Nonfinitely based ai-semirings with finitely based semigroup reducts
Logic
2021-12-30 v1 Combinatorics
Group Theory
Abstract
We present some general results implying nonfinite axiomatisability of many additively idempotent semirings with finitely based semigroup reducts. The smallest is a -element commutative example, which we show also has \texttt{NP}-hard membership for its variety. As well as being the only nonfinite axiomatisable ai-semiring on -elements, we are able to show that its nonfinite basis property infects many related semirings, including the natural ai-semiring structure on the semigroup . We also extend previous group-theory based examples significantly, by showing that any finite additively idempotent semiring with a nonabelian nilpotent subgroup is not finitely axiomatisable for its identities.
Keywords
Cite
@article{arxiv.2112.13918,
title = {Nonfinitely based ai-semirings with finitely based semigroup reducts},
author = {Marcel Jackson and Miaomiao Ren and Xianzhong Zhao},
journal= {arXiv preprint arXiv:2112.13918},
year = {2021}
}