On some varieties of ai-semirings satisfying \(x^{p+1} \approx x\)
type: Item
instance of: scholarly article
title: On some varieties of ai-semirings satisfying \(x^{p+1} \approx x\)
zbMATH Open document ID: 1454.08008
author: Yong Shao
author: Aifa Wang
published in: Open Mathematics
publication date: 2019-03-29T00:00:00Z
review text: A semiring is called additively idempotent if its additive reduct is a semilattice. \textit{M. M. Ren} et al., [Algebra Univers. 77, No. 4, 395--408 (2017; Zbl 1371.08005)] have described the lattice \(L_2\) of varieties of additively idempotent semirings satisfying the identity \(x^3\approx x\). It was proved that \(L_2\) is discributive and consists of 179 varieties, each of which is finitely based and finitely generated. If an additively idempotent semiring satisfies \(x^3\approx x\), its multiplicative reduct is a union of groups of exponent 2. The authors fix a prime number \(p\) and consider the lattice \(L_p\) of varieties of additively idempotent semirings whose multiplicative reducts are unions of abelian groups of exponent \(p\). They prove that \(L_p\) is discributive and consists of 179 varieties, each of which is finitely based and finitely generated (Theorems 3.16 and 3.17). Reviewer's remark. Even though the authors do not state this explicitly, their proofs show that the lattice \(L_p\) does not really depend on \(p\): all lattices of this form are isomorphic.
Mathematics Subject Classification ID: 08B15
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 20M07
zbMATH DE Number: 7045456
zbMATH Keywords: additively idempotent semiring
zbMATH Keywords: identity
zbMATH Keywords: variety
zbMATH Keywords: varietal lattice
MaRDI profile type: Publication
full work available at URL: math-2018-0079
OpenAlex ID: W2885170253
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DOI: 10.1515/MATH-2018-0079
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description: scientific article; zbMATH DE number 7045456