On triple derivations of partially ordered sets

Abstract

In this paper, as a generalization of derivation on a partially ordered set, the notion of a triple derivation is presented and studied on a partially ordered set. We study some fundamental properties of the triple derivation on partially ordered sets. Moreover, some examples of triple derivations on a partially ordered set are given. Furthermore, it is shown that the image of an ideal under triple derivation is an ideal under some conditions. Also, the set of fixed points under triple derivation is an ideal under certain conditions. We establish a series of further results of the following nature. Let $(P,\leq)$ be a partially ordered set. 1. If $d,s$ are triple derivations on $P,$ then $d=s$ if and only if $\mathrm{Fix}_{d}(P)=\mathrm{Fix}_{s}(P).$ 2. If $d$ is a triple derivation on $P,$ then, for all $x \in P$;$ \mathrm{Fix}_{d}(P)\cap l(x) = l(d(x)).$ 3. If $d$ and $s$ are two triple derivations on $P,$ then $d$ and $s$ commute. 4. If $d$ and $s$ are two triple derivations on $P,$ then $d \leq s$ if and only if $sd = d.$ In the end, the properties of ideals and operations related to triple derivations are examined.