The Burnside ai-semiring variety defined by xⁿ≈ x

2022/07/12 by Ren, Miaomiao, Zhao, Xianzhong, Volkov, Mikhail V.
#08B15 #08B26 #16Y60 #20M07 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2207.05490

Abstract

Let \bf Sr(n, 1) denote the ai-semiring variety defined by the identity xⁿ≈ x, where n>1. We characterize all subdirectly irreducible members of a semisimple subvariety of \bf Sr(n, 1). Based on this result, we prove that \bf Sr(n, 1) is hereditarily finitely based (resp., hereditarily finitely generated) if and only if n<4 and that the lattice of subvarieties of \bf Sr(n, 1) is countable if and only if n<4. Also, we show that the class of all locally finite members of \bf Sr(n, 1) forms a variety and so we affirmatively answer the restricted Burnside problem for \bf Sr(n, 1). In addition, we provide a simplified proof of the main result obtained by Gajdoš and Kuřil (Semigroup Forum 80: 92--104, 2010).

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