Some Results on Partially Ordered Sets Involving Permuting n-derivations

LATIFA BEDDA,1ABDELKARIM BOUA1

1Department of Mathematics, Polydisciplinary Faculty of Taza,

Sidi Mohammed Ben Abdellah University, Fez,

MOROCCO

Abstract: The main objective of the present work, as a generalization of derivation, is to give the concept of per-

muting n-derivations on partially ordered sets (posets). Several associated theorems and fondamental properties

involving permuting n-derivations are presented. Moreover, we demonstrate that if Dis a permuting n-derivation

on poset Gwith the greatest element 1and the trace δ, then δ(1) = 1 if and only if δis an identity on G. Further-

more, we discuss the relations among derivations, ideals and fixed sets in posets.

Key-Words: Partially ordered sets (posets), Upper bound, Lower bound, Directed poset, Derivation, Permuting

n-derivations, Lower homomorphism, Fixed set.

Received: March 12, 2024. Revised: July 5, 2024. Accepted: August 21, 2024. Published: September 20, 2024.

1 Introduction

In mathematics, a partially ordered set, often abbrevi-

ated as a poset, is a set provided with a binary relation

(often denoted as ≤) that is reflexive, antisymmetric

and transitive. Partially ordered sets arise naturally in

various mathematical contexts, including order rela-

tions, such as less than or equal to (≤) on numbers,

subsets of sets under inclusion (⊆), and many other

situations where there’s a notion of ’precedence’ or

’ordering’ among elements, but not necessarily every

pair of elements can be compared.

Hausdorff introduced the first comprehensive the-

ory of partially ordered sets in 1914 in his book ”

Grundzüge der Mengenlehre ”. One significant out-

come of Hausdorff’s work is the maximal chain theo-

rem, which is equivalent to Zorn’s lemma..

The derivation is consequent topic to study, [1],

defined the derivation on ring and many mathe-

maticians have developed the derivation theory in

rings and prime rings, [2], [3]. Multi- derivations

(e.g. bi-derivations, tri-derivations, in general, n-

derivations) are studied in prime and semi-prime

rings, [4], [5], [6].

In this direction, the concept of derivation on lattice

was defined and developed in [7], [8], respectively.

In [9], the study defined symmetric bi-derivations of a

lattice and proved some results, and in [10], he applied

his concepts and theorems to the n-derivation of lat-

tices. Derivations on posets have also been a subject

of study. Recently, [11], started studying the deriva-

tions in poset, establishing several fundamental prop-

erties related to ideals and operations associated with

these derivations. On partially ordered sets, the notion

of bi- and tri-derivations are provided. and the funda-

mental Characteristics are studied (see, [12], [13], for

more details). Our research was mainly inspired by

the work in [11], [12]. This research presents a gener-

alization of derivations by introducing a new concept

of permuting n-derivations of partially ordered sets.

Moreover, we present the examples that demonstrate

the existence of this class of applications and we have

proved important properties. Additionally, we give

the fixed set F ixδ(G) = {a∈G:δ(a) = a}and

proved that is an ideal of G. The final section is de-

voted to studying some properties involving permut-

ing n-derivations and their traces.

As in [11], for p, q ∈Gand X⊆P, we define

(i) ↓p={w∈G:w≤p}.

(ii) ↑q={v∈G:q≤v}.

(iii) L(X) = {λ∈G:λ≤x,∀x∈X}the Lower

cone of the set X.

(iv) U(X) = {α∈G:x≤α,∀x∈X}the Upper

cone of X.

As mentioned in [14], we write ”L(U(L(Y))) =

L(Y)” and ”U(L(U(M))) = U(M)”, for all Y, M ⊂

G. If Y={y1, ..., yn}, we write L(Y) =

L(y1, y2, ..., yn)and U(Y) = U(y1, ..., yn−1, yn).

Further, For X, Y ⊆P,L(X∪Y)will be represented

by L(X, Y )and U(X∪Y)by U(X, Y ). We write

also, ↓X={w∈P:w≤y for some y ∈X}”.

According to [15], a set Xis named a Lower set if

X=↓X. The directed set is a nonempty set Xthat

for every finite subset of X, the supremum has ex-

isted in X. Given that Xis nonempty, it suffices to

expect that every pair {a, b}of elements in Xhas the

supremum in X. For J ⊂ Gis said ideal of Gif J

is directed lower set.

2 Permuting n-Derivations on Posets

Throughout the present work, Grepresents a partially

ordered set, which will be abbreviated as poset

Definition 1. [11] Let Gbe a poset, a function D:

G→Gis called a derivation on Gif these two con-

ditions are verified, (∀a, b ∈P),

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(i) D(L(a, b)) = L(U(L(D(a), b), L(a, D(b))));

(ii) L(D(U(a, b))) = L(U(D(a), D(b)));

A mapping D:G×G→Gis called symmetric

if D(s, t) = D(t, s), ∀s, t ∈G, and a mapping

δ:G→Ggiven by δ(s) = D(s, s),∀s∈Gis

named a trace of Dunder which Dis symmetric.

This section introduces a new notion called

permuting n-derivations for a partially ordered

set, followed by the examples that demonstrates the

existence of this type of application.

Let n∈Nsuch that n⩾2and Gn=

G×G×. . . ×G

| {z }

ntimes

. The map D:G→Gis said to

be permuting if the equation

D(a1, a2, . . . , an) = D(aπ(1), aπ(2), . . . , aπ(n))(1)

holds ∀ai∈Gand for every permutation π(i),

i= 1, ..., n.

Definition 2. Let Gbe a poset and D:Gn→Gbe

a map. We say Dis n-derivation if Dis a derivation

for all components, which means:

(1) D(L(a1, w), a2, . . . , an) =

L(U(L(D(a1,. . . , an), w), L(a1, D(w, a2, . . . , an))))

D(a1, L(a2, w), a3, . . . , an) =

L(U(L(a2,D(a1,w, a3, . . . , an)), L(D(a1, a2, . . . , an), w)))

. . .

D(a1, a2, . . . , an−1, L(an, w)) =

L(U(L(an,D(a1,...,an−1, w)), L(D(a1, a2, . . . , an), w)))

(2 ) L(D(U(a1, w), a2, . . . , an)) =

L(U(D(a1,a2, . . . , an), D(w, a2, . . . , an)));

. . .

L(D(a1, a2, a3, . . . , an−1, U(an, w)) =

L(U(D(a1,a2, . . . , an), D(a1, . . . , an−1, w)));

are valid, (∀ai, w ∈P).

Example 1

Let D:Nn→Nbe a function defined by

D(m1, m2, . . . , mn) = min{m1, m2, . . . , mn}.

It is simple to confirm that Dis a permuting n-

derivation on N.

Example 2

Let 0be the least element of a poset G. A function

D:Gn→Gdefined by D(a1, a2, . . . , an)=0is a

permuting and a n-derivation on G.

In the following, we assume that Gis a poset

and Dis a permuting n-derivation on G.

Proposition 1. Let 0be the least element of a poset

Gand δbe the trace of D. Then

(i) D(a1, a2, a3, . . . , an)≤ai,∀ai∈G;

(ii) D(a1, . . . , an)∈L(a1, a2, . . . , an),∀ai∈G;

(iii) D(a1, a2, ..., an)=0 if there exist i∈

{1,2, . . . , n}which satisfy ai= 0;

(iv) For each iin {1,2, . . . , n}, if ai≤bi, then

D(a1, ..., ai, . . . , an)≤D(a1, . . . , bi, . . . , an);

(v) δ(a)≤a,∀a∈G;

(vi) δ(0) = 0;

(vii) δ(L(a)) ⊂L(δ(a)),∀a∈G;

(viii) ∀g1, g2∈G,g1≤g2implies δ(g1)≤δ(g2);

(ix) δ2(s) = δ(s),∀s∈G;

Proof. (i) From Definition 2 (i), we have

D(L(a1), a2, . . . , an) = D(L(a1, a1), a2, . . . , an)

=L(U(L(D(a1,. . .,an), a1), L(a1, D(a1, a2, . . . , an))))

=L(U(L(a1,D(a1,a2, . . . , an)))) = L(D(a1, a2, . . . , an), a1)

Then,

D(L(a1),a2, . . . , an) = L(D(a1, a2, . . . , an), a1)(2)

Since D(a1,a2, a3,...,an)∈D(L(a1), a2, . . . , an),

the above result (2) imply that

D(a1,a2, . . . , an)∈L(D(a1, a2, . . . , an), a1).

Therefore, D(a1, . . . , an)≤a1.

Similar to above processe, we can see that

D(a1, a2, . . . , an)≤ai,∀ai∈G.

It is evident that (ii) and (iii) are induced by (i).

(iv) Suppose that ai≤bifor ai, bi∈G. By using

Definition 2 (ii), we get

L(D(a1, a2, . . . , U(bi), . . . , an−1, an)) =

L(D(a1,. . .,U(ai, bi), . . . , an)) =

L(U(D(a1,. . .,ai, ai+1, . . . , an), D(a1, . . . , bi−1, bi, . . . , an))).

(3)

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Since D(a1, . . . , ai−1, ai, ai+1, . . . , an)∈

L(U(D(a1, . . . , ai, . . . , an), D(a1, . . . , bi, . . . , an))),

equation (3) proves that

D(a1,a2, . . . , ai, . . . , an)∈L(D(a1, . . . , U(bi), ai+1, . . . , an))

Therefore,

D(a1,. . .,ai, . . . , an)≤D(a1, . . . , bi, . . . , an)

(v) Let a∈G, we have δ(a) = D(a, a, . . . , a).

By (i), we get D(a, . . . , a)≤a,∀a∈G. Hence,

δ(a)≤a,∀a∈G.

(vi) Since δ(0) ≤0by (v), we get 0≤δ(0) ≤0.

This means that δ(0) = 0.

(vii) Let a∈G,

D(L(a),. . .,L(a)) = D(L(a, a), L(a), . . . , L(a))

={D(L(a, a), y2, . . . , yn)|yi∈Gand yi⩽a,∀

i= 2, ..., n}

={L(U(L(D(a, y2, . . . , yn), a), L(a, D(a, y2, . . . , yn))))

|yi∈Gand yi⩽a,∀i= 2, ..., n}

={L(U(L(D(a, y2, . . . , yn))) |yi∈Gand yi⩽a,∀

i= 2, ..., n}

={L(D(a, y2, . . . , yn)) |yi∈Gand yi⩽a,∀

i= 2, ..., n}

then

D(L(a),..., L(a)) = L(D(a, L(a), ..., L(a))).(4)

Since δ(L(a)) ⊂D(L(a),. . . , L(a)), the Equation

(4) implies that δ(L(a)) ⊂L(D(a, L(a), . . . , L(a))),

so δ(L(a)) ⊂L(D(a, a, . . . , a)).This shows that

δ(L(a)) ⊂L(δ(a)), for all a∈G.

(viii) Let g1and g2be two different elements in G

which satisfy the condition g1≤g2, then g1∈L(g2),

and this implies that δ(g1)∈δ(L(g2)). By using

(vii), we can get δ(g1)∈L(δ(g2)), so δ(g1)≤δ(g2).

(ix) According to (v) and (viii), we can see

δ(δ(a)) ≤δ(a),∀a∈P, so

δ2(a)≤δ(a).(5)

Let a∈G, combining (v) and (viii) we get δ2(a)∈

L(a)and by using (4), we obtain

δ(L(a)) ⊂L(D(a, δ2(a), . . . , δ2(a))).(6)

since d(a, δ2(a), . . . , δ2(a)) ≤δ2(a)by (i), we have

L(D(a, δ2(a), . . . , δ2(a))) ⊂L(δ2(a)).(7)

Adding the equations (6) and (7), we find that δ(L(a))

is included in L(δ2(a)),∀a∈G. Since δ(a)∈

δ(L(a)), we obtain δ(a)∈L(δ2(a)), so

δ(a)≤δ2(a),∀a∈P. (8)

Therefore, (5) and (8) imply that δ2(a) = δ(a),∀a∈

Theorem 1. Let D:Gn→Gbe a permuting map-

ping on poset G.Dis an n-derivation on Gif and

only if

(1) D(L(a1,w),a2, . . . , an) = L(D(a1, . . . , an), w)=

L(a1, D(w, a2, . . . , an));

(2) L(D(U((a1,w),a2, . . . , an))

=L(U(D(a1, . . . , an), D(w, a2, . . . , an)));

∀ai, w ∈Gand i= 1, . . . , n.

Proof. Assume that the condition (1) holds. Then,

D(L(a1,w),a2, . . . , an−1, an) = L(D(a1, a2, . . . , an), w)

=L(U(L(D(a1, . . .,an), w)))

=L(U(L(w, D(a1, a2, . . . , an))), L(D(a1, . . . , an), w)))

=L(U(L(w, D(a1, . . . , an−1, an))), L(a1, D(w, a2, . . . , an)).

In addition to the condition (2), we deduce that Dis

an n-derivation on G.

Inversement, suppose that Dis a n-derivation on G.

it holds that:

L(D(a1,. . .,an), w) = L(U(L(D(a1, . . . , an), w))

⊂L(U(L(w, D(a1, . . . , an))), L(a1, D(w, a2, a3, . . . , an))

=D(L(a1, w), a2, . . . , an),

then

L(D(a1, . . . , an), w)⊂D(L(a1, w), a2, . . . , an).

(9)

Now, let z∈D(L(a1, w), a2, . . . , an), then there ex-

ixts t∈L(a1, w)satisfying z=D(t, a2, . . . , an).

Since t∈L(a1, w), we get t≤a1and by us-

ing Proposition 1 (iv) we obtain D(t, a2, . . . , an)≤

D(a1, . . . , an), so z≤D(a1, . . . , an). From Propo-

sition 1 (i), we can get D(t, a2, . . . , an)≤t≤w, so

z≤wand this imply that z∈L(D(a1, . . . , an), w).

Therefore,

D(L(a1, w), a2, . . . , an)⊂L(D(a1, . . . , an), w).

(10)

Combining the results (9) and (10), we get

D(L(a1, w), a2, . . . , an) = L(D(a1, . . . , an), w),∀

ai, w ∈G.

Symmetrically, we can also prove the

second equality D(L(a1, w), a2, . . . , an) =

L(a1, D(w, a2, . . . , an)),∀ai, w ∈G.

Lemma 1. Let Gbe a poset. If s≤tand L(s) =

L(t), then s=t.

Proof. Assume that s≤tand L(s) = L(t). It is

evident that t∈L(t),∀t∈G, then t∈L(s), so

t≤s. By hypothesis, we conclude that s=t.

Lemma 2. Let Gbe a poset. If δbe the trace of D,

then the subsequent claims are valid:

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(1) If D(L(s), s, . . . , s, s) = L(t), then δ(s) = t,∀

s, t ∈G;

(2) If D(U(s), s, . . . , s) = U(t), then δ(s) = t,∀

s, t ∈G.

Proof. Let s, t ∈Gsuch that

D(L(s), s, . . . , s) = L(t).(11)

By using Theorem 1 (1) and Proposition 1 (v), we get

D(L(s), s, . . . , s) = D(L(s, s), s, . . . , s)

=L(D(s, . . . , s), s)

=L(D(s, . . . , s))

then

D(L(s), s, . . . , s) = L(δ(s)).(12)

For all s∈P.

Using (11) and (12), we infer that L(δ(s)) = L(t).

Since D(s, s, . . . , s)∈D(L(s), s, . . . , s) = L(t), we

find δ(s)≤tand with Lemme 1 the result holds.

Definition 3. Let Gbe a poset, a mapping

ϕ:G→Gis known as a L-homomorphism of Gif

U(ϕ(L(λ, µ))) =U(L(ϕ(λ), ϕ(µ))), (∀λ, µ ∈G).

Proposition 2. Let Gbe a poset and D:Gn→Gbe

a permuting n-derivation on G.

Then, D(L(a1,w),a2, a3, . . . , an−1, an) =

L(D(a1, . . . , an), D(w, a2, . . . , an)),∀ai, w ∈G.

Proof. Let z∈D(L(a1, w), a2, . . . , an)then, ∃t∈

L(a1, w):

z=D(t, a2, . . . , an).(13)

Since t∈L(a1, w), we get t≤a1and t≤w. Propo-

sition 1 (iv) implies that

z=D(t, a2, . . . , an)≤D(a1, a2, . . . , an−1, an)).

(14)

and

z=D(t, a2, . . . , an)≤D(w, a2, ..., an).(15)

Combining (14) and (15), we get

z∈L(D(a1, . . . , an), D(w, a2, . . . , an)). This

shows that, ∀ai, w ∈G, we have

D(L(a1, w), a2, a3, . . . , an)⊂

L(D(a1,a2, a3,...,an−1, an), D(w, a2, . . . , an)))

(16)

Moreover, we suppose that

v∈L(D(a1, a2, . . . , an), D(w, a2, . . . , an)), then

v≤D(a1, . . . , an)and v≤D(w, a2, . . . , an)≤

w, so v∈L(D(a1, . . . , an), w) =

D(L(a1, w), a2, . . . , an)by application of Theo-

rem 1 (1). Consequently,

L(D(a1, a2, . . . , an), D(w, a2, . . . , an)) ⊂

D(L(a1, w), a2, . . . , an),∀ai, w ∈G. (17)

The results (16) and (17) proves the theorem.

Theorem 2. Let Gbe a poset and 1its greatest ele-

ment and δbe the trace of D. Then,

δ(1) = 1 ⇐⇒ D(a, 1, .., 1, ., 1) = a,∀a∈G.

Proof. Suppose that D(a, 1,1,1, ..., 1) = a,∀a∈G,

then D(1, ..., 1,1) = 1, hence δ(1) = 1.

Conversely, we suppose that δ(1) = 1. Let a∈G, by

using Theorem 1, we have

D(L(a),1,1,1, ..., 1) = D(L(a, 1),1, ..., 1,1)

=L(a, D(1,1, ..., 1))

=L(a, δ(1))

=L(a, 1)

=L(a),

then

D(L(a),1,1, ..., 1,1) = L(a).(18)

Furthermore,

D(L(a),1,1, ..., 1, .., 1) = D(L(a, a),1, ..., 1)

=L(D(a, 1, ..., 1,1), a)

=L(D(a, 1,1,1, ..., 1)),

then,

D(L(a),1, ..., 1) = L(D(a, 1,1, ..., 1,1)).(19)

Hence, (18) and (19) implies that

L(D(a, 1, ..., 1)) = L(a).(20)

By using Proposition 1 (i), we get

D(a, 1, ..., 1) ≤a. (21)

In view of Lemma 1 together (20) and (21), we con-

clude that D(a, 1, ..., 1) = a,∀a∈G.

Theorem 3. Let Gbe a poset and δbe the trace of D

on G. We have,

δ(L(a1, ..., an)) ⊂L(δ(a1), .., δ(an)),∀ai∈G.

Proof. Let t∈δ(L(a1, a2, ..., an)), then there ex-

ists y∈L(a1, a2, a3, ..., an)such that t=δ(y).

The relation y∈L(a1, a2, ..., an−1, an)implies that

y≤ai,∀ai∈G, and by using Proposition 1

(viii), we get δ(y)≤δ(ai),∀i= 1, .., n, then

t=δ(y)∈L(δ(a1), .., δ(an)). This means that

δ(L(a1, a2, ..., an)) ⊂L(δ(a1), .., δ(an)),∀ai∈

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Corollary 1. Let Gbe a poset and 1be the greatest

element of Gand δbe the trace of D. If a≤δ(1),

then D(a, 1, ..., 1) = a,∀a∈G.

Proof. Let a∈G, assume that a≤δ(1), from Theo-

rem 1 we can get,

D(L(a),1,1, ..., 1) = D(L(a, 1),1, ..., 1,1)

=L(a, D(1,1, ..., 1))

=L(a, δ(1))

=L(a).

Then,

D(L(a),1, ..., 1) = L(a).(22)

In addition,

D(L(a),1,1,1, ..., 1) = D(L(a, a),1,1, ..., 1,1,1)

=L(D(a, 1, ..., 1,1), a)

=L(D(a, 1, ..., 1)).

Then,

D(L(a),1, ..., 1) = L(D(a, 1,1, ..., 1)).(23)

Therefore, (22) and (23) shows that

L(D(a, 1,1, ..., 1)) = L(a).

Combining Lemma 1 and Proposition 1 (i), we can

get D(a, 1, ..., 1) = a,∀a∈G.

Proposition 3. Let Gbe a poset and 1its greatest ele-

ment. Let δbe a the trace of a permuting n-derivation

Don G. Then δ(1) = 1 ⇐⇒ δ=idD.

Proof. It is obvious that if δ=idD, then δ(1) = 1.

Inversely, let a∈G. Combining Theorem 1 and

Proposition 1 (v) we can get

D(L(a), a, ..., a) = D(L(a, a), a, ..., a)

=L(a, D(a, ..., a))

=L(a, δ(a))

=L(δ(a))

D(L(a), a, ..., a) = L(δ(a)).(24)

Moreover,

D(L(a), a, ..., a) = D(L(a, 1), a, ..., a)

=L(a, D(1, a, ..., a, a))

=L(D(1, a, ..., a))

D(L(a), a, ..., a) = L(D(1, a, .., a)).(25)

According (24) and (25) we get

L(D(1, a, a, ..., a)) = L(δ(a)).(26)

Proposition 1 (iv) implies that δ(a) =

D(a, a, ..., a)≤D(1, a, ..., a),which, be-

cause of (26) together Lemma 1, Show that

δ(a) = D(1, a, ..., a),∀a∈G. SinceDis a per-

muting map, we get D(1, a, ..., a) = D(a, ..., a, 1).

Hence,

δ(a) = D(a, ..., a, 1).(27)

With the similar process, we show that δ(a) =

D(a, ..., a, 1,1). In fact, Combining Theorem 1 (1)

and Proposition 1 (v) we have

D(L(a), a, ..., a, 1) = D(L(a, 1), a, ..., a, 1)

=L(a, D(1, a, a, ..., a, 1))

=L(D(1, a, a, ..., a, a, 1))

D(L(a), a, a, ..., a, 1) = L(D(1, a, ..., a, a, 1)).

(28)

Moreover,

D(L(a), a, a, ..., a, 1) = D(L(a, a), a, ..., a, 1)

=L(D(a, a, ..., a, 1), a)

=L(D(a, a, ..., a, 1))

D(L(a), a, ..., a, a, 1) = L(D(a, ..., a, 1)).(29)

Adding these last tow equations (28) and (29) we see

that

L(D(1, a, ..., a, 1)) = L(D(a, ..., a, 1)) (30)

Proposition 1 (iv) implies that D(a, a, ..., a, 1) ≤

D(1, a, ..., a, 1) which, because of (30) to-

gether Lemma 1, implies that D(1, a, ..., a, 1) =

D(a, a, ..., a, 1). SinceDis a permuting map, we

have D(1, a, ..., a, 1) = D(a, ..., a, 1,1).Hence,

D(a, ..., a, 1) = D(a, ..., a, 1,1).(31)

Combining (27) and (31), we cleam that

δ(a) = D(a, ..., a, 1,1).(32)

Similarly, we get

D(L(a), a, ..., a, 1,1) = D(L(a, 1), a, a, ..., a, 1,1)

=L(a, D(1, a, a, a, ..., a, 1,1))

=L(D(1, a, ..., a, 1,1)),

then

D(L(a), a, ..., a, 1,1) = L(D(1, a, ..., a, 1,1)).

(33)

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Furthermore,

D(L(a), a, a, ..., a, 1,1) = D(L(a, a), a, ..., a, 1,1)

=L(D(a, ..., a, a, 1,1), a)

=L(D(a, a, ..., a, 1,1)),

then

D(L(a), a, ..., a, 1,1) = L(D(a, a, ..., a, 1,1)).

(34)

Combining (33) and (34), we get

L(D(1, a, ..., a, 1,1)) = L(D(a, a, ..., a, 1,1)),

and by using Proposition 1 (iv) we see that

D(a, a, ..., a, a, 1,1) ≤D(1, a, ..., a, 1,1) and

from Lemma 1, we deduce that

D(a, ..., a, 1,1) = D(1, a, ..., a, ..., a, 1,1)

=D(a, ..., a, a, 1,1,1).(35)

Therefore, (32) and (35) show that

δ(a) = D(a, a, ..., a, a, 1,1,1).(36)

According to the results (27), (32) and (36), we

get δ(a) = D(a, ..., a, 1) = D(a, ..., a, 1,1) =

D(a, ..., a, 1,1,1),∀a∈G.

Using the same method of proof, we arrive at the fol-

lowing conclusion

δ(a) = D(a, a, ..., a, 1) = D(a, a, ..., a, 1,1) =

D(a, a, ..., a, 1,1,1) = ... =D(a, 1,1, ..., 1),∀a∈

To complete this demonstration, it is enough to show

that D(a, 1,1, ..., 1) = a.

From Theorem 2, since δ(1) = 1, we get

D(a, 1,1, ..., 1) = a,∀a∈G. This means that

δ(a) = a,∀a∈G. Thus, the theorem is proved..

Proposition 4. Considered Gbe a poset and 0its

least element. Let δbe the trace of D. Denote

F ixδ(G) = {a∈G:δ(a) = a}. Then,

(1) 0∈F ixδ(G).

(2) If a∈F ixδ(G)and b≤a, then b∈F ixδ(G).

(3) If Gis directed, then ∀b1, b2∈F ixδ(P),

∃k∈F ixδ(G):b1≤kand b2≤k.

Proof. (1) It is clear that since δ(0) = 0.

(2) Let a, b ∈G. Assume that a∈F ixδ(G)and

b≤a, then δ(a) = a. By using Theorem 1 (1), we

have

D(L(b), a, ..., a) = D(L(a, b), a, ..., a, a)

=L(D(a, ..., a), b)

=L(δ(a), b)

=L(a, b)

=L(b).

Since b∈L(b), it follows that b∈

D(L(b), a, ..., a, a).Hence, ∃t∈L(b)provided

that b=D(t, a, ..., a), by using Proposition 1 (iv)

and (i), we get

b=D(t, a, ..., a)≤D(b, a, ..., a)≤b, so

D(b, a, ..., a) = b. (37)

Again,

D(b, L(b), a, ..., a) = D(b, L(a, b), a, ..., a)

=L(D(b, a, ..., a), b)

=L(b, b)using (37)

=L(b).

Since b∈L(b), we get b∈D(b, L(b), a, ..., a).

Hence, there exists t∈L(b)such that b=

D(b, t, a, ..., a), by using Proposition 1 (iv) and (i),

we get

b=D(b, t, a, ..., a)≤D(b, b, a, a, ..., a, a)≤b, so

D(b, b, a, ..., a) = b. (38)

Also by using Theorem 1 (1), we have

D(b, b, L(b), a, a, a, ..., a) = D(b, b, L(a, b), a, ..., a)

=L(D(b, b, a, a, ..., a, a), b)

=L(b, b)by , using (38)

=L(b).

Since b∈L(b), we find b∈D(b, b, L(b), a, ..., a).

Hence, we can find an t∈L(b)which b=

D(b, b, t, a, ..., a), by using Proposition 1 (iv) and (i),

we get

b=D(b, b, t, a, ..., a)≤D(b, b, b, a, a, ..., a)≤b, so

D(b, b, b, a, ..., a) = b. (39)

From the results (37), (38) and (39), we ob-

tain D(b, a, ..., a) = D(b, b, a, a, ..., a) =

D(b, b, b, a, ..., a) = b,∀a, b ∈G.

With the same method, we arrive at

D(b, a, ..., a) = D(b, b, a, ..., a) =

D(b, b, b, a, ..., a) = ... =D(b, b, ..., b, b, a) = b,∀

b∈G. So

D(b, b, ..., b, a) = b. (40)

Moreover,

D(b, ..., b, b, L(b)) = D(b, ..., b, L(a, b))

=L(D(b, ..., b, a), b)

=L(b, b)by (40)

=L(b).

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Then D(b, ..., b, L(b)) = L(b), application Lemma 2

(1) yields that δ(b) = b,∀b∈G. This shows that

b∈F ixδ(G).

(3) Let b1, b2∈G, since Gis directed, ∃c∈G:

b1≤cand b2≤c. Since b1, b2∈F ixδ(G), we

get δ(b1) = b1and δ(b2) = b2. By Proposition 1

(viii) we can get b1≤δ(c)and b2≤δ(c). Put

k=δ(c), by Proposition 1 (ix) we get δ(t) = t, hence

t∈F ixδ(G).

Corollary 2. Let 0be the least element of Gand δbe

the trace of D, then F ixδ(G)is an ideal of G.

Proposition 5. Let d1and d2be two permuting n-

derivations on Gwith traces δ1, δ2, respectively. Then

δ1=δ2⇐⇒ F ixδ1(G) = F ixδ2(G).

Proof. It is obvious that δ1=δ2implies F ixδ1(G) =

F ixδ2(G). Conversely, let F ixδ1(G) = F ixδ2(G)

and a∈G. By Proposition 1 (ix), we have δ1(a)∈

F ixδ1(G) = F ixδ2(G), so

δ2(δ1(a)) = δ1(a).(41)

Combining (v) and (viii) in Proposition 1, we get

δ2(δ1(a)) ≤δ2(a).(42)

These last two equations (41) and (42) show that

δ1(a)≤δ2(a).(43)

Similarly, we can get δ1(δ2(a)) = δ2(a)and

δ1(δ2(a)) ≤δ2(a). Then

δ2(a)≤δ2(a).U((44)

Adding these last two arguments (43) and (44), we

find that δ2(a) = δ2(a),∀a∈G. So δ1=δ2.

3 Some properties of posets

involving permuting n-derivations

Theorem 4. Let Gbe a poset and δbe the of D. If

0be the least element of G, Then kerδ ={a∈G:

δ(a) = 0}is a nonempty and a lower set of G.

Proof. By Proposition 1 (vi), we can see that δ(0) = 0

imply 0∈kerδ. Therefore kerδ =ϕ. Furthermore,

if a∈kerδ and b∈Gin which b≤a, since δ(b)≤

δ(a)by Proposition 1 (viii) and δ(a) = 0, so δ(b) = 0.

Therefore, b∈kerδ and thus forces the results.

Proposition 6. Let Gbe a poset, 0be the least ele-

ment of Gand δbe the of Don G.

If Jis an ideal of G, then δ−1(J)is an ideal of G.

Proof. Assume that Jis an ideal of G, then 0∈ J

and so, δ(0) = 0 ∈I. Hence, 0∈δ−1(J), then

δ−1(J)=ϕ. Suppose that a∈δ−1(J)and b∈G

where b≤a, then δ(a)∈ J and δ(b)≤δ(a)by

Proposition 1 (viii), this imply that δ(b)∈ J and so

b∈δ−1(J). This means that δ−1(J)is an ideal of

Proposition 7. Let Gbe a poset and δthe trace of a

permuting n-derivation Don G.

Let I1and I2be two ideals of G, we have

I1⊆I2⇒δ(I1)⊆δ(I2).

Proof. Let b∈δ(I1), then ∃a∈I1⊆I2:δ(a) = b.

Hence, b∈δ(I2). It follows that δ(I1)⊆δ(I2).

Theorem 5. Let Gbe a poset and D1,D2be two

permuting n-derivations on Gwith traces δ1,δ2, re-

spectively. Then, ∀a∈G,

δ1(a)≤δ2(a)⇐⇒ δ2(δ1(a)) = δ1(a).

Proof. Assume that δ1(a)≤δ2(a),∀a∈G, that

is, δ1(δ1(a)) ≤δ2(δ1(a)).By Proposition 1 (ix),

δ1(a) = δ1(δ1(a)). So

δ1(a)≤δ2(δ1(a)).(45)

Moreover, the Proposition 1 (v) gives that

δ2(δ1(a)) ≤δ1(a).(46)

From the above arguments (45) and (46), we can get

δ2(δ1(a)) = δ1(a),∀a∈G. Inversely, suppose that

δ2(δ1(a)) = δ1(a),∀a∈G. By using Proposition

1 (v) and (viii), we obtain δ2(δ1(a)) ≤δ2(a), and by

hypothesis, we can get δ1(a)≤δ2(a),∀a∈G.

4 Conclusion

This work has provided a comprehensive analysis of

derivations and permuting n-derivations in the con-

text of partially ordered sets (posets), which are gen-

eralizations of derivations on a poset. We have in-

troduced and studied the concept of permuting n-

derivations on posets and presented several character-

ization theorems and fundamental properties related

to permuting n-derivations. Additionally, we have in-

troduced the fixed set of permuting n-derivations in

posets and discussed the relationships among deriva-

tions, ideals and fixed sets within posets. This study

opens up further avenues for research, inviting deeper

exploration into the interactions between derivations

and poset structures. Our future research on posets

will be inspired by our recent work on lattices in [16]

which involves generalized derivations. We aim to

explore how these concepts can be applied to posets

to develop new theories and applications.

Acknowledgment:

The authors express their gratitude to the referees for

their recommendations, and counsel on the work.

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