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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