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Green's graphs of a semigroup

  • Yanliang Cheng , Yong Shao EMAIL logo und Xuanlong Ma
Veröffentlicht/Copyright: 12. August 2025
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 23 Heft 1

Abstract

Let S be a semigroup. In this study, we first introduce the Green’s digraphs and Green’s graphs related to the Green’s relations L , R , and J of S . Further, the connectedness and completeness of the Green’s graphs are discussed. For a finite semigroup S , we show that each of the Green’s graphs of S has a transitive orientation. Moreover, we obtain that these Green’s graphs are perfect. Finally, the structures of the Green’s graphs are characterized using the generalized lexicographic product.

Keywords: semigroup; Green’s relation; Green’s graph; complete graph; connected graph
MSC 2010: 05C25; 20M99

1 Introduction and preliminaries

Graphs related to groups and semigroups have been actively investigated in the literature [1–6]. Given a semigroup, there are many different ways to associate a directed or undirected graph with the semigroup, including the zero-divisor graphs [4], divisibility graphs [2], power graphs [2,5], and Cayley graphs [1,3], etc. Let S be a semigroup, and let T be a subset of S . The Cayley graph C a y ( S , T ) of S relative to T is defined as the directed graph with vertex set S and arc set consisting of those ordered pairs ( x , y ) where t x = y and x ≠ y for some t ∈ T . Kelarev and Praeger [3] characterized all vertex-transitive Cayley graphs arising from periodic semigroups. Kelarev [1] described all finite inverse semigroups and all commutative inverse semigroups with bipartite Cayley graphs. The Cayley graph of a semigroup is related to finite state automata and has many applications [7–16].

The directed power graph of a semigroup S was defined by Kelarev and Quinn in [2] as the directed graph P → ( S ) with vertex set S in which there is an arc from x to y if and only if x ≠ y and y = x m for some positive integer m . Motivated by this, Chakrabarty et al. [5] focused their study on the undirected power graphs of a semigroup S , in which distinct x and y are adjacent if one is a power of the other. They characterized the class of semigroups S for which the power graph P ( S ) is connected or complete. Based on these, Cameron and Ghosh [17,18] explored the power graphs of finite groups, obtained many profound results, and promoted the research of related problems. Nowadays, power graphs of groups and semigroups are actively investigated by researchers [6,19–23]. A detailed list of results and open problems can be found in [24,25].

Dalal et al. [21], as a generalization of power graphs of semigroups, introduced a new graph: the enhanced power graph of a semigroup S , denoted by P e ( S ) , as the graph whose vertex set is S and in which two distinct vertices x , y are adjacent if x , y ∈ ⟨ z ⟩ for some z ∈ S . They described the structure of P e ( S ) and discussed some of its graph-theoretic properties. Ma et al. [26], gave a recent survey on the enhanced power graphs of groups. An interesting notion of an E -extended power graph of a finite semigroup is studied recently in [27].

It is known that Green’s relations play a fundamental role in the study of semigroups. Let S be a semigroup. Green’s relations on S are defined as follows. The relations ℒ , ℛ , and J on S are given by

a L b ⇔ S 1 a = S 1 b , a R b ⇔ a S 1 = b S 1 , a J b ⇔ S 1 a S 1 = S 1 b S 1 ,

where S 1 is the monoid obtained from S by adjoining an identity if necessary. Let H = L ∩ R and D = L ∨ R , where L ∨ R denote the minimum equivalence relation containing L and R . Clearly, for all a , b ∈ S ,

a L b ⇔ a = x b and b = y a for some x , y ∈ S 1 .

The relations R and J can be characterized similarly. Let L a , R a , and J a denote the L -class, R -class, and J -class containing a , respectively.

It is worth noting that Green’s relations also play an important role in the study of the graph theory related to semigroups. In 2014, Gharibkhajeh and Doostie [28] introduced the Green graphs of a finite semigroup S by generalizing the notion of conjugacy graphs of groups. The left Green graph of S is an undirected graph whose vertices are the L -classes of S , and two vertices L i and L j are adjacent if and only if gcd ( ∣ L i ∣ , ∣ L j ∣ ) > 1 . The other Green graphs are defined in a similar way. They gave a necessary condition for the Green graphs related to L , R , J , and H of S to coincide. Moreover, Sorouhesh et al. [29] obtained a sufficient condition on non-group semigroups that implies the coinciding of these Green graphs. In 2023, Nupo and Chaiya [15] investigated the Cayley digraphs of full transformation semigroups with respect to Green’s equivalence classes, and presented structural properties and isomorphism theorems of these digraphs. Recently, Ashegh Bonabi and Khosravi [23] gave a characterization of a completely simple semigroup in terms of its power graph and Green’s relations. Cheng et al. [22] explored the power graphs of certain completely 0-simple semigroups, and they showed that a G 0 -normal completely 0-simple orthodox semigroup with abelian group H -classes is characterized by its power graph.

Recall that the power graph P ( S ) of a semigroup S is an undirected graph whose vertex set is S such that two vertices a , b ∈ S are adjacent if and only if a ≠ b and a = b m or b = a m for some positive integer m . This means that S 1 a ⊆ S 1 b or S 1 b ⊆ S 1 a (resp. a S 1 ⊆ b S 1 or b S 1 ⊆ a S 1 , S 1 a S 1 ⊆ S 1 b S 1 or S 1 b S 1 ⊆ S 1 a S 1 ). That is, a = x b or b = y a (resp. a = b x or b = a y , a = x b y or b = x a y ) for some x , y ∈ S 1 . As a generalization of power graphs of semigroups, we introduce new types of graphs on semigroups called Green’s digraphs and Green’s graphs.

As usual, a graph means an undirected simple graph, and a digraph means a directed graph without loops. Given a graph (resp. digraph) Γ , we always use V ( Γ ) and E ( Γ ) to denote the vertex set and the edge set (resp. the arc set), respectively. The digraph O is an orientation for Γ if V ( O ) = V ( Γ ) and ∣ { ( u , v ) , ( v , u ) } ∩ E ( O ) ∣ = 1 for all { u , v } ∈ E ( Γ ) . A transitive orientation for Γ is an orientation O such that { ( u , v ) , ( v , w ) } ⊆ E ( O ) implies ( u , w ) ∈ E ( O ) . A comparability graph is a graph that admits a transitive orientation. It has been characterized in [30,31].

The study is structured as follows. In Section 2, we first give the definitions of Green’s digraphs and Green’s graphs of a semigroup. We then characterize the connected components of the Green’s graphs and the class of semigroups for which Green’s graphs are complete. For a finite semigroup S , we construct transitive orientations for the Green’s graphs of S . Moreover, we prove that the Green’s graphs are perfect graphs. In Section 3, we use the generalized lexicographic product to characterize the structures of the Green’s graphs of a finite semigroup.

For other notations and terminologies not given in this article, the reader is referred to the books [32] and [33].

2 Green’s graphs of a semigroup

The aim of this section is to give the definitions of Green’s digraphs and Green’s graphs of a semigroup related to Green’s relations L , R , and J . Further, we shall discuss some graph-theoretic properties of the Green’s graphs.

Definition 2.1

Let S be a semigroup. The Green’s L -digraph (resp. R -digraph, J -digraph) of S , denoted by D L ( S ) (resp. D R ( S ) , D J ( S ) ), is a directed graph with vertex set S such that there is an arc from a to b if and only if a ≠ b and b ∈ S a (resp. b ∈ a S , b ∈ S 1 a S 1 ).

It is easy to see that D L ( S ) and D R ( S ) are two distinct subdigraphs of D J ( S ) , and the directed power graph P → ( S ) of S is a subdigraph of each of the Green’s digraphs of S . Obviously, the Green’s L -digraph D L ( S ) and the Cayley graph C a y ( S , S ) are the same, and the divisibility graph D i v ( S ) and Green’s J -digraph are the same.

In the following, we shall give the definitions of Green’s graphs of a semigroup.

Definition 2.2

Let S be a semigroup. The Green’s L -graph of S , denoted by Γ L ( S ) , is an undirected graph whose vertex set is S such that two vertices a , b ∈ S are adjacent if and only if a ≠ b and b ∈ S a or a ∈ S b , i.e.,

V ( Γ L ( S ) ) = S , E ( Γ L ( S ) ) = { { a , b } ⊆ S ∣ a ≠ b , and b ∈ Sa or a ∈ Sb } .

The other Green’s graphs Γ R ( S ) and Γ J ( S ) are defined in a similar way: V ( Γ ℛ ( S ) ) = V ( Γ J ( S ) ) = S and

E ( Γ R ( S ) ) = { { a , b } ⊆ S ∣ a ≠ b and b ∈ a S or a ∈ b S } , E ( Γ J ( S ) ) = { { a , b } ⊆ S ∣ a ≠ b and b ∈ S 1 a S 1 or a ∈ S 1 b S 1 } .

For a semigroup S , it is clear that both Γ L ( S ) and Γ R ( S ) are obtained by deleting some of the edges from E ( Γ J ( S ) ) . Hence, Γ L ( S ) and Γ R ( S ) are two spanning subgraphs of Γ J ( S ) . It is easy to check that the power graph P ( S ) of S is a spanning subgraph of each of the Green’s graphs. The principal left ideal graph of S is the graph S G with V ( S G ) = S such that two vertices a and b ( a ≠ b ) are adjacent in S G if and only if S 1 a ∩ S 1 b ≠ ∅ . The principal right ideal graph is defined similarly. Moreover, it is clear that Γ L ( S ) (resp. Γ R ( S ) ) is a spanning subgraph of the principal left (resp. right) ideal graph of S .

Now, we shall discuss some graph-theoretic properties of the Green’s graphs for a semigroup S . We only consider the Green’s L -graph Γ L ( S ) . There are analogous results for Γ R ( S ) and Γ J ( S ) . We define two binary relations τ 1 and τ 2 on S by

a τ 1 b ⇔ S 1 a ∩ S 1 b ≠ ∅ , a τ 2 b ⇔ S 1 a ∪ S 1 b ⊆ S 1 c for some c ∈ S .

Clearly, τ 1 ∪ τ 2 is also a binary relation. Let τ denote the equivalence relation on S generated by τ 1 ∪ τ 2 , i.e., the minimum equivalence relation on S containing τ 1 ∪ τ 2 . By [33, Proposition 4.25], we have the following lemma.

Lemma 2.3

Let S be a semigroup. Then, τ = ( τ 1 ∪ τ 2 ) ∞ .

Moreover, we have the following result.

Theorem 2.4

Let S be a semigroup, and let a , b ∈ S with a ≠ b . Then, a and b are connected in Γ L ( S ) if and only if a τ b .

Proof

Suppose that a and b are connected by a path, say ( a , c 1 , c 2 , … , c k , b ) in Γ L ( S ) , where c 1 , c 2 , … , c k ∈ S . Letting a = c 0 and b = c k + 1 , for each i ∈ { 0 , … , k } , we have c i ∈ S c i + 1 or c i + 1 ∈ S c i , and hence c i τ c i + 1 . It follows by transitivity that a τ b .

Conversely, let a τ b . By Lemma 2.3, we have ( a , b ) ∈ ( τ 1 ∪ τ 2 ) ∞ . That is, ( a , b ) ∈ ( τ 1 ∪ τ 2 ) n for some positive integer n . Hence, there exist c 1 , c 2 , … , c n − 1 ∈ S such that ( a , c 1 ) ∈ τ 1 ∪ τ 2 , ( c 1 , c 2 ) ∈ τ 1 ∪ τ 2 , … , ( c n − 1 , b ) ∈ τ 1 ∪ τ 2 . Let a = c 0 and b = c n , and consider i ∈ { 0 , … , n − 1 } . If ( c i , c i + 1 ) ∈ τ 1 , then there exists some d ∈ S 1 c i ∩ S 1 c i + 1 . If ( c i , c i + 1 ) ∈ τ 2 , then there exists some d ∈ S such that S 1 c i ∩ S 1 c i + 1 ⊆ S 1 d . In either case, we have { c i , d } , { d , c i + 1 } ∈ E ( Γ L ( S ) ) , and hence c i and c i + 1 are connected by the path ( c i , d , c i + 1 ) . It follows that a and b are connected.□

The following corollary is immediate.

Corollary 2.5

Suppose that S is a semigroup. Then, the connected components of Γ L ( S ) are precisely { τ a ∣ a ∈ S } , where τ a is the equivalence class containing a.

It follows from Corollary 2.5 that Γ L ( S ) is a connected graph (resp. a null graph) if and only if τ is the universal relation (resp. the equality relation) on S .

Recall that a partially ordered set, or simply poset, P is an ordered pair ( V ( P ) , ≤ P ) , where V ( P ) is called the vertex set of P , and ≤ P is a partial order on V ( P ) . As usual, we write x < P y if x ≤ P y and x ≠ y . For two elements x , y ∈ V ( P ) , x and y are comparable in P if x ≤ P y or y ≤ P x ; otherwise, x and y are incomparable. A chain (resp. antichain) is a partially ordered set such that all elements are pairwise comparable (resp. incomparable).

Recall that a graph is complete if any two vertices are adjacent. The next result characterizes the class of semigroups S for which Γ L ( S ) is complete.

Proposition 2.6

Let S be a semigroup. Then, Γ L ( S ) is complete if and only if the principal left ideals of S form a chain with respect to the usual inclusion.

Proof

Γ L ( S ) is complete if and only if for any a , b ∈ S with a ≠ b either a ∈ b S or b ∈ a S if and only if the principal left ideals of S form a chain.□

Example

For a monogenic semigroup S = ⟨ a ⟩ , every principal (left) ideal of S is of the form S a t for some positive integer t or S 1 a . It is easy to check that the principal left (resp. right, two-sided) ideals of S form a chain under usual inclusion. Hence, by Proposition 2.6, Γ L ( S ) , Γ R ( S ) , and Γ J ( S ) are complete.

Note that if S is a left simple (resp. right simple, simple) semigroup, i.e., L = S × S (resp. R = S × S , J = S × S ), then Γ L ( S ) (resp. Γ R ( S ) , Γ J ( S ) ) is a complete graph. In particular, if S is a left zero semigroup, then Γ L ( S ) is complete and Γ R ( S ) is a null graph. Dually, for a right zero semigroup S , Γ R ( S ) is complete and Γ L ( S ) is a null graph.

Now, let S be a finite semigroup. As usual, for each a ∈ S , L a = { b ∈ S ∣ ( a , b ) ∈ L } . Write

(2.1) L ( S ) ≔ { L a ∣ a ∈ S } = { L a 11 , L a 21 , … , L a m 1 } ,

where m is the number of L -classes of S , and let L a i 1 = { a i 1 , a i 2 , … , a i s i } for each i ∈ [ m ] (= { 1,2 , … , m } ), where s i denote the cardinality of the L -class containing a i 1 .

Definition 2.7

Let S be a finite semigroup and let a , b ∈ S with a ≠ b . Define a ≺ b if one of the following conditions holds.

  1. L a < L b , i.e., S 1 a ⊊ S 1 b .

  2. For some i ∈ [ m ] , a = a i ℓ , b = a i k , and ℓ < k .

Define a ≼ b if a ≺ b or a = b .

The proof of the following lemma is obvious.

Lemma 2.8

Suppose that S is a finite semigroup. With reference to (2.1), if there exist i , j ∈ [ m ] and i ≠ j such that a i ℓ 0 ≺ a j k 0 for some ℓ 0 ∈ [ s i ] and k 0 ∈ [ s j ] , then a i ℓ ≺ a j k for each ℓ ∈ [ s i ] and each k ∈ [ s j ] .

Now, we define O S as the digraph with vertex set S such that there is an arc from b to a if a ≺ b . It is easy to see that O S is an orientation of Γ L ( S ) . Moreover, we have the following result.

Theorem 2.9

Let S be a finite semigroup. Then, the following statements hold.

  1. O S is a transitive orientation of Γ L ( S ) and a subdigraph of D L ( S ) .

  2. If O is a transitive orientation of Γ L ( S ) and a subdigraph of D L ( S ) , then the graphs O and O S are isomorphic.

Proof

( i ) . Assume that { ( a , b ) , ( b , c ) } ⊆ E ( O S ) for distinct a , b , c ∈ S . Then b ≺ a and c ≺ b . This implies that L b ≤ L a and L c ≤ L b . First, suppose that ( a , c ) ∉ L in S . Then, L c < L a , and so c ≺ a . It follows that ( a , c ) ∈ E ( O S ) . Now, suppose that ( a , c ) ∈ L , i.e., L a = L c . Then, by ( 2.1 ) , there exists an index i ∈ [ m ] such that a = a i ℓ , b = a i r , and c = a i k for some ℓ , r , k ∈ [ s i ] and k < r < ℓ . Thus, c ≺ a , and so ( a , c ) ∈ E ( O S ) . This shows that O S is a transitive orientation of Γ L ( S ) . Clearly, O S is a subdigraph of D L ( S ) .

( i i ) . Suppose that a subdigraph O of D L ( S ) is another transitive orientation of Γ L ( S ) . Since the induced subgraph on L a i 1 of Γ L ( S ) is a complete graph and all transitive orientations of a fixed completed graph are isomorphic, we need only to show that ( a i 1 , a j 1 ) ∈ E ( O S ) if and only if ( a i 1 , a j 1 ) ∈ E ( O ) for any i , j ∈ [ m ] and i ≠ j .

If ( a i 1 , a j 1 ) ∈ E ( O S ) , that is, L a j 1 < L a i 1 , then a j 1 = x a i 1 for some x ∈ S , and so { a i 1 , a j 1 } ∈ E ( Γ L ( S ) ) , ( a i 1 , a j 1 ) ∈ E ( D L ( S ) ) and ( a j 1 , a i 1 ) ∉ E ( D L ( S ) ) . Since O is a subdigraph of D L ( S ) , we have ( a i 1 , a j 1 ) ∈ E ( O ) .

Conversely, if ( a i 1 , a j 1 ) ∈ E ( O ) , then ( a i 1 , a j 1 ) ∈ D L ( S ) . That is, a j 1 = x a i 1 for some x ∈ S , and so L a j 1 < L a i 1 . It follows that a j 1 ≺ a i 1 , and so ( a i 1 , a j 1 ) ∈ E ( O S ) . This shows that O and O S are isomorphic.□

The following result is an immediate consequence of Theorem 2.9.

Corollary 2.10

The Green’s L -graph Γ L ( S ) of a finite semigroup S is a comparability graph.

A graph Γ is perfect if for each induced subgraph Λ of Γ , the chromatic number and the clique number of Λ are equal. It is well known that comparability graphs are perfect [34, Chapter V, Theorem 17]. Therefore, we have the following corollary.

Corollary 2.11

The Green’s L -graph Γ L ( S ) of a finite semigroup S is perfect.

3 Structure of Green’s graphs for a finite semigroup

In this section, by means of the generalized lexicographic product of certain graphs, we characterize the structures of Green’s graphs for a finite semigroup S . We shall only characterize the Green’s L -graph Γ L ( S ) ; analogous results hold for Γ R ( S ) and Γ J ( S ) .

Let P be a poset. For any subset U ⊆ V ( P ) , the subposet of P induced by U , denoted by P ( U ) , is a poset ( U , ≤ P ( U ) ) , where for any x , y ∈ U , x ≤ P ( U ) y if and only if x ≤ P y . It follows from Definition 2.7 that ( S , ≼ ) is a poset. In the remainder of this study, we use L S to denote this poset.

The comparability graph of P , denoted by C P , is the graph with the vertex set V ( P ) , where two distinct vertices are adjacent if and only if they are comparable.

Lemma 3.1

Let S be a finite semigroup. Then, C L S = Γ L ( S ) .

Proof

Clearly, V ( C L S ) = V ( Γ L ( S ) ) = S . For any a , b ∈ S , we have { a , b } ∈ E ( C L S ) if and only if L a ≤ L b or L b ≤ L a if and only if { a , b } ∈ E ( Γ L ( S ) ) . Thus, E ( C L S ) = E ( Γ L ( S ) ) .□

From [6], let P be a poset. A subset Q of P is homogeneous if for any y ∈ P \ Q , one of the following holds:

  1. For all x ∈ Q , x ≤ P y .

  2. For all x ∈ Q , y ≤ P x .

  3. For all x ∈ Q , x and y are incomparable.

A homogeneous chain (resp. antichain) in P is a chain (resp. an antichain) that is homogeneous. An equivalence relation ρ of P is homogeneous if all its equivalence classes are homogeneous in P , and the partition Ω corresponding to ρ is called a homogeneous partition of P . The quotient P ⁄ ρ is the poset ( Ω , ≤ P ⁄ ρ ) such that two subsets Ω 1 , Ω 2 ∈ Ω satisfy Ω 1 ≤ P ⁄ ρ Ω 2 if and only if Ω 1 = Ω 2 or x < P y for each x ∈ Ω 1 and each y ∈ Ω 2 .

L S is defined in the second paragraph of Section 3. The following lemma is an immediate consequence of Lemma 2.8.

Lemma 3.2

Let S be a finite semigroup. Any element in L ( S ) is a homogeneous chain in L S . Moreover, L ( S ) and L are a homogeneous partition and a homogeneous equivalence relation of L S , respectively.

Recall that the lexicographical sum [35] is defined as follows. Let P be a poset and P = { Q x ∣ x ∈ V ( P ) } be a family of posets indexed by V ( P ) . The lexicographical sum of P over P , denoted by P [ P ] , is the poset with the vertex set V ( P [ P ] ) = { ( x , y ) ∣ x ∈ V ( P ) , y ∈ V ( Q x ) } , where ( x 1 , y 1 ) ≤ P [ P ] ( x 2 , y 2 ) , provided that either x 1 = x 2 and y 1 ≤ Q x 1 y 2 or x 1 < P x 2 .

Let ρ be a homogeneous equivalence relation of a poset P , and let R = P ⁄ ρ and P = { P ( Q ) ∣ Q ∈ R } . Then, P is isomorphic to R [ P ] [6, Lemma 2.8]. Hence, the following result is an immediate consequence of Lemma 3.2.

Theorem 3.3

Let S be a finite semigroup and let P = { L S ( L a ) ∣ L a ∈ L ( S ) } . Then, L S ≅ ( L S ⁄ L ) [ P ] .

Now, in order to characterize the structure of Green’s L -graph Γ L ( S ) of a finite semigroup S , we need the definition of the generalized lexicographic product [36]. Given a graph Γ and a family of graphs F = { F v ∣ v ∈ V ( Γ ) } , indexed by V ( Γ ) , their generalized lexicographic product, denoted by Γ [ F ] , is defined as the graph with vertex set

V ( Γ [ F ] ) = { ( v , w ) ∣ v ∈ V ( Γ ) , w ∈ V ( F v ) }

and edge set

E ( Γ [ F ] ) = { { ( v 1 , w 1 ) , ( v 2 , w 2 ) } ∣ { v 1 , v 2 } ∈ E ( Γ ) , or v 1 = v 2 and { w 1 , w 2 } ∈ E ( F v 1 ) } .

Given a poset P , let P be a family of posets indexed by V ( P ) . Suppose that C P consists of all comparability graphs of posets in P . Then, C P [ P ] = C P [ C P ] [6, Lemma 2.12].

Lemma 3.4

Given a finite semigroup S, let P = { L S ( L a ) ∣ L a ∈ L ( S ) } and C P = { C L S ( L a ) ∣ L S ( L a ) ∈ P } . Then,

C ( L S ⁄ L ) [ P ] = C ( L S ⁄ L ) [ C P ] .

Now, for a finite semigroup S and a ∈ S , let K a denote the complete graph of order ∣ L a ∣ , and write K S = { K a ∣ L a ∈ L ( S ) } . Thus, we have the following result.

Theorem 3.5

Let S be a finite semigroup. Then, the Green’s L -graph Γ L ( S ) is isomorphic to the generalized lexicographic product C ( L S ⁄ L ) [ K S ] .

Proof

It follows from Lemma 3.1 that C L S = Γ L ( S ) . Hence, we only need to show that C L S ≅ C ( L S ⁄ L ) [ K S ] .

By (2.1) and Definition 2.7, for each a ∈ S , the subposet L S ( L a ) is a chain, i.e., every pair of distinct elements in L a are comparable. Hence, the comparability graph C L S ( L a ) is the complete graph of order ∣ L a ∣ . That is, C L S ( L a ) ≅ K a for each a ∈ S . Moreover,

C ( L S ⁄ L ) [ { C L S ( L a ) ∣ L a ∈ L ( S ) } ] ≅ C ( L S ⁄ L ) [ K S ] .

It follows from Theorem 3.3 that

L S ≅ ( L S ⁄ L ) [ P ] = ( L S ⁄ L ) [ { L S ( L a ) ∣ L a ∈ L ( S ) } ] .

Therefore, by Lemma 3.4,

C L S ≅ C ( L S ⁄ L ) [ { L S ( L a ) ∣ L a ∈ L ( S ) } ] ≅ C ( L S ⁄ L ) [ { C L S ( L a ) ∣ L a ∈ L ( S ) } ] ≅ C ( L S ⁄ L ) [ K S ] .

Thus, Γ L ( S ) ≅ C ( L S ⁄ L ) [ K S ] . This completes the proof.□

Acknowledgments

The authors are grateful for the reviewers’ valuable comments that improved the manuscript.

  1. Funding information: This work was supported by National Natural Science Foundation of China (11971383), Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSY023) and Shaanxi Provincial Natural Science Basic Research Program Project (No. 2025JC-YBMS-086).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. YLC and YS proposed the research idea and XM participated in the discussion. YLC prepared the manuscript with contributions from all co-authors.

  3. Conflict of interest: Prof. Xuanlong Ma is an Editor of the Open Mathematics journal and was not involved in the review and decision-making process of this article.

  4. Data availability statement: All data generated or analyzed during this study are included in this published work.

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Received: 2024-04-23
Revised: 2025-06-16
Accepted: 2025-07-02
Published Online: 2025-08-12

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

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  2. Forbidden subgraphs of TI-power graphs of finite groups
  3. Finite group with some c#-normal and S-quasinormally embedded subgroups
  4. Classifying cubic symmetric graphs of order 88p and 88p 2
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  7. Further results on permanents of Laplacian matrices of trees
  8. Special Issue on Convex Analysis and Applications - Part II
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  15. Some results on value distribution concerning Hayman's alternative
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https://doi.org/10.1515/math-2025-0187
Schlagwörter für diesen Artikel
semigroup; Green’s relation; Green’s graph; complete graph; connected graph
Creative Commons
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Artikel in diesem Heft

  1. Special Issue on Contemporary Developments in Graphs Defined on Algebraic Structures
  2. Forbidden subgraphs of TI-power graphs of finite groups
  3. Finite group with some c#-normal and S-quasinormally embedded subgroups
  4. Classifying cubic symmetric graphs of order 88p and 88p 2
  5. Simplicial complexes defined on groups
  6. Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
  7. Further results on permanents of Laplacian matrices of trees
  8. Special Issue on Convex Analysis and Applications - Part II
  9. A generalized fixed-point theorem for set-valued mappings in b-metric spaces
  10. Research Articles
  11. Dynamics of particulate emissions in the presence of autonomous vehicles
  12. The regularity of solutions to the Lp Gauss image problem
  13. Exploring homotopy with hyperspherical tracking to find complex roots with application to electrical circuits
  14. The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
  15. Some results on value distribution concerning Hayman's alternative
  16. 𝕮-inverse of graphs and mixed graphs
  17. A note on the global existence and boundedness of an N-dimensional parabolic-elliptic predator-prey system with indirect pursuit-evasion interaction
  18. On a question of permutation groups acting on the power set
  19. Chebyshev polynomials of the first kind and the univariate Lommel function: Integral representations
  20. Blow-up of solutions for Euler-Bernoulli equation with nonlinear time delay
  21. Spectrum boundary domination of semiregularities in Banach algebras
  22. Statistical inference and data analysis of the record-based transmuted Burr X model
  23. A modified predictor–corrector scheme with graded mesh for numerical solutions of nonlinear Ψ-caputo fractional-order systems
  24. Dynamical properties of two-diffusion SIR epidemic model with Markovian switching
  25. Classes of modules closed under projective covers
  26. On the dimension of the algebraic sum of subspaces
  27. Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems
  28. On tangent bundles of Walker four-manifolds
  29. Regularity of weak solutions to the 3D stationary tropical climate model
  30. A new result for entire functions and their shifts with two shared values
  31. Freely quasiconformal and locally weakly quasisymmetric mappings in metric spaces
  32. On the spectral radius and energy of the degree distance matrix of a connected graph
  33. Solving the quartic by conics
  34. A topology related to implication and upsets on a bounded BCK-algebra
  35. On a subclass of multivalent functions defined by generalized multiplier transformation
  36. Local minimizers for the NLS equation with localized nonlinearity on noncompact metric graphs
  37. Approximate multi-Cauchy mappings on certain groupoids
  38. Multiple solutions for a class of fourth-order elliptic equations with critical growth
  39. A note on weighted measure-theoretic pressure
  40. Majorization-type inequalities for (m, M, ψ)-convex functions with applications
  41. Recurrence for probabilistic extension of Dowling polynomials
  42. Unraveling chaos: A topological analysis of simplicial homology groups and their foldings
  43. Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator
  44. A characterization of the translational hull of a weakly type B semigroup with E-properties
  45. Some new bounds on resolvent energy of a graph
  46. Carmichael numbers composed of Piatetski-Shapiro primes in Beatty sequences
  47. The number of rational points of some classes of algebraic varieties over finite fields
  48. Singular direction of meromorphic functions with finite logarithmic order
  49. Pullback attractors for a class of second-order delay evolution equations with dispersive and dissipative terms on unbounded domain
  50. Eigenfunctions on an infinite Schrödinger network
  51. Boundedness of fractional sublinear operators on weighted grand Herz-Morrey spaces with variable exponents
  52. On SI2-convergence in T0-spaces
  53. Bubbles clustered inside for almost-critical problems
  54. Classification and irreducibility of a class of integer polynomials
  55. Existence and multiplicity of positive solutions for multiparameter periodic systems
  56. Averaging method in optimal control problems for integro-differential equations
  57. On superstability of derivations in Banach algebras
  58. Investigating the modified UO-iteration process in Banach spaces by a digraph
  59. The evaluation of a definite integral by the method of brackets illustrating its flexibility
  60. Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces
  61. Tilings, sub-tilings, and spectral sets on p-adic space
  62. The higher mapping cone axiom
  63. Continuity and essential norm of operators defined by infinite tridiagonal matrices in weighted Orlicz and l∞ spaces
  64. A family of commuting contraction semigroups on l 1 ( N ) and l ∞ ( N )
  65. Pullback attractor of the 2D non-autonomous magneto-micropolar fluid equations
  66. Maximal function and generalized fractional integral operators on the weighted Orlicz-Lorentz-Morrey spaces
  67. On a nonlinear boundary value problems with impulse action
  68. Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
  69. Decompositions of the extended Selberg class functions
  70. Subharmonic functions and associated measures in ℝn
  71. Some new Fejér type inequalities for (h, g; α - m)-convex functions
  72. The robust isolated calmness of spectral norm regularized convex matrix optimization problems
  73. Multiple positive solutions to a p-Kirchhoff equation with logarithmic terms and concave terms
  74. Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
  75. Green's graphs of a semigroup
  76. Some new Hermite-Hadamard type inequalities for product of strongly h-convex functions on ellipsoids and balls
  77. Infinitely many solutions for a class of Kirchhoff-type equations
  78. On an uncertainty principle for small index subgroups of finite fields
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