Nonfinitely based ai-semirings with finitely based semigroup reducts
Nonfinitely based ai-semirings with finitely based semigroup reducts
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DOI：
10.1016/j.jalgebra.2022.07.042
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发表时间：
2021-12
影响因子：
0.9
通讯作者：
M. Jackson;M. Ren;X. Zhao
中科院分区：
文献类型：
--
作者：
M. Jackson;M. Ren;X. Zhao
本文给出了一些一般性结果，这些结果暗示了许多加性幂等半环具有基于半群约简的非有限公理化性。最小的是一个3元交换的例子，我们也有NP-硬成员的品种。以及作为唯一的非有限axiomatisable ai-半环上的3-元素，我们能够证明，它的非有限基性质感染许多相关的半环，包括自然ai-半环结构上的半群B 2 1。我们还扩展了以前的群论为基础的例子显着，通过显示，任何有限的加法幂等半环的非交换幂零子群是不可公理化的身份。
We present some general results implying nonfinite axiomatisability of many additively idempotent semirings with finitely based semigroup reducts. The smallest is a 3-element commutative example, which we show also has NP-hard membership for its variety. As well as being the only nonfinite axiomatisable ai-semiring on 3-elements, we are able to show that its nonfinite basis property infects many related semirings, including the natural ai-semiring structure on the semigroup B 2 1. 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.