Geodesic
Algebraic geometry over the monoid of natural numbers. Irreducible algebraic sets
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 2, pp. 258-269

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Irreducible coordinate monoids of algebraic sets over the additive monoid of natural numbers are described. An arbitrary algebraic set over $\mathbb N$ is decomposed into the union of irreducible components, and the properties of the irreducible components are investigated.
Keywords: natural numbers, algebraic geometry, monoid, universal closure. 
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     title = {Algebraic geometry over the monoid of natural numbers. {Irreducible} algebraic sets},
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A. Shevlyakov. Algebraic geometry over the monoid of natural numbers. Irreducible algebraic sets. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 2, pp. 258-269. http://geodesic.mathdoc.fr/item/TIMM_2010_16_2_a22/