The lattice of ai-semiring varieties satisfying $$x^n\approx x$$ x n
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The lattice of ai‑semiring varieties satisfying x n ≈ x and xy ≈ yx Miaomiao Ren1 · Xianzhong Zhao1 · Yong Shao1 Received: 9 August 2018 / Accepted: 2 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the lattice L(𝐂𝐒𝐫(n, 1)) of subvarieties of the ai-semiring variety 𝐂𝐒𝐫(n, 1) defined by xn ≈ x and xy ≈ yx . We divide L(𝐂𝐒𝐫(n, 1)) into five intervals and provide an explicit description of each member of these intervals except [𝐂𝐒𝐫(2, 1), 𝐂𝐒𝐫(n, 1)] . Based on these results, we show that if n − 1 is square-free, then L(𝐂𝐒𝐫(n, 1)) is a distributive lattice of order 2 + 2r+1 + 3r , where r denotes the number of prime divisors of n − 1 . Also, all members of L(𝐂𝐒𝐫(n, 1)) are finitely based and finitely generated and so 𝐂𝐒𝐫(n, 1) is a Cross variety. Moreover, the axiomatic rank of each member of L(𝐂𝐒𝐫(n, 1)) is less than four. Keywords Ai-semiring · Variety · Lattice · Identity · Finitely based variety · Finitely generated variety
1 Introduction A variety 𝐕 is called finitely based if it can be defined by finitely many identities. Otherwise, 𝐕 is called nonfinitely based. If 𝐕 is finitely based, then it is easy to see that all subvarieties of 𝐕 are finitely based if and only if the lattice L(𝐕) of subvarieties of 𝐕 satisfies the descending chain condition, i.e., there is no infinite descending
To the memory of Professor Yuqi Guo.
Communicated by Mikhail Volkov. * Xianzhong Zhao [email protected] Miaomiao Ren [email protected] Yong Shao [email protected] 1
School of Mathematics, Northwest University, Xi’an 710127, Shaanxi, People’s Republic of China
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chain in L(𝐕) . In particular, if 𝐕 is finitely based and L(𝐕) is a finite lattice, then all subvarieties of 𝐕 are finitely based. Recall that 𝐕 is said to be finitely generated if it can be generated by a finite number of finite algebras and that 𝐕 is said to be locally finite if each finitely generated member of 𝐕 is finite. It is well-known that for a locally finite variety 𝐕 , all subvarieties of 𝐕 are finitely generated if and only if L(𝐕) satisfies the ascending chain condition, i.e., there is no infinite ascending chain in L(𝐕) (see [12]). This implies that if 𝐕 is locally finite and L(𝐕) is finite, then all subvarieties of 𝐕 are finitely generated. We say that 𝐕 is a Cross variety (see [27]) if it is finitely based and finitely generated and L(𝐕) is finite. It is well-known that the variety generated by every finite group (finite ring) is a Cross variety (see [9, 11, 15]). But it is not true for every finite semigroup or finite semiring (see [2, 18]). By a semiring we mean an algebra (S, +, ⋅) of the type (2, 2) such that – the additive reduct (S, +) is a commutative semigroup; – the multiplicative reduct (S, ⋅) is a semigroup; – (S, +, ⋅) satisfies the identities x(y + z) ≈ xy + xz and (x + y)z ≈ xz + yz. Such algebras have broad applications in information science and theoretical computer science (see [5, 6]). A semiring is said to be an additively idempoten
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