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Ideal-theoretic characterizations of valuation and Prüfer monoids

Franz Halter-Koch (2004)

Archivum Mathematicum

It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system (Lorenzen r -system of finite character). We prove an analogous result for root-closed (cancellative) monoids and apply it to give several new characterizations of Prüfer (multiplication) monoids and integral domains.

Idempotent semigroups and tropical algebraic sets

Zur Izhakian, Eugenii Shustin (2012)

Journal of the European Mathematical Society

The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup structure. We address the question of the geometry of idempotent semigroups, in particular, tropical algebraic sets carrying the structure of a commutative idempotent semigroup. We show that commutative idempotent semigroups are contractible, that systems of tropical...

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