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On \((r,t)\)-commutativity of \(n_{(2)}\)-permutable semigroups. (English) Zbl 0828.20042
A semigroup \(S\) is \(n_{(2)}\)-permutable \((n \in \mathbb{N}\), \(n \geq 2)\) if for every \(n\)-tuple \((s_1, \dots, s_n)\) of elements of \(S\) there exists an integer \(t\), with \(1 \leq t \leq n-1\), such that \((s_1 \dots s_t) (s_{t+1} \dots s_n)=(s_{t+1} \dots s_n) (s_1 \dots s_t)\). Also, a semigroup \(S\) is \((r,t)\)-commutative \((r, t \in \mathbb{N}^*)\) if, for every \((r+t)\)-tuple \((s_1,\dots, s_{r+t})\) of elements of \(S\), \((s_1 \dots s_r)(s_{r+1} \dots s_{r+t})=(s_{r+1} \dots s_{r+t}) (s_1 \dots s_r)\). These kinds of conditions are interesting because they generalize in a natural manner commutativity and also they are specializations of a more general notion, the \(n\)-permutability, which intervines in the Burnside problem for semigroups [see A. Restivo, C. Reutenauer, J. Algebra 89, 102- 104 (1984; Zbl 0545.20051)].
A. Nagy asked if the \(n_{(2)}\)-permutability of a semigroup implies its \((r,t)\)-commutativity, for some \(r\) and \(t\). The authors proved that this is true in the case of finite semigroups. Later, a positive answer for arbitrary semigroups was given by the reviewer who proved that every \(n_{(2)}\)-permutable semigroup \((n \geq 4)\) is also \((1,2n- 4)\)- commutative [C. R. Acad. Sci., Paris, Sér. I 317, 923-924 (1993; Zbl 0795.20042)].
Thus, for every \(n \geq 2\) it makes sense to ask about the least integer \(m=\varphi(n) \geq 2\) having the property that for every \(n_{(2)}\)- permutable semigroup \(S\) \((n \geq 2)\) there exist \(r\), \(t\) in \(\mathbb{N}^*\), with \(r+t=m\), such that \(S\) is \((r,t)\)-commutative. It is obvious that \(\varphi(n) \leq 2n-3\). Also it was already proved by the authors that \(\varphi(n) \geq n\). In this paper, they establish that \(\varphi(n) \geq 2n-4\). In order to prove this inequality, they obtain, by an ingenious construction, a family of \(n_{(2)}\)-permutable and \((1,n+k)\)-commutative semigroups \(S_{n+k}\) \((n \geq 4\), \(0 \leq k \leq n-4)\) which are neither \((n -1)_{(2)}\)-permutable nor \((1,n+k-1)\)- commutative. We mention that the problem of the existence of \(n_{(2)}\)- permutable semigroups which are not \((r,t)\)-commutative, for any \(r\), \(t\) with \(r+t=2n-4\), is still open.
20M05 Free semigroups, generators and relations, word problems
20M14 Commutative semigroups
20M10 General structure theory for semigroups
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