# The finite basis problem for endomorphism semirings of finite semilattices with zero

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## Abstract

If \({\fancyscript{I}}\) is a join-semilattice with a distinguished least element, then all its endomorphisms form an additively idempotent semiring End(\({\fancyscript{I}}\)); conversely, it is known that any additively idempotent semiring embeds into an endomorphism semiring of such kind. If \({|\fancyscript{I}| \leq 2}\), then End(\({\fancyscript{I}}\)) is readily seen to be finitely based. On the other hand, if \({{\fancyscript{I}}}\) is finite and either contains the square of a two-element chain or is a chain with at least four elements, then End(\({\fancyscript{I}}\)) is shown to be inherently nonfinitely based. This leaves only the case when \({{\fancyscript{I}}}\) is a three-element chain as an open problem.

## 2000 Mathematics Subject Classification

Primary: 16Y60 Secondary:08B05## Keywords and phrases

semiring endomorphism semiring of a semilattice finite basis problem INFB## References

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