Prime congruences containing a bend congruence on toric semirings are strongly characterized, yielding a Nullstellensatz analogue, cancellativity of certain quotients, identification with tropical function semirings, and polyhedron closure descriptions.
Extension of valuations in characteristic one
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abstract
We develop an extension of valuations theorem for suitable extensions of idempotent semirings. As an application, we give a new proof for the classical case of fields. Along the way, we develop characteristic one analogues of some central results in the theory of valuation rings.
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2026 1verdicts
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Geometric classification of primes modulo a (bend) congruence
Prime congruences containing a bend congruence on toric semirings are strongly characterized, yielding a Nullstellensatz analogue, cancellativity of certain quotients, identification with tropical function semirings, and polyhedron closure descriptions.