Additive Derivations of Incidence Modules

Abstract

Let R be an associative ring and M a left R-module. This paper examines the structural properties of the incidence module $I(\mathcal{P},M)$, associated with a module M over a ring R and a locally finite poset $\mathcal{P}$. We provide a complete characterization of when an additive derivation on $I(\mathcal{P},M)$ is inner, for the case where $\mathcal{P}$ is a finite and connected poset. These criteria are then generalized to arbitrary posets, revealing a profound connection between the algebraic properties of the module and the graph-theoretic structure of $\mathcal{P}$ as a directed graph.

1. Introduction

Ring theory provides a unifying language across diverse mathematical disciplines, offering a powerful structural framework for investigating problems of both classical and contemporary importance. While rings with derivations do not undergo substantial modifications, their study has attracted considerable attention over the past five decades. Scholars have extensively explored the connections between derivations and the structural properties of rings, leading to profound insights.

This paper devotes significant attention to the derivation of incidence ring, referencing a highly informative introduction in [1], which presents a historical overview of derivation studies, along with a comprehensive list of references. Specifically, we investigate the incidence modules $I(\mathcal{P},M)$ for an arbitrary partially ordered set $\mathcal{P}$ over an associative ring R. The foundational results on the derivation of incidence rings are detailed in [2], with the key outcome presented in ([2], Theorem 7.1.4). This theorem characterizes the derivations of $I(\mathcal{P},R)$, where R is assumed to be a commutative ring with identity and $\mathcal{P}$ is regarded as a partially ordered set.

Baclawski [3] examined the $I(\mathcal{P},F)$ of a locally finite poset $\mathcal{P}$ over a field F using derivations. He showed that every derivation can be decomposed as the sum of an inner derivation and a derivation induced by an additive function on the intervals of $\mathcal{P}$. Although such decompositions are generally non-unique, the author provided a complete characterization of derivations that are simultaneously inner and generated by additive functions. Moreover, it was shown that the existence of either a minimal element 0 or a maximal element 1 in the poset $\mathcal{P}$ implies that all derivations of $I(\mathcal{P},F)$ is necessarily inner.

Khripchenko [4] further generalized these findings by examining derivations in broader classes of incidence rings and applying the theoretical framework established by Baclawski and Spiegel and O’Donnell to these structures. His work extended the applicability of these results to more general algebraic contexts.

Suppose R is a commutative ring and $\mathcal{P}$ is a connected, finite partially ordered set. While not every additive derivation of the $I(\mathcal{P},R)$ is necessarily inner, it is possible to identify those additive derivations that exhibit this property. This work characterizes the inner derivations of $I(\mathcal{P},R)$ for arbitrary locally finite posets $\mathcal{P}$.

Stanley [5] initiated the study of the isomorphism problem for incidence algebra. Baclawski [3] further explored the structural properties of derivations and automorphisms in this context. More recently, Xiao [6] extended Herstein’s program on Lie-type mappings [7] to incidence algebras. He showed Jordan derivations coincide with derivations for 2-torsion-free R. Moreover, it was subsequently demonstrated that Lie derivations and Lie triple derivations of $I(\mathcal{P},R)$ are of the standard form in [8,9], respectively.

An additive mapping $D:R\to R$ is a derivation if it fulfills the Leibniz rule:

$$D\left(pq\right)=D\left(p\right)q+pD\left(q\right)\mathrm{for}\mathrm{all}p,q\in R.$$

Thus, $Der\left(R\right)$ denotes all derivations of R, and $Inn\left(R\right)\subseteq Der\left(R\right)$ is a subring. A straightforward example is the classical derivative operator applied to algebras of differentiable functions. In the context of noncommutative rings, however, the fundamental examples are conceptually different. For any elements $c,p,q\in R$, the following identity holds:

$$[c,pq]=[c,p]q+p[c,q]$$

For a fixed $c\in R$, define $D:R\to R$ by

$$D\left(P\right)=[p,c]\mathrm{for}\mathrm{all}p\in R.$$

For any two elements $p,c\in R$, the commutator $[p,c]$ is defined as

$$[p,c]=pc-cp$$

The derivation function $D\left(p\right)$ is defined as

$$D\left(p\right)=[p,c]=pc-cp,\mathrm{for}\mathrm{all}p\in R$$

where $[p,c]$ represents the commutator of p and c, measuring the extent to which they fail to commute. If $[p,c]=0$, then p and c commute.

A straightforward computation confirms that D is additive and satisfies

$$D\left(pq\right)=[pq,c]=p[q,c]+[p,c]q=pD\left(q\right)+D\left(p\right)q\mathrm{for}\mathrm{all}p,q\in R.$$

Thus, D is a derivation, specifically referred to as an inner derivation of R associated with c, often denoted as $I_c$. While every inner derivation on R is a derivation, there exist numerous examples of derivations that are not inner.

Recall the definition of an incidence ring $I(\mathcal{P},R)$. Let $\mathcal{P}$ be a locally finite partially ordered set, where for any $p\le q$, there are only finitely many elements $z\in \mathcal{P}$ such that $p\le z\le q$. The incidence ring $I(\mathcal{P},R)$, as defined in [2], consists of the set of functions.

$$I(\mathcal{P},R)=\{g:\mathcal{P}\times \mathcal{P}\to R:g(p,q)=0\mathrm{unless}p\le q\},$$

with the following operations

$$(g+h)(p,q)=f(p,q)+h(p,q)$$

$$gh(p,q)=\sum _{p\le z\le q}g(p,z)h(z,q).$$

for all $g,h$ in $I(\mathcal{P},R)$ and $p,q$ in X, if R is commutative, then $I(\mathcal{P},R)$ forms an R-algebra, where the operation is defined as $\left(rg\right)(p,q)=rg(p,q)$ for any $r\in R$.

An incidence ring $I(\mathcal{P},R)$ is a potent tool in algebraic combinatorics that utilizes algebraic structures to represent the combinatorial interaction between components, such as points and lines. The incidence ring formalizes these interactions often by using functions or matrices that specify the components that are incident to one another. The study of incidence rings in algebraic systems is both fascinating and challenging. Module theories and posets interact to form incidence rings, providing a fruitful ground for research. Continuing our investigation, we examine incidence modules. We focus on the incidence module $I(\mathcal{P},M)$, where M is a left R-module and $\mathcal{P}$ is a locally finite poset defined over M. However, the need for a more refined and adaptable structure resulted in the implementation of incidence modules. Incidence modules expand upon the fundamental notion of incidence rings by integrating module theory, thus enabling a more refined and flexible methodology for investigating combinatorial combinations. This progress is especially beneficial for additive derivation in incidence modules, as the additive characteristics of modules allow for a more thorough and adaptable examination of combinatorial structures. This allows mathematicians to investigate more intricate interactions and obtain results that are not readily achievable within incidence rings alone.

The goal of this paper is to introduce the idea of incidence modules as a way to study derivations in a structured way. This is because the author wants to learn more about the algebraic property of $I(\mathcal{P},R)$. We want to investigate the operation of additive and inner derivations inside directed graphs by using $I(\mathcal{P},M)$, which are fundamental in combinatorial and graph theory. Directed graphs, due to their intricate connectivity characteristics, serve as an optimal framework for analyzing the behavior of these derivatives. This method provides a toolkit for examining the algebraic structure of $I(\mathcal{P},R)$ and facilitates future applications in graph theory, network analysis, and other disciplines. This paper presents $I(\mathcal{P},M)$ to provide a formal method for comprehending additive derivations, inner derivations, and their operation inside a directed graph.

This section presents foundational definitions and known results concerning $I(\mathcal{P},M)$ and their associated derivations.

2. Preliminaries

Throughout this paper, let R be an associative ring with identity. A partially ordered set (poset) P is called locally finite if for every $p,q\in P$ with $p\le q$, the interval $[p,q]=\{z\in P\mid p\le z\le q\}$ is a finite set. This condition is essential for the convolution product in the incidence ring $I(P,R)$ and module $I(P,M)$ to be well-defined. The Hasse diagram of a poset P is a directed graph whose vertices are the elements of P, and there is a directed edge from p to q if and only if q covers p (i.e., $p<q$ and there is no z such that $p<z<q$).

Let M be a left R-module, and all R-modules are unital.

Let R be a commutative ring with identity and $\mathcal{P}$ be a locally finite poset. Throughout this paper, all R-modules are assumed to be unital, meaning that the identity element $1\in R$ acts as the identity map on the module (i.e., $1·m=m$ for all $m\in M$). Under these conditions, $I(\mathcal{P},R)$ forms an incidence ring. Now, let M be a left R-module and $I(\mathcal{P},M)$, be the incidence module. By definition, $I(\mathcal{P},M)$ is the set of all functions defined as

$$I(\mathcal{P},M)=\{\kappa :\mathcal{P}\times \mathcal{P}⟶M\mathrm{with}\kappa (p,q)=0\mathrm{unless}p\le q\},$$

with the following operations:

$$(\kappa +\beta )(p,q)=\kappa (p,q)+\beta (p,q)$$

$$\left(f\kappa \right)(p,q)=\sum _{p\le t\le q}f(p,t)\kappa (t,q)$$

for all $\kappa ,\beta$ in $I(\mathcal{P},M)$, f in $I(\mathcal{P},R)$ and $p,q$ in $\mathcal{P}$, the module $I(\mathcal{P},M)$ becomes an R-module under the operations

$$\left(rf\right)(p,q)=r·f(p,q)$$

for any $r\in R$. When R is commutative, $I(\mathcal{P},M)$ acquires the structure of an $I(\mathcal{P},R)$-bimodule. Indeed, in this case, M may be regarded as an R-bimodule by defining $m·r=r·m$ for all $r\in R$ and $m\in M$. Consequently, we obtain the scalar multiplication $kf$ for $k\in I(\mathcal{P},M)$ and $f\in I(\mathcal{P},R)$.

In an incidence module the element ${\delta }_m$ is defined as follows:

$${\delta }_m(p,q)=\left\{\begin{array}{cc}m\hfill & \mathrm{if}p=q\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

In particular, if $M=R$ and $m=1\in R$, let $\delta ={\delta }_1$.

The function ${\delta }_m\in I(\mathcal{P},M)$ as an element of $I(\mathcal{P},M)$ and, for any $(p,q)\in \mathcal{P}\times \mathcal{P}$ with $p\le q$, we define $e_{pq}\in I(\mathcal{P},R)$ as

$$e_{pq}(u,v)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}(u,v)=(p,q)\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

For convenience of statements we write $m{e}_{pq}$, i.e.,

$$m{e}_{pq}(u,v)=\left\{\begin{array}{cc}m\hfill & \mathrm{if}(u,v)=(p,q)\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

Definition 1.

Let $\mathcal{P}$ be a locally finite poset and R a commutative ring with identity. An element $\alpha \in I(\mathcal{P},M)$ is additive if it satisfies the condition:

$$\alpha (p,q)=\alpha (p,z)+\alpha (z,q)\mathit{for}\mathit{each}\mathit{triple}p\le z\le q.$$

In addition, $\alpha (p,p)=0$ for all $p\in \mathcal{P}$.

Now, suppose that $\alpha \in I(\mathcal{P},R)$ is additive. We define the map $L_{\alpha }:I(\mathcal{P},M)\to I(\mathcal{P},M)$ by convolution, where for any $f\in I(\mathcal{P},M)$, $L_{\alpha }\left(f\right)=\alpha \ast f$ is given by

$$\left(L_{\alpha }\left(f\right)\right)(p,q)=\alpha \ast f(p,q)=\alpha (p,q)f(p,q)$$

for all $p,q\in \mathcal{P}$.

The map $L_{\alpha }$ satisfies the following two properties for all $f,g\in I(\mathcal{P},M)$ and $r\in R$:

1.

$L_{\alpha }(f+g)=L_{\alpha }\left(f\right)+L_{\alpha }\left(g\right)$;

2.

$L_{\alpha }\left(rf\right)=r{L}_{\alpha }\left(f\right)$.

Definition 2.

Let A be an R-algebra and M be an A-bimodule. A linear mapping $D:M\to M$ is called a derivation if for all $m_1,m_2\in M$, $a\in A$, and $r\in R$, the following conditions hold:

1.

$D(m_1+m_2)=D\left(m_1\right)+D\left(m_2\right)$;

2.

$D\left(rm\right)=rD\left(m\right)$;

3.

$D\left(am\right)=d\left(a\right)m+aD\left(m\right)$.

where $d:A\to A$ is a derivation of the algebra A. If d is an inner derivation, meaning $d\left(a\right)=[a,b]=ab-ba$ for some fixed $b\in A$, then D is called an inner derivation of the module M.

For each $b\in A$, the map $D_b:M\to M$ defined by

$$D_b\left(m\right)=b·m-m·b,\mathit{for}\mathit{all}m\in M$$

is an inner derivation with respect to b.

Definition 3.

Let A be an R-algebra and M be an A-bimodule. A linear mapping $D:M\to M$ is called a derivation if for all $m_1,m_2\in M$, $a\in A$, and $r\in R$, the following conditions hold:

1.

$D(m_1+m_2)=D\left(m_1\right)+D\left(m_2\right)$;

2.

$D\left(rm\right)=rD\left(m\right)$;

3.

$D\left(am\right)=d\left(a\right)m+aD\left(m\right)$.

where $d:A\to A$ is a derivation of the algebra A. If d is an inner derivation, meaning $d\left(a\right)=[a,b]=ab-ba$ for some fixed $b\in A$, then D is called an inner derivation of the module M.

For each $b\in A$, the map $D_b:M\to M$ defined by

$$D_b\left(m\right)=b·m-m·b,\mathit{for}\mathit{all}m\in M$$

is an inner derivation with respect to b.

The following result is a consequence which follows from [2] (Theorem 7.1.4, Propositions 7.1.6, 7.1.8).

Proposition 1.

Let M be an R-bimodule and $\mathcal{P}$ a locally finite partially ordered set. If $\mathcal{P}$ contains an element that is comparable with every element of $\mathcal{P}$, then every additive derivation of $I(\mathcal{P},M)$ is inner.

An element comparable to all in $\mathcal{P}$ is not needed for every additive derivation of $I(\mathcal{P},M)$ to be inner. To illustrate, consider the incidence algebra associated with the poset ${\mathcal{P}}_1$, whose Hasse diagram is shown below.

                                            

Clearly, there is no element in ${\mathcal{P}}_1$ that is comparable with all other elements in ${\mathcal{P}}_1$. Example 1 shows all additive derivations of $I({\mathcal{P}}_1,M)$ are inner.

However, in some cases, in which the $I(\mathcal{P},M)$ allows for additive derivations that are not inner. Such a situation occurs for the $I(\mathcal{P},M)$ of the partially ordered set ${\mathcal{P}}_2$ shown below.

                                            

• Spiegel and O’Donnell presented this example in ([2], page 257). Consequently, this result also follows from the subsequent discussion (see Example 2).

3. Conditions for an Additive Derivation to Be Inner

3.1. Finite and Connected Partially Ordered Sets

In this subsection, we assume that the poset $\mathcal{P}$ is finite and connected. A poset $\mathcal{P}$ is defined as connected if, for any $p,q$ in $\mathcal{P}$, there exists a sequence $p_0,p_1,\dots ,p_n\in \mathcal{P}$ such that $p=p_0,q=p_n,\mathrm{and}{p}_i\le p_{i+1}\mathrm{or}{p}_{i+1}\le p_i,\mathrm{for}\mathrm{all}0\le i\le n-1$. According to this condition, a chain in $\mathcal{P}$ connects p and q, with all pairs in the chain comparable, even if p and q are not directly comparable. Based on the partially ordered set, comparisons can be conducted in either direction (increasing or decreasing), allowing traversal from p to q within the structure of the partially ordered set. This implies that $\mathcal{P}$ has a connected Hasse diagram. In a Hasse diagram, the process of navigating between comparable elements in a partially ordered set by following the edges of the diagram is called traversal.

In short, a finite and connected partially ordered sets allows movement between elements using order relations, and its Hasse diagram visually represents this connectivity. Understanding these structures is essential for both theoretical insights and practical applications across various domains.

In this subsection, we demonstrate how viewing $\mathcal{P}$ as a directed graph facilitates the formulation of a condition under which an additive derivation of $I(\mathcal{P},M)$ is inner.

Let $G=(\mathcal{P},A)$ be a directed graph, where $\mathcal{P}$ represents the set of vertices and A the set of edges. The edge $(p,q)\in A$ whenever q covers p, i.e., $p<q$ and $\{h\in \mathcal{P}:p<h<q\}=\varnothing$, which is used to define the partially ordered set $\mathcal{P}$. Below, we provide a restatement of key definitions relevant to directed graphs. For further results and definitions from graph theory, see [10].

Definition 4.

Let $G=(V,A)$ be a directed graph, where V constitutes a collection of vertices and A comprises ordered pairs known as arcs between vertices in V. For $v,w\in V$, let $\mathcal{W}$ represent a sequence of vertices $v_0{v}_1\dots v_k$ such that $v_0=v$, $v_k=w$, and $v_i\ne v_{i+1}$ for vertices $0\le i\le k-1$. Let $\mathcal{W}:v=v_0{v}_1\dots v_k=w$ represent this sequence. A sequence $\mathcal{W}$ is called a walk from v to w in G if for every $0\le i\le k-1$, either $(v_i,v_{i+1})$ or $(v_{i+1},v_i)$ belongs to A. If $v=w$, the walk is closed at v. A semipath is a walk where each vertex is distinct. If G has a unique semipath for every pair $p,q\in V$, then G is classified as a tree. As noted in [11], a path in $G=(V,A)$ is a sequence of distinct vertices $(v_0,v_1,\dots ,v_k)$ such that $(v_i,v_{i+1})\in A$ for $0\le i<k$, with forward arcs. A semipath from v to w is a path (or counter-path) if $(v_i,v_{i+1})$ or $(v_{i+1},v_i)\in A$.

Let $G=(V,A)$ be a directed graph, and let $\mathcal{W}$ and $\mathcal{U}$ be paths in G. Each path induces a subgraph of G; we denote their vertex sets by $V\left(\mathcal{W}\right)$ and $V\left(\mathcal{U}\right)$, and their edge sets by $A\left(\mathcal{W}\right)$ and $A\left(\mathcal{U}\right)$, respectively. The intersection of the paths $\mathcal{W}$ and $\mathcal{U}$ is defined as the subgraph $G^{\prime }=(V^{\prime },A^{\prime })$ of G, where

$$V^{\prime }=V\left(\mathcal{W}\right)\cap V\left(\mathcal{U}\right),A^{\prime }=A\left(\mathcal{W}\right)\cap A\left(\mathcal{U}\right).$$

This subgraph, denoted $G^{\prime }=\mathcal{W}\cap \mathcal{U}$, consists precisely of the vertices and edges common to both paths.

Let $G=(V,A)$ be a directed graph and T a set. A weight function on G is a mapping $w:A\to T$, and the graph G equipped with w is termed a weighted directed graph. Now, let $G=(V,A)$ be a weighted directed graph with a weight function $w:A\to M$. We define the set of reverse edges of A as $A^{\ast }=\{(u,v)\mid (v,u)\in A\}$. The weight function w then induces a function ${\eta }_w:A\cup A^{\ast }\to M$ given by

$${\eta }_w(u,v)=\left\{\begin{array}{cc}w(u,v),\hfill & \mathrm{if}(u,v)\in A,\hfill \\ -w(v,u),\hfill & \mathrm{if}(u,v)\in A^{\ast }.\hfill \end{array}\right.$$

Let $\alpha$ be an additive element of $I(X,M)$. We can transform $G=(\mathcal{P},A)$ into a weighted directed graph with weights in M as follows: for each edge $(u,v)$ in A, the weight of the edge $(u,v)$ is $\alpha (u,v)$. Observe that if $p=v_0{v}_1\dots v_k=q$ is a path in $\mathcal{P}$, then $\alpha (p,q)={\displaystyle \sum _{i=0}^{k-1}}\eta (v_i,v_{i+1})$. When there are two distinct semipaths connecting p to q in $\mathcal{P}$, the sum of the directed edge weights may differ. For instance, consider the poset ${\mathcal{P}}_2$ related to the Hasse diagram below:

                                            

If $\alpha$ is an additive element, then ${\eta }_{\alpha }(a,b)+{\eta }_{\alpha }(b,d)=1\ne 0={\eta }_{\alpha }(a,c)+{\eta }_{\alpha }(c,d)$. The next theorem shows that this equation gives a complete characterization of inner derivations $L_{\alpha }$.

Let $\mathcal{W}:p=v_0{v}_1\dots v_k=q$ be a walk from p to q. For an additive element $\alpha \in I(\mathcal{P},M)$, we define the sum of directed edge weights along $\mathcal{W}$ as

$${{\rm Y}}_{\mathcal{W}}^{\alpha }(x,y)=\sum _{i=0}^{k-1}{\eta }_{\alpha }(v_i,v_{i+1}).$$

The following lemma holds:

Lemma 1.

Let $\mathcal{P}$ be a partially ordered set, and let M be an R-bimodule. Suppose that $\mathcal{W}:p=v_0{v}_1\dots v_k=q$ represents a walk connecting p to q in $\mathcal{P}$, and let $\alpha \in I(\mathcal{P},M)$ be additive. The function ${{\rm Y}}_{\mathcal{W}}^{\alpha }$ is a map from the set of walks in $\mathcal{P}$ to $I(\mathcal{P},M)$, which assigns a value in the bimodule to each walk.

(a)

If $\mathcal{W}$ is a path, then ${{\rm Y}}_{\mathcal{W}}^{\alpha }=\alpha (p,q)$. If $\mathcal{W}$ is a counter-path, then ${{\rm Y}}_{\mathcal{W}}^{\alpha }=-\alpha (p,q)$.

(b)

If ${\mathcal{W}}_1:p=v_0\dots v_l=z$ and ${\mathcal{W}}_2:z=v_l\dots v_k=q$, where $0<l<k$, then the composition of the walks $\mathcal{W}={\mathcal{W}}_1{\mathcal{W}}_2$ satisfies

$${{\rm Y}}_{\mathcal{W}}^{\alpha }(p,q)={{\rm Y}}_{{\mathcal{W}}_1}^{\alpha }(p,z)+{{\rm Y}}_{{\mathcal{W}}_2}^{\alpha }(z,q).$$

Specifically, if $\mathcal{W}$ is a closed walk (i.e., $p=q$), then

$${{\rm Y}}_{\mathcal{W}}^{\alpha }(p,p)={{\rm Y}}_{{\mathcal{W}}_1{\mathcal{W}}_2}^{\alpha }(p,p)=0.$$

(c)

If ${\mathcal{W}}^{-1}:q=v_k\dots v_1{v}_0=p$ is the inverse walk of $\mathcal{W}$, then ${\mathcal{W}}^{-1}$ is a walk connecting q to p in $\mathcal{P}$, and

$${{\rm Y}}_{{\mathcal{W}}^{-1}}^{\alpha }(q,p)=-{{\rm Y}}_{\mathcal{W}}^{\alpha }(p,q),$$

where $-{{\rm Y}}_{\mathcal{W}}^{\alpha }(p,q)$ represents the additive inverse of ${{\rm Y}}_{\mathcal{W}}^{\alpha }(p,q)$ in the bimodule $I(\mathcal{P},M)$. If $\mathcal{W}$ is a semipath, then so is its inverse ${\mathcal{W}}^{-1}$.

Theorem 1.

Let $\mathcal{P}$ be a partially ordered set, M an R-bimodule, and α in $I(\mathcal{P},M)$ an additive element. Then additive derivation $L_{\alpha }$ is inner if and only if, for every pair of distinct elements $p,q\in \mathcal{P}$ and any semipaths ${\mathcal{H}}_1,{\mathcal{H}}_2$ connecting p to q, the following holds:

$${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(p,q)={{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(p,q).$$

Proof. 

First, assume that $\alpha$ is inner. According to [2] (Theorem 7.1.6), since $L_{\alpha }$ is a potential, there exists a function $g:\mathcal{P}\to M$ satisfying $\alpha (p,q)=g\left(q\right)-g\left(p\right)$, for $p\le q$ in $\mathcal{P}$. Consider $p,q$ in $\mathcal{P}$, $p\ne q$, and suppose that $\mathcal{H}:v_0{v}_1\dots v_k$ is a semipath joining p to q. We will demonstrate this by induction on k${{\rm Y}}_{\mathcal{H}}^{\alpha }(p,q)=g\left(q\right)-g\left(p\right)$. Let $k=1$. If $p<q$, then ${{\rm Y}}_{\mathcal{H}}^{\alpha }(p,q)=\alpha (p,q)=g\left(q\right)-g\left(p\right)$. If $q<p$, then ${{\rm Y}}_{\mathcal{H}}^{\alpha }(p,q)=-\alpha (q,p)=-(g\left(p\right)-g\left(q\right))=g\left(q\right)-g\left(p\right)$. Let $k>1$, and assume result holds for $k-1$. Let $\mathcal{H}={\mathcal{H}}_1{\mathcal{H}}_2$ be the semipath joining p to q, where ${\mathcal{H}}_1$ joins $v_0$ to $v_{k-1}$, and ${\mathcal{H}}_2$ joins $v_{k-1}$ to $v_k$. Then,

$$\begin{array}{cc}\hfill {{\rm Y}}_{\mathcal{H}}^{\alpha }(p,q)& ={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_0,v_{k-1})+{{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(v_{k-1},v_k)\hfill \\ \hfill =& g\left(v_{k-1}\right)-g\left(v_0\right)+g\left(v_k\right)-g\left(v_{k-1}\right)\hfill \\ \hfill =& g\left(v_k\right)-g\left(v_0\right)\hfill \\ \hfill =& g\left(q\right)-g\left(p\right).\hfill \end{array}$$

Therefore, for every semipath $\mathcal{H}$ connecting p to q, ${{\rm Y}}_{\mathcal{H}}^{\alpha }(p,q)=g\left(q\right)-g\left(p\right)$. Now, assume that for any distinct $p,q\in \mathcal{P}$ and for any semipaths ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ connecting p to q, the equality ${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(p,q)={{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(p,q)$ holds. Let $g:\mathcal{P}\to M$ be defined as follows: Choose a maximal element $z\in \mathcal{P}$ and set $g\left(z\right)=0$. For each other element $p\in \mathcal{P}$, define $g\left(p\right)={{\rm Y}}_{{\mathcal{H}}_p}^{\alpha }(z,P)$, where ${\mathcal{H}}_{z,p}$ is a semipath from z to p. By assumption, the function g is well-defined, meaning it does not depend on the choice of ${\mathcal{H}}_p$. We will use Lemma 1 to show that $\alpha (p,q)=g\left(q\right)-g\left(p\right)$ for any $p\le q\in \mathcal{P}$. First, observe that for all $p\in \mathcal{P}$, we have $\alpha (p,p)=0$, which immediately yields $g\left(p\right)-g\left(p\right)=0$. Now, for $p<q$, let $\mathcal{H}:w_0{w}_1\dots w_l$ be a path from p to q. Given the maximality of z, it follows that $z\ne p$. In the case where $q=z$, we obtain

$$g\left(z\right)-g\left(p\right)=-{{\rm Y}}_{{\mathcal{H}}_p}^{\alpha }(z,p)={{\rm Y}}_{{\mathcal{H}}_p^{-1}}^{\alpha }(p,z)={{\rm Y}}_{\mathcal{H}}^{\alpha }(p,z)=\alpha (p,z).$$

Assume that $q\ne z$, and let ${\mathcal{H}}_p:u_0{u}_1\dots u_k$ represent a semipath connecting z to p. Let n be the smallest index for which $u_n=w_m$, with $m\in \{0,1,\dots ,l\}$. Note that $n\ne 0$, as this would imply $z=u_0=w_m<w_l=q$, which contradicts the assumption that $\mathcal{H}$ is a path. Now assume $0<n<k$. Then $m\ne 0$ by distinctness of the vertices $u_0,\dots ,u_k$. If $0<m<l$, consider the semipaths ${\mathcal{H}}_1:w_0\dots w_m$, ${\mathcal{H}}_2:w_m\dots w_l$, and ${\mathcal{H}}^{\prime }:u_0\dots u_n$. Then ${\mathcal{H}}^{\prime }{\mathcal{H}}_2$ connects z to q, and ${\mathcal{H}}^{\prime }{\mathcal{H}}_1^{-1}$ connects z to p. By Lemma 1,

$$g\left(z\right)-g\left(p\right)={{\rm Y}}_{{\mathcal{H}}^{\prime }{\mathcal{H}}_2}^{\alpha }(z,q)-{{\rm Y}}_{{\mathcal{H}}^{\prime }{\mathcal{H}}_1^{-1}}^{\alpha }(z,p)={{\rm Y}}_{{\mathcal{H}}_1{\mathcal{H}}_2}^{\alpha }(p,q)={{\rm Y}}_{\mathcal{H}}^{\alpha }(p,q)=\alpha (p,q).$$

If $m=l$, then ${\mathcal{H}}_1=\mathcal{H}$ suffices. For $n=k$, $m\ne l$ since $p\ne q$. If $0<m<l$, then $w_0=u_n=w_m$, which leads to a contradiction since the vertices $w_0,\dots ,w_l$ are pairwise distinct. Therefore, we must have $m=0$. In this case, ${\mathcal{H}}_2=\mathcal{H}$ and ${\mathcal{H}}^{\prime }={\mathcal{H}}_p$, so the preceding argument holds without the need for ${\mathcal{H}}_1$. Thus, we conclude that $\alpha (p,q)=g\left(q\right)-g\left(p\right)$ for any $p\le q\in \mathcal{P}$. According to [2] (Proposition 7.1.6), it follows that $L_{\alpha }$ is inner. □

Corollary 1.

Let $\mathcal{P}$ be a tree (as a partially ordered set), and let M be an R-bimodule. Then every additive derivation $D:I(\mathcal{P},M)\to I(\mathcal{P},M)$ is inner.

Proof. 

Given that for every distinct pair p and $q\in \mathcal{P}$, there exists exactly one semipath connecting p to q, we can conclude that the proof of the statement is straightforward and follows directly from Theorem 1. □

Corollary 2.

Let $\mathcal{P}$ be a partially ordered set whose Hasse diagram is a tree, and let M be an R-bimodule. Then every derivation $D:I(\mathcal{P},M)\to I(\mathcal{P},M)$ is inner.

Proof. 

This follows from [2] (Theorem 7.1.4) and Corollary 4. □

Example 1.

Consider the partially ordered set ${\mathcal{P}}_1$ from Section 2, a tree. Since every additive derivation of $I({\mathcal{P}}_1,M)$ is inner, it follows from the previous corollary that all derivations are inner.

Example 2.

Consider the partially ordered set ${\mathcal{P}}_2$ from Section 2, with $\alpha =e_{bd}\in I({\mathcal{P}}_2,M)$, which is additive. Let the semipaths $\mathcal{H}:abd$ and $\zeta :acd$ connect a to d. Since ${{\rm Y}}_{\mathcal{H}}^{\alpha }(a,d)=1$ and ${{\rm Y}}_{\zeta }^{\alpha }(a,d)=0$, it follows from Theorem 1 that $L_{\alpha }$ is not an inner additive derivation.

Definition 5.

Let $G=(V,A)$ be a directed graph. A cycle is a closed walk $\mathcal{C}:v_0{v}_1\dots v_k$ with $k\ge 3$, $v_0=v_k$, and distinct vertices for $0\le i\le k-1$.

Remark 1.

(a) For a directed graph $G=(V,A)$, if $\mathcal{C}:v_0{v}_1\dots v_k$ constitutes a cycle based at vertex $v_0$, then necessarily $k\ge 4$. This follows from the property that the corresponding Hasse diagram excludes triangular configurations.

(b) A poset $\mathcal{P}$ possesses a tree structure precisely when its associated graph contains no cycles.

Corollary 3.

Let M be an R-bimodule and $\alpha \in I(\mathcal{P},M)$ an additive element. The additive derivation $L_{\alpha }$ induced by α is an inner derivation if and only if, for every vertex $p\in \mathcal{P}$ and every cycle $\mathcal{C}$ based at p, the cycle evaluation map satisfies ${{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)=0$.

Proof. 

Consider a cycle $\mathcal{C}:v_0{v}_1\dots v_k$ at vertex p, assuming $L_{\alpha }$ is an inner derivation. Decompose this cycle into two segments $\mathcal{C}={\zeta }_1{\zeta }_2$, where ${\zeta }_1:v_0{v}_1\dots v_{k-1}$ and ${\zeta }_2:v_{k-1}{v}_k$. In this decomposition, both the inverse path ${\zeta }_1^{-1}$ and the path ${\zeta }_2$ connect the vertex $v_{k-1}$ to x. By Theorem 1, it follows that ${{\rm Y}}_{{\zeta }_1^{-1}}^{\alpha }(v_{k-1},p)={{\rm Y}}_{{\zeta }_2}^{\alpha }(v_{k-1},p)$. Consequently, according to Lemma 1, we have

$$\begin{array}{cc}\hfill {{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)& ={{\rm Y}}_{{\zeta }_1{\zeta }_2}^{\alpha }(p,p)\hfill \\ & ={{\rm Y}}_{{\zeta }_1}^{\alpha }(x,v_{k-1})+{{\rm Y}}_{{\zeta }_2}^{\alpha }(v_{k-1},p)\hfill \\ & ={{\rm Y}}_{{\zeta }_1}^{\alpha }(x,v_{k-1})+{{\rm Y}}_{{\zeta }_1^{-1}}^{\alpha }(v_{k-1},p)=0.\hfill \end{array}$$

Conversely, consider two distinct vertices p and q in $\mathcal{P}$, and let ${\mathcal{H}}_1:v_0{v}_1\dots v_k$ and ${\mathcal{H}}_2:w_0{w}_1\dots w_l$ be two distinct semipaths from p to q. We first analyze the scenario where these semipaths share no vertices other than their endpoints p and q. Assume that the closed walk ${\mathcal{H}}_1{\mathcal{H}}_2^{-1}:v_0{v}_1\dots v_k{w}_{l-1}\dots w_0$ wherein the vertices $v_0,v_1,v_2,\dots ,v_k,w_1,w_2,\dots ,w_{l-1}$ are all distinct. If ${\mathcal{H}}_1{\mathcal{H}}_2^{-1}$ does not constitute a cycle based at p, then by Definition 5 and Remark 1, it follows that $k+1\le 3$. On the other hand, $\mathcal{P}$ has no triangles, thus $k+1<3$ and so ${\mathcal{H}}_1{\mathcal{H}}_2^{-1}:pqp.$ Thus, ${\mathcal{H}}_1={\mathcal{H}}_2:pq$ and, in this case, ${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(p,q)={{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(p,q)$. However, if ${\mathcal{H}}_1{\mathcal{H}}_2^{-1}$ is a cycle, then by hypothesis, ${{\rm Y}}_{{\mathcal{H}}_1{\mathcal{H}}_2^{-1}}^{\alpha }(p,p)=0$. Consequently, ${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(p,q)={{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(p,q)$, and it then follows from Lemma 1.

It is necessary to examine the case when ${\mathcal{H}}_1\cap {\mathcal{H}}_2$ has vertices other than the extremities of ${\mathcal{H}}_1$. Assume that $v_{i_0},v_{i_1},\dots ,v_{i_n}$ denote pairwise distinct vertices of ${\mathcal{H}}_1$, where $n\ge 1$, $v_{i_0}=v_0$, however it is not requried that $i_1<i_2<\dots <i_{n-1}<i_n$.

By abusing notation, we will indicate the sum ${{\rm Y}}_{{\mathcal{H}}_1^{\prime }}^{\alpha }(v_i,v_{i_{j+1}})$ when we have ${\mathcal{H}}_1^{\prime }:v_{i_j}=v_r{v}_{r+1}\dots v_s=v_{i_{j+1}}$ as a semipath of ${\mathcal{H}}_1$ by ${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_i,v_{i_{j+1}})$. Even if $i_{j+1}<i_j$, specifically when ${\mathcal{H}}_1^{\prime }:v_{i_{j+1}}=v_r{v}_{r-1}\dots v_s=v_{i_j}$, we will denote ${{\rm Y}}_{{\mathcal{H}}_1^{\prime }}^{\alpha }(v_i,v_{i_{j+1}})$ by ${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_i,v_{i_{j+1}})$, focussing on the opposites in this case.

Claim ${\displaystyle \sum _{j=0}^{n-1}}{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_j},v_{i_j+1})={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_n})$. For $n=1$, it’s obvious. For $n=2$, we assume $i_1<i_2$ and proceed by induction on n, so we have

$${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_1})+{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_1},v_{i_2})={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_2}),$$

according to Lemma 1(b). Now, if $i_2<i_1$, then

$$\begin{array}{cc}\hfill {{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_1})+{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_1},v_{i_2})& ={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_2})+{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_2},v_{i_1})+{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_1},v_{i_2})\hfill \\ & ={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_2}),\hfill \end{array}$$

because ${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_1},v_{i_2})=-{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_2},v_{i_1})$.

Now, for $n>2$, the inductive hypothesis assumes the equality for $n-1$. We have

$$\sum _{j=0}^{n-1}{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_j},v_{i_{j+1}})=\sum _{j=0}^{n-2}{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_j},v_{i_{j+1}})+{{\rm Y}}_{\mathcal{H}}^{\alpha }(v_{i_{n-1}},v_{i_n}).$$

By induction hypothesis, ${\displaystyle \sum _{j=0}^{n-2}}{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_j},v_{i_{j+1}})={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_{n-1}})$ and, by applying the result established for the base case $n=2$,

$$\sum _{j=0}^{n-1}{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_j},v_{i_{j+1}})={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_{n-1}})+{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_{n-1}},v_{i_n})={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_0},v_{i_n}).$$

Let $V\left({\mathcal{H}}_1\right)\cap V\left({\mathcal{H}}_2\right)=\{v_{i_o}=w_{j_0},v_{i_1}=w_{j_1},\dots ,v_{i_t}=w_{j_t}\}$, with $v_{i_0}=v_0$, $v_{i_t}=v_k$ and $i_0<i_1<\dots <i_t$ (note that not necessarily $j_1<j_2<\dots <j_{t-1}$). Then

$${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_0,v_k)=\sum _{r=0}^{t-1}{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_{i_r},v_{i_{r+1}})=\sum _{r=0}^{t-1}{{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(w_{j_r},w_{j_{r+1}})={{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(w_0,w_l),$$

The second equality is justified by the property that the corresponding subsemipaths of ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ connecting $v_{i_r}=w_{j_r}$ to $v_{i_{r+1}}=w_{j_{r+1}}$ intersect only at their endpoints, while the first and final equalities are direct consequences of the aforementioned claim. □

One can apply Corollary 3 to demonstrate that proving an additive derivation is inner is more straightforward than using Theorem 1, particularly in the case of small partially ordered sets. This is exemplified by the partially ordered set ${\mathcal{P}}_3$, shown below. Specifically, ${\mathcal{P}}_3$ consists of six elements that are not vertices of any cycle.

                                            

Additionally, Corollary 3 need not be checked for every cycle of the partially ordered set, as will be shown subsequently. Let $\mathcal{C}:v_0{v}_1\dots v_k$ be a cycle represent in $v_0$. Define i such that $1\le i\le k-1$ and assume the semipaths ${\mathcal{H}}_1:v_0{v}_1\dots v_i$ and ${\mathcal{H}}_2:v_i{v}_{i+1}\dots v_k$. Consequently, ${\mathcal{H}}_2^{-1}{\mathcal{H}}_1^{-1}$ are likewise cycles in $v_0$, but ${\mathcal{H}}_2{\mathcal{H}}_1$ and ${\mathcal{H}}_1^{-1}{\mathcal{H}}_2^{-1}$ are cycles in $v_i$. All four cycles $\mathcal{C}={\mathcal{H}}_1{\mathcal{H}}_2,{\mathcal{H}}_2^{-1}{\mathcal{H}}_1^{-1}$, ${\mathcal{H}}_2{\mathcal{H}}_1$ and ${\mathcal{H}}_1^{-1}{\mathcal{H}}_2^{-1}$, have identical edges with $\mathcal{C}$. According to Lemma 1 (b) and (c), we have

$${{\rm Y}}_{{\mathcal{H}}_1{\mathcal{H}}_2}^{\alpha }(v_0,v_0)={{\rm Y}}_{{\mathcal{H}}_2{\mathcal{H}}_1}^{\alpha }(v_i,v_i)=-{{\rm Y}}_{{\mathcal{H}}_1^{-1}{\mathcal{H}}_2^{-1}}^{\alpha }(v_i,v_i)=-{{\rm Y}}_{{\mathcal{H}}_2^{-1}{\mathcal{H}}_1^{-1}}^{\alpha }(v_0,v_0)$$

(1)

Let $\xi$ represent the collection of all cycles in the structure $\mathcal{P}$. A binary relation ∼ is defined on $\xi$ according to the following: for ${\mathcal{C}}_1,{\mathcal{C}}_2\in \xi$,

$${\mathcal{C}}_1\sim {\mathcal{C}}_2\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if},A\left({\mathcal{C}}_1\right)=A\left({\mathcal{C}}_2\right).$$

(2)

where $A\left(\mathcal{C}\right)$ represents a certain characteristic of the cycle $\mathcal{C}$. The characteristic of a cycle $\mathcal{C}$, denoted $A\left(\mathcal{C}\right)$, is the sum of the weights of all edges in the cycle:

$$A\left(\mathcal{C}\right)=\sum _{e\in \mathcal{C}}w\left(e\right)$$

It is evident that ∼ constitutes an equivalence relation. The equivalence class of a cycle $\mathcal{C}\in \xi$ will be denoted by $\overline{\mathcal{C}}$. Now, suppose $\mathcal{C}$ is a cycle based at p and ${\mathcal{C}}^{\prime }$ is a cycle based at q, with the property that $\overline{\mathcal{C}}=\overline{{\mathcal{C}}^{\prime }}$. As a consequence of Equation (1), we derive the following relation:

$${{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)=0\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{{\rm Y}}_{{\mathcal{C}}^{\prime }}^{\alpha }(q,q)=0.$$

(3)

Theorem 2.

Let $\mathcal{P}$ be a locally finite partially ordered set, M be an R-bimodule, and $\alpha \in I(\mathcal{P},M)$ an additive element. Let $\{{\mathcal{C}}_1,{\mathcal{C}}_2,\dots ,{\mathcal{C}}_l\}$ represent a transversal of ξ for the equivalence relation defined in Lemma 1, where each ${\mathcal{C}}_i$ is a cycle corresponding to $p_i$ for $i=1,2,\dots ,l$. Then, the additive derivation $L_{\alpha }$ is inner if and only if

$${{\rm Y}}_{{\mathcal{C}}_i}^{\alpha }(p_i,p_i)=0\mathit{for}\mathit{all}i=1,2,\dots ,l.$$

Proof. 

This is an immediately consequence of Theorem 1 and Corollary 3. □

In order to make an additive derivation inner, we now attempt to weaken the requirements stated in Theorems 1 and 2.

Lemma 2.

Let M be an R-bimodule and $\alpha \in I(\mathcal{P},M)$ an additive element. Let $\mathcal{H}:v_0{v}_1\dots v_k$ be a semipath such that every vertices is comparable with $v_0$. Then

(a) $v_0<v_k$ (resp. $v_k<v_0$) $\Rightarrow v_0<v_i$ (resp. $v_i<v_0$), for all $i=1,2,\dots ,k-1$.

(b) The function ${{\rm Y}}_{\mathcal{H}}^{\alpha }(v_0,v_k)$ is given by:

$${{\rm Y}}_{\mathcal{H}}^{\alpha }(v_0,v_k)=\left\{\begin{array}{cc}\alpha (v_0,v_k)\hfill & \mathrm{if}{v}_0<v_k,\hfill \\ -\alpha (v_k,v_0)\hfill & \mathrm{if}{v}_k<v_0.\hfill \end{array}\right.$$

Proof. 

(a) This result corresponds precisely to part (a) of [12] (Lemma 11).

(b) Let $v_0<v_k$. By induction on k, we show that $v_0$ is the minimum of $\{v_0,v_1,\dots ,v_k\}$, which is given by condition (a). For $k=1$, ${{\rm Y}}_{\mathcal{H}}^{\alpha }(v_0,v_k)={\eta }_{\alpha }(v_0,v_1)=\alpha (v_0,v_1)$. Let $k>1$ and assume that the result is true for $k-1$. If we write $\mathcal{H}={\mathcal{H}}_1{\mathcal{H}}_2$, where ${\mathcal{H}}_1:v_0{v}_1\dots v_{k-1},{\mathcal{H}}_2:v_{k-1}{v}_k$, then

$${{\rm Y}}_{\mathcal{H}}^{\alpha }(v_0,v_k)={{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_0,v_{k-1})+{{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(v_{k-1},v_k)=\alpha (v_0,v_{k-1})+{{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(v_{k-1},v_k).$$

If $v_k$ covers $v_{k-1}$, then ${\eta }_{\alpha }(v_{k-1},v_k)=\alpha (v_{k-1},v_k)$. Thus,

$${{\rm Y}}_{\mathcal{H}}^{\alpha }(v_0,v_k)=\alpha (v_0,v_{k-1})+\alpha (v_{k-1},v_k)=\alpha (v_0,v_k).$$

However, if $v_{k-1}$ covers $v_k$, then ${\eta }_{\alpha }(v_{k-1},v_k)=-\alpha (v_k,v_{k-1})$; consequently,

$${{\rm Y}}_{\mathcal{H}}^{\alpha }(v_0,v_k)=\alpha (v_0,v_{k-1})-\alpha (v_k,v_{k-1})=\alpha (v_0,v_k),$$

since $\alpha (v_0,v_{k-1})=\alpha (v_0,v_k)+\alpha (v_k,v_{k-1})$. The case when $v_k<v_0$ is analogous. □

Definition 6.

Consider a cycle $\mathcal{C}$ within the structure $\mathcal{P}$. Let $v\in V\left(\mathcal{C}\right)$ and $p\in \mathcal{P}\setminus V\left(\mathcal{C}\right)$ be comparable, with $q<v$ or $v<q$. In both cases, a path (or counter-path) connects q to v.. Suppose we have a counter-path $u_0{u}_1\dots u_t$ with $u_0=q$ and $u_t=v$. This counter-path may intersect $\mathcal{C}$ at a vertex preceding v; that is, there might exist an index j with $0<j<t$ such that $u_j\in V\left(\mathcal{C}\right)$. Define $b:=min\{i\in \{1,2,\dots ,t\}:u_i\in V\left(\mathcal{C}\right)\}$. The resulting sequence $u_0{u}_1\dots u_{b-1}{u}_b$ forms a counter-path joining q to a vertex on $\mathcal{C}$, satisfying the condition that either $v<u_i$ holds for all $i=0,\dots ,b-1$ with $v\le u_b$, or $u_i<v$ holds for all $i=0,\dots ,b-1$ with $u_b\le v$. Such a counter-path is called a v-tail from q to v. (It is important to note that this v-tail may not necessarily reach v itself.) We adopt the convention that the tail originates at q and terminates at $u_b$. To illustrate, let $\mathcal{C}:v_1{v}_2{v}_3{v}_4{v}_1$. In the accompanying Hasse diagram, the $v_2$-tail from q to $v_2$ is given by the counter-path $q{v}_1$.

                                            

(See ([12], page 732) for a definition of v-tail.)

Lemma 3.

Let M be an R-bimodule and $\alpha \in I(\mathcal{P},M)$ an additive element. Consider a vertex $p\in \mathcal{P}$ and a cycle $\mathcal{C}$ contaning p. If there exists an element $q\in \mathcal{P}$ that is comparable to every vertex in $\mathcal{C}$, then ${{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)=0$.

Proof. 

Let p in $\mathcal{P}$, and suppose $\mathcal{C}:v_0{v}_1\dots v_k$ forms a cycle based at p as per the hypothesis. Assume $q\notin V\left(\mathcal{C}\right)$. Since p and q are comparable, let $\mathcal{T}:u_0{u}_1\dots u_b$ be an p-tail connecting q to p, and suppose $u_b=v_i$ for some index i.

Now, consider an index i satisfying $0<i<k$, and analyze the two semipaths ${\mathcal{H}}_1:v_i{v}_{i+1}\dots v_k$ and ${\mathcal{H}}_2:v_i{v}_{i-1}\dots v_0$ that both connect $v_i$ to p. It follows that the concatenated paths $\mathcal{T}{\mathcal{H}}_1$ and $\mathcal{T}{\mathcal{H}}_2$ represent two distinct semipaths from q to p. Furthermore, every vertex along these composite semipaths is comparable with the element q.

By applying the result of the previous lemma in conjunction with Lemma 1(b), we obtain the following:

$${{\rm Y}}_{\mathcal{T}}^{\alpha }(u_0,u_b)+{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_i,v_k)={{\rm Y}}_{\mathcal{T}}^{\alpha }(u_0,u_b)+{{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(v_i,v_0).$$

Canceling ${{\rm Y}}_{\mathcal{T}}^{\alpha }(u_0,u_b)$ from both sides, we obtain

$${{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_i,v_k)-{{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(v_i,v_0)=0.$$

Since $-{{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(v_i,v_0)={{\rm Y}}_{{\mathcal{H}}_2^{-1}}^{\alpha }(v_0,v_i)$, we have

$${{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)={{\rm Y}}_{{\mathcal{H}}_2^{-1}{\mathcal{H}}_1}^{\alpha }(v_0,v_k)=-{{\rm Y}}_{{\mathcal{H}}_2}^{\alpha }(v_i,v_0)+{{\rm Y}}_{{\mathcal{H}}_1}^{\alpha }(v_i,v_k)=0.$$

In the special case where $i=0$ or $i=k$, we examine the semipaths ${\mathcal{H}}_1^{\prime }:v_0{v}_1\dots v_{k-1}$ and ${\mathcal{H}}_2^{\prime }:v_k{v}_{k-1}$. In this case, $\mathcal{T}{\mathcal{H}}_1^{\prime }$ and $\mathcal{T}{\mathcal{H}}_2^{\prime }$ are distinct semipaths from q to $v_{k-1}$ such that every vertex on them is comparable to q. Likewise, we obtain ${{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)=0$. If $q\in V\left(\mathcal{C}\right)$ (say $q=v_i$), the previous arguments hold without $\mathcal{T}$.

Therefore, in all cases, we conclude that ${{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)=0$. □

Corollary 4.

Let M be an R-bimodule and $\mathcal{P}$ a partially ordered set such that for every cycle $\mathcal{C}$ contained in $\mathcal{P}$, there exists an element $q\in \mathcal{P}$ that is comparable to each vertex of $\mathcal{C}$. Then the incidence module $I(\mathcal{P},M)$ possesses the property that all of its derivations are inner.

Proof. 

Let $\alpha \in I(\mathcal{P},M)$ be additive. Consider any vertex p in $\mathcal{P}$ and an arbitrary cycle $\mathcal{C}$ based at p. By Lemma 3, it follows immediately that ${{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)=0$. This condition, by virtue of Corollary 3, implies that the associated additive derivation $L_{\alpha }$ is inner. Since $\alpha$ was chosen arbitrarily, and invoking the general characterization provided in [2] (Theorem 7.1.4), we conclude that all derivations of $I(\mathcal{P},M)$ are inner. □

Example 3.

Suppose that the partially ordered set ${\mathcal{P}}_4$ below. According to Corollary 4, all derivations of $I({\mathcal{P}}_4,M)$ are inner.

                                            

Example 4.

Suppose that the partially ordered set ${\mathcal{P}}_5$ is as shown below. Although no single element exists within ${\mathcal{P}}_5$ that is comparable to every vertex comprising the cycle ${\mathcal{C}}_1:v_0{v}_1\dots v_9{v}_0$, the incidence module $I(P_5,M)$ nevertheless possesses the property that all its derivations are inner. In fact, let α be an additive element of $I(P_5,M)$, and assume that the transversal $\{{\mathcal{C}}_1,{\mathcal{C}}_2,{\mathcal{C}}_3\}$ of ξ with respect to the equivalence relation defined in (2), where the representative cycles are given by ${\mathcal{C}}_2:v_0{v}_1\dots v_4{v}_9{v}_0$ and ${\mathcal{C}}_3:v_4{v}_5\dots v_9{v}_4$.

                                            

According to Lemma 3, ${{\rm Y}}_{{\mathcal{C}}_2}^{\alpha }(v_0,v_0)={{\rm Y}}_{{\mathcal{C}}_3}^{\alpha }(v_4,v_4)=0$. It follows that

$$\begin{array}{cc}\hfill {{\rm Y}}_{{\mathcal{C}}_1}^{\alpha }(v_0,v_0)& =\sum _{i=0}^8{\eta }_{\alpha }(v_i,v_{i+1})+{\eta }_{\alpha }(v_9,v_0)\hfill \\ & =\sum _{i=0}^8{\eta }_{\alpha }(v_i,v_{i+1})+{\eta }_{\alpha }(v_9,v_0)+\alpha (v_4,v_9)-\alpha (v_4,v_9)\hfill \\ & =\sum _{i=0}^8{\eta }_{\alpha }(v_i,v_{i+1})+{\eta }_{\alpha }(v_9,v_0)+{\eta }_{\alpha }(v_4,v_9)+{\eta }_{\alpha }(v_9,v_4)\hfill \\ & =\sum _{i=0}^3{\eta }_{\alpha }(v_i,v_{i+1})+{\eta }_{\alpha }(v_4,v_9)+{\eta }_{\alpha }(v_9,v_0)\hfill \\ & +\sum _{i=4}^8{\eta }_{\alpha }(v_i,v_{i+1})+{\eta }_{\alpha }(v_9,v_4)\hfill \\ & ={{\rm Y}}_{{\mathcal{C}}_2}^{\alpha }(v_0,v_0)+{{\rm Y}}_{{\mathcal{C}}_3}^{\alpha }(v_4,v_4)=0.\hfill \end{array}$$

Thus, by Theorem 2, the derivation $L_{\alpha }$ is inner.

Definition 7.

A subset $\mathcal{Q}$ of a partially ordered set $\mathcal{P}$ is an antichain if no two distinct elements of $\mathcal{Q}$ are comparable. That is, for any $p,q$ in $\mathcal{Q}$ where $p\ne q$, we have neither $p\le q$ nor $q\le p$. An antichain $\mathcal{Q}$ is an all-comparable subset if every element of $\mathcal{P}$ is comparable to some element of $\mathcal{Q}$. In other words, for each $p\in \mathcal{P}$, there exists $q\in \mathcal{Q}$ such that $p\le q$ or $q\le p$. For finite $\mathcal{P}$, all-comparable subsets exist. Examples include $\mathcal{P}$’s maximal and minimal elements. By Proposition 1 shows that if an all-comparable subset $\mathcal{Q}$ consists of a single element (i.e., $\mathcal{Q}=\left\{q\right\}$), then all additive derivations on $I(\mathcal{P},M)$ are inner.

Theorem 3.

Let M be an R-bimodule and α be an additive element of the incidence module $I(\mathcal{P},M)$. Assume there exists an all-comparable subset $\mathcal{Q}\subseteq \mathcal{P}$ such that, for any distinct elements $q_1,q_2\in \mathcal{Q}$ and for any ${\xi }_1$ and ${\xi }_2$ semipaths joining $q_1$ to $q_2$, we have ${{\rm Y}}_{{\xi }_1}^{\alpha }(q_1,q_2)={{\rm Y}}_{{\xi }_2}^{\alpha }(q_1,q_2)$. Therefore, $L_{\alpha }$ is an inner additive derivation.

Proof. 

Let $\mathcal{C}:v_0{v}_1\dots v_k$ denote a cycle in $\mathcal{P}$. By Corollary 4, it suffices to prove that ${\Gamma }_{\mathcal{C}}^{\alpha }(p,p)=0$. The preceding lemma implies that there exists an element of $\mathcal{C}$ are comparable with the same $q\in \mathcal{Q}$. Therefore, no $q\in \mathcal{Q}$ is comparable to every elements of $\mathcal{C}$. Consequently, there exists successive vertices $v_i,v_{i+1}\in V\left(\mathcal{C}\right)$ and elements $q_1,q_2\in \mathcal{Q}$ such that $v_i$ is comparable to $q_1$ and $v_{i+1}$ is comparable to $q_2$ and not comparable to $q_1$.

Now, suppose that $q_1,q_2\notin V\left(\mathcal{C}\right)$, and consider ${\mathcal{T}}_1$ a $v_i$-tail joining $q_1$ to $v_i$ and ${\mathcal{T}}_2$ a $v_{i+1}$-tail joining $q_2$ to $v_{i+1}$. There are now two situations to take into account: when $V\left({\mathcal{T}}_1\right)\cap V\left({\mathcal{T}}_2\right)\ne \varnothing$ and when $V\left({\mathcal{T}}_1\right)\cap V\left({\mathcal{T}}_2\right)=\varnothing$. We will first show that Case 1 leads to a contradiction, and is therefore impossible.

Case 1: When $V\left({\mathcal{T}}_1\right)\cap V\left({\mathcal{T}}_2\right)\ne \varnothing$, we assert that either $v_i<q_1$ and $q_2<v_{i+1}$ or $v_{i+1}<q_2$ and $q_1<v_i$.

Indeed, assume that $v_i<q_1$ and $v_{i+1}<q_2$, and consider $u\in V\left({\mathcal{T}}_1\right)\cap V\left({\mathcal{T}}_2\right)$. Thus, $v_i\le u\le q_1$ and $v_{i+1}\le u\le q_2$, which implies that $v_{i+1}\le u\le q_1$. This contradicts the assumption that $v_{i+1}$ is not comparable with $q_1$.

Analogously, it can be shown that neither $q_1<v_i$ nor $q_2<v_{i+1}$ occurs. We will assume, without loss of generality, that $v_i<q_1$ and $q_2<v_{i+1}$. If $u\in V\left({\mathcal{T}}_1\right)\cap V\left({\mathcal{T}}_2\right)$, then $v_1\le u\le q_1$ and $q_2\le u\le v_{i+1}$. Hence, we have $q_2\le u\le q_1$. However, this is impossible because $q_1$ and $q_2$ are distinct elements of the antichain $\mathcal{Q}$, and are therefore incomparable.

Case 2: When $V\left({\mathcal{T}}_1\right)\cap V\left({\mathcal{T}}_2\right)=\varnothing$. We now examine the case where $V\left({\mathcal{T}}_1\right)\cap V\left({\mathcal{T}}_2\right)=\varnothing$. Assume the tail ${\mathcal{T}}_1$ ends at $v_j\in V\left(\mathcal{C}\right)$ and ${\mathcal{T}}_2$ at $v_m\in V\left(\mathcal{C}\right)$. If $0<j<m<k$, define the paths ${\mathcal{H}}_1:v_j{v}_{j-1}\dots v_0$, ${\mathcal{H}}_2:v_k{v}_{k-1}\dots v_m$, and ${\mathcal{H}}_3:v_j{v}_{j+1}\dots v_m$. Consequently, ${\mathcal{T}}_1{\mathcal{H}}_1{\mathcal{H}}_2{\mathcal{T}}_2^{-1}$ and ${\mathcal{T}}_1{\mathcal{H}}_3{\mathcal{T}}_2^{-1}$ constitute two semipaths joining $q_1$ to $q_2$. According to the hypothesis,

$${{\rm Y}}_{{\mathcal{T}}_1{\mathcal{H}}_1{\mathcal{H}}_2{\mathcal{T}}_2^{-1}}^{\alpha }(q_1,q_2)={{\rm Y}}_{{\mathcal{T}}_1{\mathcal{H}}_3{\mathcal{T}}_2^{-1}}^{\alpha }(q_1,q_2).$$

From Lemma 1, it follows that

$${{\rm Y}}_{{\mathcal{H}}_1{\mathcal{H}}_2}^{\alpha }(v_j,v_m)={{\rm Y}}_{{\mathcal{H}}_3}^{\alpha }(v_j,v_m).$$

Hence,

$${{\rm Y}}_{\mathcal{C}}^{\alpha }(p,p)={{\rm Y}}_{\mathcal{C}}^{\alpha }(v_0,v_k)={{\rm Y}}_{{\mathcal{H}}_1^{-1}{\mathcal{H}}_3{\mathcal{H}}_2^{-1}}^{\alpha }(v_0,v_k)=0.$$

In the special cases where $0=j<m<k$, the argument omits ${\mathcal{H}}_1$; for $0<j<m=k$, it omits ${\mathcal{H}}_2$. The case $0=j<m=k$ is impossible, as $V\left({\mathcal{T}}_1\right)\cap V\left({\mathcal{T}}_2\right)=\varnothing$.

Finally, by considering semipaths without the tails ${\mathcal{T}}_1$ or ${\mathcal{T}}_2$, the reasoning holds if $q_1\in V\left(\mathcal{C}\right)$ or $q_2\in V\left(\mathcal{C}\right)$.

We have thus shown that ${\Gamma }_{\mathcal{C}}^{\alpha }(p,p)=0$ in all cases. By Corollary 3, it follows that the additive derivation $L_{\alpha }$ is inner, which completes the proof. □

Example 5.

The incidence module of the partially ordered set ${\mathcal{P}}_6$ below does not satisfy the condition that all additive derivations are inner. However, by applying our previous result, we can construct an additive derivation that is inner, as demonstrated below.

                                            

In this example, $\mathcal{Q}=\{q_1,q_2\}$ is the all comparable subset of ${\mathcal{P}}_6$ that satisfies Theorem 3. Although there are seven semipaths joining $q_1$ to $q_2$, the direct weights sum for each of these seven semipaths is the same.

3.2. P Locally Finite

In this subsection, we address the situation in which $\mathcal{P}$ denotes an arbitrary locally finite poset, meaning that every interval $[p,q]$ in $\mathcal{P}$ is finite. Consider the decomposition $\mathcal{P}={\bigcup }_{j\in J}{\mathcal{P}}_j$ of $\mathcal{P}$ into connected components, with each ${\mathcal{P}}_j$ being a maximal connected subset. It is easy to show that the map $\delta :I(\mathcal{P},M)\to {\displaystyle \prod _{j\in J}}I({\mathcal{P}}_j,M)$, defined by $\delta {\left(f\right)}_j{=f|}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}$ for every $j\in J$, is an isomorphism of modules. Intuitively, $\delta$ restricts a function $f\in I(\mathcal{P},M)$ to each connected component ${\mathcal{P}}_j$. Since the connected components ${\left\{{\mathcal{P}}_j\right\}}_{j\in J}$ form a partition of $\mathcal{P}$, this map $\delta$ is well-defined and making it an isomorphism of module.

Lemma 4.

Let $\mathcal{P}={\displaystyle \bigcup _{j\in J}}{\mathcal{P}}_j$ be the decomposition of $\mathcal{P}$ into its connected components, and let $\alpha \in I(\mathcal{P},M)$, where M is an R-bimodule. Then, α is potential (additive) if and only if ${\alpha |}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}$ is a potential (additive) element of $I({\mathcal{P}}_j,M)$ for all $j\in J$.

Proof. 

Suppose $\alpha$ is potential. Then, there exists a function $g:\mathcal{P}\to M$ such that $\alpha (p,q)=g\left(q\right)-g\left(p\right)$ for all $p\le q\in \mathcal{P}$. For each $j\in J$, let $g_j{=g|}_{{\mathcal{P}}_j}$ be the restriction of g to ${\mathcal{P}}_j$. Since ${\mathcal{P}}_j$ is connected, for all $p\le q\in {\mathcal{P}}_j$, we have

$${\alpha |}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}(p,q)=\alpha (p,q)=g\left(q\right)-g\left(p\right)=g_j\left(q\right)-g_j\left(p\right).$$

Thus, For each $j\in J$, ${\alpha |}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}$ is a potential element of $I({\mathcal{P}}_j,M)$.

Conversely, suppose that ${\alpha |}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}$ is a potential element of $I({\mathcal{P}}_j,M)$ for every j in J. Thus, for each $j\in J$, there exists a function $f_j:{\mathcal{P}}_j\to M$ such that

$${\alpha |}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}(p,q)=f_j\left(q\right)-f_j\left(p\right),\mathrm{for}\mathrm{all}p\le q\mathrm{in}{\mathcal{P}}_j.$$

Define $f:\mathcal{P}\to M$ by $f\left(p\right)=f_j\left(p\right)$ if $p\in {\mathcal{P}}_j$. This is well-defined since ${\left\{{\mathcal{P}}_j\right\}}_{j\in J}$ forms a disjoint partition of $\mathcal{P}$. Therefore, if $p,q$ in $\mathcal{P}$ satisfy $p\le q$, then both p and q belong to the same connected component ${\mathcal{P}}_j$ of $\mathcal{P}$ for some $j\in J$. Hence,

$${\alpha (p,q)=\alpha |}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}(p,q)=f_j\left(q\right)-f_j\left(p\right)=f\left(q\right)-f\left(p\right).$$

Thus, $\alpha$ is a potential element of $I(\mathcal{P},M)$.

It follows directly that $\alpha$ is additive if and only if ${\alpha |}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}$ is additive for every $j\in J$. To see this, note that if $p\le q\le z$ in $\mathcal{P}$, then p, q, and z must all lie in the same connected component ${\mathcal{P}}_j$. Hence, the additivity of $\alpha$ on each ${\mathcal{P}}_j$ implies the additivity of $\alpha$ on $\mathcal{P}$. □

Combining the preceding lemma with ([2], Proposition 7.1.6), we obtain the following result.

Theorem 4.

Let $\mathcal{P}={\bigcup }_{j\in J}{\mathcal{P}}_j$ be the decomposition of $\mathcal{P}$ into connected components, and let M be an R-bimodule. The additive derivation $L_{\alpha }$ of $I(\mathcal{P},M)$ is inner if and only if, for each $j\in J$, $L_{\alpha }{|}_{{\mathcal{P}}_j\times {\mathcal{P}}_j}$ is inner.

Corollary 5.

Let $\mathcal{P}={\bigcup }_{j\in J}{\mathcal{P}}_j$ be the decomposition of $\mathcal{P}$ into connected components, and M an R-bimodule. If each ${\mathcal{P}}_j$ has an element comparable to all others in ${\mathcal{P}}_j$, then every derivation of $I(\mathcal{P},M)$ is inner.

Proof. 

This follows from Theorem 4 and Proposition 1. □

Combining the preceding lemma with [2] (Proposition 7.1.6) yields the following result.

Example 6.

The poset $\mathcal{P}$ below has only inner derivations on $I(\mathcal{P},M)$, by Theorem 4, Corollary 2, and Example 4.

                              

Example 7.

Let $\mathcal{P}={\displaystyle \bigcup _{j\in J}}{\mathcal{P}}_j$ be the decomposition of the locally finite poset $\mathcal{P}$ into its connected components as shown below.

                                       

As a result of Theorem 4 and Corollary 4, it follows that all derivations of $I(\mathcal{P},M)$ are inner.