Nonfinitely based ai-semirings with finitely based semigroup reducts

2021/12/27 by Jackson, Marcel, Ren, Miaomiao, Zhao, Xianzhong · 5 citations
#05C65 #16Y60 #20M07 #Combinatorics (math.CO) #FOS: Mathematics #G.2.2 #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.2112.13918

Abstract

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 B₂¹. 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.

Cited by

• Every additively idempotent semiring satisfying xy≈ xz is finitely based
• The finite basis problem for the endomorphism semirings of finite semilattices
• Embedding lattices of quasivarieties of periodic groups into lattices of additively idempotent semiring varieties: An algebraic proof
• On the variety generated by all semirings of order two
• The flat semirings with nilpotent multiplicative reducts

Related

• The additively idempotent semiring S₇⁰ is nonfinitely based
• The finite basis problem for additively idempotent semirings of order four, I
• The finite basis problem for additively idempotent semirings that relate to S₇
• The finite basis problem for additively idempotent semirings of order four, III
• The finite basis problem for additively idempotent semirings of order four, II