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arXiv · 2312.01770

The finite basis problem for endomorphism semirings of finite chains

Sergey V. Gusev·Mikhail V. Volkov
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Abstract

For every semilattice $\mathcal{S}=(S,+)$, the set $\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the 3-element chain has no finite identity basis. This, combined with earlier results by Dolinka (The finite basis problem for endomorphism semirings of finite semilattices with zero, Algebra Universalis 61, 441-448 (2009)), gives a complete solution to the finite basis problem for semirings of the form $\mathrm{End}(\mathcal{S})$ where $\mathcal{S}$ is a finite chain.

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Sergey V. Gusev, Mikhail V. Volkov. 2023-12-04. The finite basis problem for endomorphism semirings of finite chains. https://doi.org/10.1142/s0218196726400084

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