© Wuhan University 2024

0 Introduction

Weighted homology is a generalization of homology theory. In general, one of the main motivations of weighted homology is distinguishing different elements in the data set, which was initially defined on a simplicial complex where each simplex is assigned a weight value^[1]. Later, Horak and Jost^[2] first developed a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. Ren et al^[3] developed the theory of weighted persistent homology of weighted simplicial complex, extended the homology of weighted simplicial complexes to the embedded homology of weighted hypergraphs^[4], and proved that weighted persistent homology can tell apart filtrations that ordinary persistent homology does not distinguish^[5].

Different from simplicial complexes and hypergraphs, the directed edges in a digraph may have two directions, and there may be directed loops in a digraph. Therefore, in this sense, the topological study of digraphs has more general significance in mathematics. In 2020, Wang et al^[6] studied the persistent homology of vertex-weighted digraphs. They proved the persistent weighted path homology with coefficients in a filed is independent of the choices of weights.

According to Ref. [7], a simplicial complex can be viewed as a Δ-set, and a hypergraph can be represented as a graded subset of Δ-sets. Moreover, the set of allowed elementary paths on a digraph can be expressed as a graded subset of Δ-sets. Hence, our motivation for this paper is to define the weighted embedded homology of graded sets of Δ-sets which are called as super-hypergraphs^[8], study the structure of the torsion part of weighted homology groups with coefficients in $\mathbb{Z}$, and consider the relationship between the torsion parts of weighted embedded homology groups with integer coefficients of super-hypergraphs with different weights. The framework of the paper is as follows. Firstly, we respectively review the definition of embedded homology on super-hypergraphs in Section 1 and define the weighted embedded homology on super-hypergraphs in Section 2. Secondly, in Section 3, we give a quasi-partial order and a pseudo-metric on the set of all weights on a finite set. Notably, Subsection 3.2 provides the relationship between the torsion parts of weighted embedded homology with integer coefficients of super-hypergraphs under certain weights in Theorem 1 and Theorem 2.

1 Preliminaries

Let $R$ be a communicative ring with a unit. Let $X=\{X_n{\}}_{n\ge \mathrm{0}}$ be a $\mathrm{\Delta }$-set with face maps

$d_i:X_n\to X_{n-\mathrm{1}},\mathrm{0}\le i\le n,$(1)

$(x_{\mathrm{0}}{x}_{\mathrm{1}}\cdots x_n)\mapsto x_{\mathrm{0}}\cdots {\widehat{x}}_i\cdots x_n,x_{\mathrm{0}}{x}_{\mathrm{1}}\cdots x_n\in X_n,$

such that

$d_i{d}_j=d_j{d}_{i+\mathrm{1}}\mathrm{f}\mathrm{o}\mathrm{r}i\ge j.$(2)

A graded subset of $X$ is a sequence of sets $U=\{U_n{\}}_{n\ge \mathrm{0}}$ such that each $U_n$ is a subset of $X_n$. By Ref. [8, Definition 2.9], a super-hypergraph is a pair $(U,X)$, where $X$ is a $\mathrm{\Delta }$-set and $U$ is a graded subset of $X$. By Refs. [7-9], the infimum chain complex $\mathrm{I}\mathrm{n}{\mathrm{f}}_{\mathrm{*}}(U,X)$

$\mathrm{I}\mathrm{n}{\mathrm{f}}_{\mathrm{*}}(U,X)=R(U_n)\bigcap {\partial }_n^{-\mathrm{1}}R(U_{n-\mathrm{1}}),$

and the supremum chain complex $\mathrm{S}\mathrm{u}{\mathrm{p}}_{\mathrm{*}}(U,X)$

$\mathrm{S}\mathrm{u}{\mathrm{p}}_{\mathrm{*}}(U,X)=R(U_n)+{\partial }_{n+\mathrm{1}}R(U_{n+\mathrm{1}}),$

induce the same homology groups, which is called the $n$-th embedded homology on super-hypergraphs with coefficient in $R$. That is,

$H_n(U,X;R)=H_n(\mathrm{I}\mathrm{n}{\mathrm{f}}_{\mathrm{*}}(U,X))\cong H_n(\mathrm{S}\mathrm{u}{\mathrm{p}}_{\mathrm{*}}(U,X)).$

Let $X$ and $Y$ be $\mathrm{\Delta }$-sets. A $\mathrm{\Delta }$-map $f:X\to Y$ is a sequence of functions $f:X_n\to Y_n,n\ge \mathrm{0}$, such that $f\circ d_i=d_i\circ f$ for each $\mathrm{0}\le i\le n$. Suppose $X,Y$ are $\mathrm{\Delta }$-sets and $f:X\to Y$ is a $\mathrm{\Delta }$-map. If $U,V$ are graded subsets of $X,Y$ respectively such that $f(U)\subseteq V$, $f:(U,X)\to (V,Y)$ is called a morphism of super-hypergraphs. Then, by Ref. [8, Proposition 2.11], $f$ induces chain maps

$\begin{array}{l}{f}_{\#}^{\mathrm{I}\mathrm{n}\mathrm{f}}:\mathrm{I}\mathrm{n}{\mathrm{f}}_{\mathrm{*}}(U,X)\to \mathrm{I}\mathrm{n}{\mathrm{f}}_{\mathrm{*}}(V,Y),\\ f_{\#}^{\mathrm{S}\mathrm{u}\mathrm{p}}:\mathrm{S}\mathrm{u}{\mathrm{p}}_{\mathrm{*}}(U,X)\to \mathrm{S}\mathrm{u}{\mathrm{p}}_{\mathrm{*}}(V,Y).\end{array}$

The chain maps $f_{\#}^{\mathrm{I}\mathrm{n}\mathrm{f}}$ and $f_{\#}^{\mathrm{S}\mathrm{u}\mathrm{p}}$ induce homomorphisms $f_{\#}^{\mathrm{I}\mathrm{n}\mathrm{f}}$ and $f_{\#}^{\mathrm{S}\mathrm{u}\mathrm{p}}$ of homology groups, respectively, such that the following diagram commutes

Here l is the canonical inclusion of $\mathrm{I}\mathrm{n}{\mathrm{f}}_{\mathrm{*}}(U,X)$ into $\mathrm{S}\mathrm{u}{\mathrm{p}}_{\mathrm{*}}(U,X)$, and $l^{\text{'}}$ is the canonical inclusion of $\mathrm{I}\mathrm{n}{\mathrm{f}}_{\mathrm{*}}(V,Y)$ into $\mathrm{S}\mathrm{u}{\mathrm{p}}_{\mathrm{*}}(V,Y)$. Letting $f_{\mathrm{*}}=f_{\mathrm{*}}^{\mathrm{I}\mathrm{n}\mathrm{f}}$ (or alternatively, letting $f_{\mathrm{*}}=f_{\mathrm{*}}^{\mathrm{S}\mathrm{u}\mathrm{p}}$), we obtain a homomorphism of the embedded homology groups

$f_{\mathrm{*}}:H_{\mathrm{*}}(U;R)\to H_{\mathrm{*}}(V;R).$

The following example gives a homomorphism between embedded homology groups with coefficients in $\mathbb{Z}$.

Example 1 Let $R=$$\mathbb{Z}$. Consider the following two $\mathbb{Z}$-moudules(homology groups)

$\begin{array}{l}M=\mathbb{Z}\oplus \mathbb{Z}/\mathrm{2}\oplus \mathbb{Z}/\mathrm{3}\oplus \mathbb{Z}/\mathrm{9}=<x>\oplus <y(\mathrm{2,1})>\oplus <y(\mathrm{3,1})>\\ \oplus <y(\mathrm{3,2})>\end{array}$

$M^{\text{'}}=\mathbb{Z}\oplus \mathbb{Z}/\mathrm{9}\oplus \mathbb{Z}/\mathrm{4}=<x^{\text{'}}>\oplus <y^{\text{'}}(\mathrm{3,2})>\oplus <y^{\text{'}}(\mathrm{2,2})>,$

Let

$\begin{array}{c}f(x)=\lambda x^{\text{'}}+\mu y^{\text{'}}(\mathrm{3,2})+\nu y^{\text{'}}(\mathrm{2,2}),\lambda ,\mu ,\nu \in \mathbb{Z},\\ f(y(\mathrm{2,1}))=\mathrm{2}{y}^{\text{'}}(\mathrm{2,2}),\\ f(y(\mathrm{3,1}))=\mathrm{3}{y}^{\text{'}}(\mathrm{3,2}),\\ f(y(\mathrm{3,2}))=\kappa y^{\text{'}}(\mathrm{3,2}),\kappa =\mathrm{0}\begin{array}{c}\mathrm{o}\mathrm{r}\mathrm{3}\end{array},\end{array}$

then f is a homomorphism between $M$ and $M^{\text{'}}$.

2 Weighted Embedded Homology of Super-Hypergraphs

Let $X$ be a $\mathrm{\Delta }$-set. Let $w:X_{\mathrm{0}}\to \mathbb{Z}$ be an integer-valued weight. Let $p_i:X_n\to X_{\mathrm{0}}$ be the projection sending $x_{\mathrm{0}}{x}_{\mathrm{1}}\cdots x_n$ to the $i$-th point $x_i$.

We define the weighted boundary map as:

${\partial }_n^w={\displaystyle \sum _{i=\mathrm{0}}^n}{(-\mathrm{1})}^i(w\circ p_i)\cdot d_i,$

where $d_i$ are the face maps as defined in Eq. (1). Then,

${\partial }_n^w:\mathbb{Z}<X_n>\to \mathbb{Z}<X_{n-\mathrm{1}}>$

is a homomorphism of free $\mathbb{Z}$-modules given by

${\partial }_n^w(x_{\mathrm{0}}x)={\displaystyle \sum _{i=\mathrm{0}}^n}{(-\mathrm{1})}^{i}w(x_i)x_{\mathrm{0}}\cdots {\widehat{x}}_i\cdots x_n$

for any $x_{\mathrm{0}}{x}_{\mathrm{1}}\cdots x_n\in X_n$, and extends linearly over $\mathbb{Z}$. Moreover,

${\partial }_{n-\mathrm{1}}^w\circ {\partial }_n^w=\mathrm{0},$

for any $n\ge \mathrm{1}$. Hence, $\mathbb{Z}$-modules

$C_n=\mathbb{Z}<X_n>,$

with a weighted boundary map

${\partial }_n^w:C_n\to C_{n-\mathrm{1}}$

is a chain complex.

Let $U$ be a graded subset of $X$. Let

$\mathrm{I}\mathrm{n}{\mathrm{f}}_n^w(U,X)=R(U_n)\bigcap ({\partial }_n^w{)}^{-\mathrm{1}}R(U_{n-\mathrm{1}}),$(3)

$\mathrm{S}\mathrm{u}{\mathrm{p}}_n^w(U,X)=R(U_n)+{\partial }_{n+\mathrm{1}}^{w}R(U_{n+\mathrm{1}}).$(4)

Then Eq. (3) and Eq. (4) are both sub-chain complexes of $\{C_n,{\partial }_n^w\}$, called the weighted infimum chain complex of $U$ in $X$ and the weighted supremum chain complex of $U$ in $X$ , respectively. By Ref. [9], the canonical inclusion

$\tau :\mathrm{I}\mathrm{n}{\mathrm{f}}_n^w(U,X)\to \mathrm{S}\mathrm{u}{\mathrm{p}}_n^w(U,X)$

induces an isomorphism in homology;

${\tau }_{\mathrm{*}}:H_n(\mathrm{I}\mathrm{n}{\mathrm{f}}_n^w(U,X))\to H_n(\mathrm{S}\mathrm{u}{\mathrm{p}}_n^w(U,X)),$

which is called the weighted embedded homology of $(U,X)$, denoted as $H_n^w(U,X;R).$

For each $n\ge \mathrm{0}$, let

$W_n^w=\{x\in C_n|\exists \lambda \in \mathbb{Z},\lambda \ne \mathrm{0}\begin{array}{c}\mathrm{a}\mathrm{n}\mathrm{d}\begin{array}{c}\exists y\in C_{\mathrm{n}+\mathrm{1}}\begin{array}{c}\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\begin{array}{c}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\begin{array}{c}\lambda x={\partial }_{n+\mathrm{1}}^{w}y\end{array}\end{array}\end{array}\end{array}\end{array}\},$

where $\{C_n,{\partial }_n^w{\}}_{n\ge \mathrm{0}}$ is a chain complex. Then by Ref. [1, Theorem 11.4], for each $n\ge \mathrm{0}$, there exists $e_{\mathrm{1}}^w,e_{\mathrm{2}}^w,\cdots ,e_{l(w)}^w\in C_n$ linearly independent on $\mathbb{Z}$, and there exists $e^{\text{'}}{}_{\mathrm{1}}^w,e^{\text{'}}{}_{\mathrm{2}}^w,\cdots ,e^{\text{'}}{}_{l(w)}^w\in W_{n-\mathrm{1}}^w$ linearly-independent on $\mathbb{Z}$, such that;

1)$U_n^w=<e_{\mathrm{1}}^w,e_{\mathrm{2}}^w,\cdots ,e_{l(w)}^w>,$

2) $W_{n-\mathrm{1}}^w=<e^{\text{'}}{}_{\mathrm{1}}^w,e^{\text{'}}{}_{\mathrm{2}}^w,\cdots ,e^{\text{'}}{}_{l(w)}^w>,$

3) ${\partial }_n^w{|}_{U_n^w}:U_n^w\to W_{n-\mathrm{1}}^w,$

4) ${\partial }_n^w\left(\begin{array}{c}{e}_{\mathrm{1}}^w\\ ⋮\\ e_{l(w)}^w\end{array}\right)=\left(\begin{array}{ccc}{b}_{\mathrm{1}}^w& & \\ & \ddots & \\ & & b_{l(w)}^w\end{array}\right)\left(\begin{array}{c}{e}^{\text{'}}{}_{\mathrm{1}}^w\\ ⋮\\ e^{\text{'}}{}_{l(w)}^w\end{array}\right),$

where $b_i^w\ge \mathrm{1}$ and $b_{\mathrm{1}}^w|b_{\mathrm{2}}^w|\cdots |b_{l(w)}^w$. Note that $l(w),e_i^w,e^{\text{'}}{}_i^w,b_i^w,\mathrm{1}\le i\le l(w)$ are all depending on the weight $w:X_{\mathrm{0}}\to \mathbb{Z}$.

The torsion part of the $n$-homology group can be expressed as

$\mathrm{T}\mathrm{o}\mathrm{r}(H_n(\{C_k,{\partial }_k^w{\}}_{k\ge \mathrm{0}}))=\mathbb{Z}/b_{\mathrm{1}}^w\oplus \cdots \oplus \mathbb{Z}/b_{l(w)}^w.$(5)

By substituting $\{C_k,{\partial }_k^w{\}}_{k\ge \mathrm{0}}$ with $\mathrm{I}\mathrm{n}{\mathrm{f}}_{\mathrm{*}}^w(U,X)$ or $\mathrm{S}\mathrm{u}{\mathrm{p}}_{\mathrm{*}}^w(U,X)$ in Eq. (5), we have the structure of the torsion part of $H_n^w(U,X;R)$.

By Ref. [7], the path homology of digraphs can be seen as the embedded homology on super-hypergraphs. Hence, we take the path homology of digraphs as an example to illustrate that the weighted embedded homology is generally different from the unweighted embedded homology. The readers may refer to Refs. [10-13] for details on path homology of digraphs.

Example 2 As shown in Fig. 1, let $G$ be a cycle with vertex set $V=\{v_{\mathrm{0}},v_{\mathrm{1}},v_{\mathrm{2}}\}$ and directed edge set:

$E=\{v_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}\}.$

Fig. 1 Example 2

Let $w:V\to \mathbb{Z}$ be a weight on $G$ with $w(v_{\mathrm{0}})=w_{\mathrm{0}},w(v_{\mathrm{1}})=w_{\mathrm{1}}$ and $w(v_{\mathrm{2}})=w_{\mathrm{2}}$. Then,

$\begin{array}{l}{\mathrm{\Lambda }}_{\mathrm{0}}(V)=\mathbb{Z}\{v_{\mathrm{0}},v_{\mathrm{1}},v_{\mathrm{2}}\},\\ {\mathrm{\Lambda }}_{\mathrm{1}}(V)=\mathbb{Z}\{v_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{1}}{v}_{\mathrm{0}},v_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{1}},v_{\mathrm{0}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}\},\\ {\mathrm{\Lambda }}_{\mathrm{2}}(V)=\mathbb{Z}\{v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{0}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{1}},v_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{1}},\\ v_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{2}},v_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{0}}\},\\ {\mathrm{\Lambda }}_{\mathrm{3}}(V)=\mathbb{Z}\{v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{1}},v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{2}},v_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},\\ v_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{2}},v_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{0}},v_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{0}},v_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},\\ v_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{1}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{2}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{0}},\\ v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{0}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{2}}{v}_{\mathrm{1}},\\ v_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{0}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{2}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{1}}\}\end{array}$

and

$\begin{array}{l}{{A}}_{\mathrm{0}}(G)=\mathbb{Z}<v_{\mathrm{0}},v_{\mathrm{1}},v_{\mathrm{2}}>,\\ {{A}}_{\mathrm{1}}(G)=\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}>,\\ {{A}}_{\mathrm{2}}(G)=\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}>,\\ {{A}}_{\mathrm{3}}(G)=\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}>,\end{array}$

Since,

$\begin{array}{l}{\partial }_{\mathrm{0}}^w(v_i)=\mathrm{0},{\partial }_{\mathrm{1}}^w(v_i{v}_j)=w(v_i)v_j-w(v_j)v_i=w_i{v}_j-w_j{v}_i,{\partial }_{\mathrm{2}}^w(v_i{v}_j{v}_k)=w_i{v}_j{v}_k-w_j{v}_i{v}_k+w_k{v}_i{v}_j,\\ {\partial }_{\mathrm{3}}^w(v_i{v}_j{v}_k{v}_l)=w_i{v}_j{v}_k{v}_l-w_j{v}_i{v}_k{v}_l+w_k{v}_i{v}_j{v}_l-w_l{v}_i{v}_j{v}_k,\end{array}$

it follows that,

$\begin{array}{l}{\mathrm{\Omega }}_{\mathrm{0}}^w(G)=\mathbb{Z}<v_{\mathrm{0}},v_{\mathrm{1}},v_{\mathrm{2}}>,\\ {\mathrm{\Omega }}_{\mathrm{1}}^w(G)={{A}}_{\mathrm{1}}(G)\bigcap ({\partial }_{\mathrm{1}}^w{)}^{-\mathrm{1}}{{A}}_{\mathrm{0}}(G)=\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}>\bigcap ({\partial }_{\mathrm{1}}^w{)}^{-\mathrm{1}}\mathbb{Z}<v_{\mathrm{0}},v_{\mathrm{1}},v_{\mathrm{2}}>=\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}>\\ {\mathrm{\Omega }}_{\mathrm{2}}^w(G)={{A}}_{\mathrm{2}}(G)\bigcap ({\partial }_{\mathrm{2}}^w{)}^{-\mathrm{1}}{{A}}_{\mathrm{1}}(G)=\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}>\bigcap ({\partial }_{\mathrm{2}}^w{)}^{-\mathrm{1}}\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}>\\ =\{\begin{array}{ccc}\mathrm{0},& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{0}},w_{\mathrm{1}},w_{\mathrm{2}}\ne \mathrm{0},\end{array}& \\ \mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{1}}=\mathrm{0},w_{\mathrm{0}},w_{\mathrm{2}}\ne \mathrm{0},\end{array}& \\ \mathbb{Z}<v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{2}}=\mathrm{0},w_{\mathrm{0}},w_{\mathrm{1}}\ne \mathrm{0},\end{array}& \\ \mathbb{Z}<v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{0}}\end{array}=\mathrm{0},w_{\mathrm{1}},w_{\mathrm{2}}\ne \mathrm{0},& \\ \mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{1}}=w_{\mathrm{2}}=\mathrm{0},w_{\mathrm{0}}\ne \mathrm{0},\end{array}& \\ \mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{1}}=w_{\mathrm{0}}=\mathrm{0},w_{\mathrm{2}}\ne \mathrm{0},\end{array}& \\ \mathbb{Z}<v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{0}}=w_{\mathrm{2}}=\mathrm{0},w_{\mathrm{1}}\ne \mathrm{0},\end{array}& \\ \mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{0}}=w_{\mathrm{1}}=w_{\mathrm{2}}=\mathrm{0},\end{array}& \end{array}\end{array}$

and

$\begin{array}{l}{\mathrm{\Omega }}_{\mathrm{3}}^w(G)={{A}}_{\mathrm{3}}(G)\bigcap ({\partial }_{\mathrm{3}}^w{)}^{-\mathrm{1}}{{A}}_{\mathrm{2}}(G)\\ =\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}>\bigcap ({\partial }_{\mathrm{3}}^w{)}^{-\mathrm{1}}\mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}>\\ =\{\begin{array}{ccc}\mathrm{0},& \mathrm{i}\mathrm{f}\begin{array}{c}\mathrm{a}\mathrm{t}\begin{array}{c}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{t}\begin{array}{c}\mathrm{t}\mathrm{w}\mathrm{o}\begin{array}{c}\mathrm{o}\mathrm{f}\begin{array}{c}{w}_{\mathrm{0}},w_{\mathrm{1}},w_{\mathrm{2}}\ne \mathrm{0},\end{array}\end{array}\end{array}\end{array}\end{array}& \\ \mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{1}}=w_{\mathrm{2}}=\mathrm{0},w_{\mathrm{0}}\end{array}\ne \mathrm{0},& \\ \mathbb{Z}<v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{2}}=w_{\mathrm{0}}=\mathrm{0},w_{\mathrm{1}}\ne \mathrm{0},\end{array}& \\ \mathbb{Z}<v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{0}}=w_{\mathrm{1}}=\mathrm{0},w_{\mathrm{2}}\ne \mathrm{0},\end{array}& \\ \mathbb{Z}<v_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}},v_{\mathrm{1}}{v}_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}},v_{\mathrm{2}}{v}_{\mathrm{0}}{v}_{\mathrm{1}}{v}_{\mathrm{2}}>,& \mathrm{i}\mathrm{f}\begin{array}{c}{w}_{\mathrm{0}}=w_{\mathrm{1}}=w_{\mathrm{2}}=\mathrm{0}.\end{array}& \end{array}\end{array}$

Hence,

$H_{\mathrm{1}}(G,w;\mathbb{Z})=\mathrm{K}\mathrm{e}\mathrm{r}{\partial }_{\mathrm{1}}^w{|}_{{\mathrm{\Omega }}_{\mathrm{1}}^{\mathrm{w}}(G)}/\mathrm{I}\mathrm{m}{\partial }_{\mathrm{2}}^w{|}_{{\mathrm{\Omega }}_{\mathrm{2}}^{\mathrm{w}}(G)}=\mathrm{0},$(6)

for $w_{\mathrm{1}}=\mathrm{0}$ and $w_{\mathrm{0}},w_{\mathrm{2}}\ne \mathrm{0}$. By Ref. [10, Proposition 4.7], Eq. (6) is different from the unweighted 1-dimensional path homology group $H_{\mathrm{1}}(G;\mathbb{Z})\cong \mathbb{Z}$.

However, by Refs. [3,6,7], the following lemma holds.

Lemma 1   (Ref. [6, Theorem 2.2]) Let $\mathrm{ℱ}$ be any field. Let $C_n(U,X;\mathrm{ℱ})=C_n(U,X;\mathbb{Z})\otimes \mathrm{ℱ}$. Then for any nonvanishing weight $w$ on $X$,

$H_{\mathrm{*}}^w(U,X;\mathrm{ℱ})=H_{\mathrm{*}}(U,X;\mathrm{ℱ}).$

This implies that the weighted homology with $\mathrm{ℱ}$-coefficients does not depend on the choice of $w$.

Hence, by Lemma 1, we have

Corollary 1   The free part of $H_{\mathrm{*}}^w(U,X;\mathbb{Z})$ does not depend on the choice of non-vanishing weight $w$.

Proof   The free part of $H_{\mathrm{*}}^w(U,X;\mathbb{Z})$ is ${\mathbb{Z}}^{\oplus r}$, where $r=\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{\Theta }}{H}_{\mathrm{*}}^w(U,X;{Q})$. By Lemma 1, $r$ does not depend on $w$. Hence, the free part of $H_{\mathrm{*}}^w(U,X;\mathbb{Z})$ does not depend on the choice of $w$.

3 Main Results

In this section, we give a quasi-partial order and a pseudo-metric on the set made up of all non-vanishing weights on a finite set and consider the morphisms of weighted $\mathrm{\Delta }$-sets.

3.1 Vertex Weights on a Finite Set

Let $V$ be a finite set. Let $w,w^{\text{'}},w^{″}:V\to \mathbb{Z}\backslash \{\mathrm{0}\}$ be non-vanishing weights on $V$. Using the following notations:

(i) $w=w^{\text{'}}$ if for an arbitrary vertex $v\in V$, $w(v)=w^{\text{'}}(v)$;

(ii) $w\ne w^{\text{'}}$ if there exists a vertex $v\in V$ such that $w(v)\ne w^{\text{'}}(v)$;

(iii) $w\approx w^{\text{'}}$ if for an arbitrary vertex $v\in V$, $w(v)=\pm w^{\text{'}}(v)$;

(iv) $w\not\approx w^{\text{'}}$ if there exists a vertex $v\in V$ such that $\left|w(v)\right|\ne \left|w^{\text{'}}(v)\right|$ ;

(v) $w\lesssim w^{\text{'}}$ if for an arbitrary vertex $v\in V$, $w(v)|w^{\text{'}}(v)$ ($w(v)$ divides $w^{\text{'}}(v)$);

(vi) $w<w^{\text{'}}$ if $w\lesssim w^{\text{'}}$ and $w\not\approx w^{\text{'}}$.

Then

(i) for any weight $w$ on $V$, $w\approx w$ and $w\lesssim w$ and $w=w$ ;

(ii) for any weights $w$ and $w^{\text{'}}$ on $V$, if $w\lesssim w^{\text{'}}$ and $w^{\text{'}}\lesssim w$, then $w\approx w^{\text{'}}$;

(iii) for any weights $w,w^{\text{'}},w^{″}$ on $V$, if $w\lesssim w^{\text{'}}$ and $w^{\text{'}}\lesssim w^{″}$ , then $w\lesssim w^{″}$ .

Hence, "$\lesssim$" is a quasi-partial order on the set $S_V$ of all the weights on $V$. A weight $w$ on $V$ is said minimal, if for any weight $w^{\text{'}}$ on $V$, $w\lesssim w^{\text{'}}$. That is,

$w$ is minimal $\leftrightarrow$$w(v)=\pm \mathrm{1}$ for any vertex $v\in V$.

Two weights $w$ and $w^{\text{'}}$ with $w<w^{\text{'}}$ are said adjacent if there is no weight $w^{″}$ on $V$ such that $w<w^{″}<w^{\text{'}}$. Hence, $w<w^{\text{'}}$ are adjacent if and only if there exists a unique vertex $v\in V$ such that the following both hold:

(i) $w^{\text{'}}(v)=\pm pw(v)$ for some prime $p$;

(ii) $w^{\text{'}}(u)=\pm pw(u)$ for any $u\ne v,u\in V$.

Therefore, we have a pseudo-metric $d$ on $S_V$ such that for any two weights $w$ and $w^{\text{'}}$ on $V$ ,

(i) $d(w,w^{\text{'}})=\mathrm{0}$ if $w\approx w^{\text{'}}$ ;

(ii) $d(w,w^{\text{'}})=\mathrm{1}$ if $w$ and $w^{\text{'}}$ are adjacent;

(iii) $d(w,w^{\text{'}})=\mathrm{i}\mathrm{n}\mathrm{f}\{|\gamma ||\gamma =w_{\mathrm{0}}{w}_{\mathrm{1}}\cdots w_{k-\mathrm{1}}{w}_k,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{w}_{\mathrm{0}}=w,$$\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}{w}_k=w^{\text{'}}\begin{array}{c},\mathrm{a}\mathrm{n}\mathrm{d}\begin{array}{c}{w}_{i-\mathrm{1}}\begin{array}{c}\mathrm{a}\mathrm{n}\mathrm{d}\begin{array}{c}{w}_i\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\begin{array}{c}\mathrm{a}\mathrm{r}\mathrm{e}\begin{array}{c}\mathrm{a}\mathrm{d}\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\end{array}\end{array}\}$ .

Here $|\gamma |=k$ if $\gamma =w_{\mathrm{0}}{w}_{\mathrm{1}}\cdots w_{k-\mathrm{1}}{w}_k$ .

Note that any $w$ is connected to the minimal weights by certain $\gamma$. Hence, $d(w,w^{\text{'}})$ make sense for any weights $w$ and $w^{\text{'}}$.

3.2 Morphisms of Super-Hypergraphs

Let $X$ and $X^{\text{'}}$ be $\mathrm{\Delta }$-sets. Let $w:X_{\mathrm{0}}\to \mathbb{Z}\backslash \{\mathrm{0}\}$ and $w^{\text{'}}:X_{\mathrm{0}}^{\text{'}}\to \mathbb{Z}\backslash \{\mathrm{0}\}$ be the weights on $X$ and $X^{\text{'}}$, respectively. De-fine a map $f:X_{\mathrm{0}}\to X_{\mathrm{0}}^{\text{'}}$ such that for each $x_i\in X_{\mathrm{0}}$,

$w^{\text{'}}(f(x_i))|w(x_i),$(7)

and the weight of $(x_{\mathrm{0}}{x}_{\mathrm{1}}\cdots x_n)\in X_n$ is defined to be the product of weights of all vertices $x_i$, $\mathrm{0}\le i\le n$. Obviously, $f$ is a $\mathrm{\Delta }$-map between $X$ and $X^{\text{'}}$. Furthermore, according to Ref. [6, Definition 1.1], since $f$ satisfies Eq. (7), it is referred to as the morphism between weighted $\mathrm{\Delta }$-sets. Let $U,U^{\text{'}}$ be graded subsets of $X,X^{\text{'}}$, respectively. Suppose $f(U)\subseteq U^{\text{'}}$, then, $f$ is a weighted morphism of super-hypergraphs.

Lemma 2   (Ref. [6, Theorem 1.1]) A morphism of weighted $\mathrm{\Delta }$-sets induces a homomorphism of weighted embedded homology on super-hypergraphs

$f_{\mathrm{*}}:H_n^w(U,X;\mathbb{Z})\to H_n^{w^{\text{'}}}(U^{\text{'}},X^{\text{'}};\mathbb{Z}),n\ge \mathrm{0}.$

Notably, in Lemma 2, let $X=X^{\text{'}}$,$U=U^{\text{'}}$ and $f$ be the identity map $\tau$ on $X$. We have that

Corollary 2   Let $w$ and $w^{\text{'}}$ be two weights on $X$ and $w^{\text{'}}\lesssim w$. Then, the identity map $\tau$ on $X$ induces a homomorphism

${\tau }_{\mathrm{*}}:H_n^w(U,X;\mathbb{Z})\to H_n^{w^{\text{'}}}(U,X;\mathbb{Z}),n\ge \mathrm{0}.$

The following results are based on Corollary 1 and Corollary 2.

Theorem 1   Let $w$ and $w^{\text{'}}$ be two weights on $X$ and $w^{\text{'}}\lesssim w$. Then, the homomorphism ${\tau }_{\mathrm{*}}$ splits into a direct sum

${\tau }_{\mathrm{*}}=\mathrm{i}\mathrm{d}\oplus \left({\oplus }_{p\begin{array}{c}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\end{array}}\mathrm{T}\mathrm{o}{\mathrm{r}}_p({\tau }_{\mathrm{*}})\right),$

where $\mathrm{i}\mathrm{d}$ is the identity map on the free part of $H_n^w(U,X;\mathbb{Z})$ and $\mathrm{T}\mathrm{o}{\mathrm{r}}_p({\tau }_{\mathrm{*}})$ is the restriction of ${\tau }_{\mathrm{*}}$ to the $p$-torsion part