Indian Journal of Pure and Applied Mathematics ( IF 0.7 ) Pub Date : 2021-06-30 , DOI: 10.1007/s13226-021-00036-5 Ahmed Y. Abdelwanis , Shakir Ali
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.
中文翻译:
偏序集上的偏斜推导
设P是一个偏序集,\(\alpha :P\rightarrow P\)是一个函数。本文的目的是介绍和研究P上的偏斜导数的概念。我们证明了涉及偏斜推导的偏序集的一些基本属性。特别地,除了证明其他结果之外,我们证明如果d和g是与自同构\(\alpha \)相关的P 的两个偏导,使得\(d\alpha =\alpha d\)和\(g \alpha =\alpha g,\)然后\(d \le g \)当且仅当\(gd =\alpha d\)。同时,我们证明\( Fix_{\alpha ,d}(P)\cap l(\alpha (x)) = l(d(x))\) for all \(x\in P.\)此外,我们给出一些例子证明在我们的结果假设中施加的各种限制并不是多余的。