Skew derivations on partially ordered sets

Abstract

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.