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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...

Idempotents and the multiplicative group of some totally bounded rings

Mohamed A. Salim, Adela Tripe (2011)

Czechoslovak Mathematical Journal

In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power 2 0 commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.

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