Categories and General Algebraic Structures with Applications

Categories and General Algebraic Structures with Applications

A note on idempotent semirings

Document Type : Research Paper

Authors
Manuela Sobral 1
George Janelidze 2
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.
Keywords
Commutative semiring
non-empty colimit
coreflective subcategory
Boolean algebra
distributive lattice
Subjects
[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
Volume 22, Issue 1
Dedicated to Professor Themba Dube (II)
January 2025
Pages 175-180