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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.
##### MSC:
 20M05 Free semigroups, generators and relations, word problems 20M14 Commutative semigroups 20M10 General structure theory for semigroups
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