is attached a surpassing relation. A further step was taken in [8], where a theory of Clifford semialgebra is proposed. In more traditional contexts, Clifford algebras are examples of Lie super-algebras, so [8] may be considered as the first relevant example of Lie semi-(super)algebras obtained within the already collocated framework of triples and systems.…”

“…A further step was taken in [8], where a theory of Clifford semialgebra is proposed. In more traditional contexts, Clifford algebras are examples of Lie super-algebras, so [8] may be considered as the first relevant example of Lie semi-(super)algebras obtained within the already collocated framework of triples and systems. It was applied to extend to the tropical framework the polynomial representation of Lie algebras of endomorphisms of a vector space, in the same spirit of [10].…”

$\mathcal{T}$-semiring pairs

“…It is uniquely quasi-negated if (A, A 0 ) is uniquely quasinegated, seen componentwise. (v) For M an A-module, take N to be A 0 M. (vi) The following example from [8, Definitions 5.4, 5.6] played the main role in [8]. Suppose (A, A 0 ) is a T -semiring pair with a surpassing map , and (M, N ) is a module over (A, A 0 ).…”