A note on idempotent semirings | ||
| Categories and General Algebraic Structures with Applications | ||
| دوره 22، شماره 1، فروردین 2025، صفحه 175-180 اصل مقاله (439.46 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.48308/cgasa.2024.235337.1484 | ||
| نویسندگان | ||
| Manuela Sobral* 1؛ George Janelidze2 | ||
| 1Department of Mathematics, Faculty of Science and Technology, University of Coimbra, Portugal. | ||
| 2Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa. | ||
| چکیده | ||
| For a commutative semiring $S$, by an $S$-algebra we mean a commutative semiring $A$ equipped with a homomorphism $S\to A$. We show that the subvariety of $S$-algebras determined by the identities $1+2x=1$ and $x^2=x$ is closed under non-empty colimits. The (known) closedness of the category of Boolean rings and of the category of distributive lattices under non-empty colimits in the category of commutative semirings both follow from this general statement. | ||
| کلیدواژهها | ||
| Commutative semiring؛ non-empty colimit؛ coreflective subcategory؛ Boolean algebra؛ distributive lattice | ||
| مراجع | ||
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[1] Cs´ak´any, B., Primitive classes of algebras which are equivalent to classes of semimodules and modules (Russian), Acta Sci. Math. (Szeged) 24 (1963), 157-164. [2] Johnson, J.S., Manes, E.G., On modules over a semiring, J. Algebra 15 (1970), 57-67. [3] Johnstone, P.T., “Sketches of an elephant: a topos theory compendium”, Vol. 2. Oxford Logic Guides 44, The Clarendon Press, Oxford University Press, Oxford, 2002. | ||
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