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Lie-type derivations of finitary incidence algebras. (English) Zbl 1445.16039

The authors suppose that \(P\) is an arbitrary partially ordered set, \(R\) is a commutative ring with identity and \(\operatorname{FI}(P,R)\) is the finitary incidence algebra of \(P\) over \(R\). Under some natural assumption on \(R\) they prove that each Lie-type derivation of \(\operatorname{FI}(P,R)\) is proper, which partially generalizes the main results of X. Zhang and the second author [Linear Algebra Appl. 513, 69–83 (2017; Zbl 1349.16074)], D. Wang and Z. Xiao [Commun. Algebra 47, No. 5, 1841–1852 (2019; Zbl 1456.16040)], and Z. Xiao and Y. Yang [Commun. Algebra 48, No. 1, 105–118 (2020; Zbl 1437.16041)]. In fact, the main goal of this paper is to describe Lie-type derivations of the finitary incidence algebra \(\operatorname{FI}(P,R)\). Some of the results of this paper represent generalizations of other results. Also, they introduce some definitions which employ in their results. Taking into account the definitions of Lie derivations and Lie triple derivations, one naturally expects to extend them in one more general way via a Lie \(n\)-derivation. The authors employ Lie \(n\)-derivation on \(\operatorname{FI}(P,R)\). The authors close their article by an open problem. They point out that all involved Lie-type derivations in this work are linear. They state the following theorem.
Theorem 2.13. Let \(L\) be a Lie \(n\)-derivation of \(\operatorname{FI}(P,R)\). If \(R\) is \((n-1)\)-torsion-free, then \(L=d+\tau\), where \(d\) is a derivation of \(\operatorname{FI}(P,R)\) and \(\tau\) is a central-valued linear mapping which annihilates all the \(n\)-th commutators.
It is natural to ask whether Theorem 2.13 without the assumption of additivity. That is, it is interesting to investigate multiplicative Lie-type derivations of finitary incidence algebras, which are motivated by the following papers: [D. Benkovič and D. Eremita, Linear Algebra Appl. 436, No. 11, 4223–4240 (2012; Zbl 1247.16040); A. Fošner et al., Colloq. Math. 132, No. 1, 53–71 (2013; Zbl 1306.47047); Y. Wang and Y. Wang, Linear Algebra Appl. 438, No. 5, 2599–2616 (2013; Zbl 1272.16039)].

MSC:

16W25 Derivations, actions of Lie algebras
16S60 Associative rings of functions, subdirect products, sheaves of rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
47L35 Nest algebras, CSL algebras
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References:

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