A note on idempotent semirings

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science and Technology, University of Coimbra, Portugal.

2 Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa.

10.48308/cgasa.2024.235337.1484

Abstract

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.

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References

[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.
Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 03 September 2024

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