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Skew derivations on partially ordered sets. (English) Zbl 1484.06003

Summary: Let \(P\) be a poset and \(\alpha :P\rightarrow P\) be a function. The aim of this paper is to introduce and study the notion of skew derivations on \(P\). We prove some fundamental properties of posets involving skew derivations. In particular, apart from proving the other results, we prove that if \(d\) and \(g\) are two skew derivations of \(P\) associated with an automorphism \(\alpha\) such that \(d\alpha =\alpha d\) and \(g\alpha =\alpha g,\) then \(d \le g\) if and only if \(g d =\alpha d\). Also, we prove that \(Fix_{\alpha ,d}(P)\cap l(\alpha (x)) = l(d(x))\) for all \(x\in P.\) Furthermore, we give some examples to demonstrate that various restrictions imposed in the hypotheses of our results are not superfluous.

MSC:

06A06 Partial orders, general
06A07 Combinatorics of partially ordered sets
13N15 Derivations and commutative rings
16W25 Derivations, actions of Lie algebras
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