Article
Ufa Mathematical Journal
Volume 11, Number 4, pp. 108-114
On triple derivations of partially ordered sets
Abdelwanis A.Y.
DOI:10.13108/2019-11-4-108
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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.