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Graphs and Combinatorics https://doi.org/10.1007/s00373-019-02106-2 ORIGINAL PAPER
Cayley Graphs Over Green ∗ Relations of Abundant Semigroups Chunhua Li1 · Baogen Xu1 · Huawei Huang2 Received: 13 April 2019 / Revised: 21 September 2019 © Springer Japan KK, part of Springer Nature 2019
Abstract In this paper, we first introduce the concept of Cayley graphs over Green ∗ relations of abundant semigroups by using Green ∗ relations. After obtaining some basic properties, we get some conditions for Cayley graphs over Green ∗ relations of abundant semigroups to be transmissible. Finally, we give necessary and sufficient conditions for a Cayley graph over Green ∗ relations of an abundant semigroup to be linear and complete, respectively. Keywords Abundant semigroups · Cayley graphs over Green ∗ relations · Transmissible · Linear graphs · Complete graphs Mathematics Subject Classification 06F05 · 20M10
1 Introduction The Cayley graph of a group was introduced by Arthur Cayley in 1878 to interpret the notion of abstract groups. Since then, a lot of interesting results have been obtained by using the concept of Cayley graphs of groups (see, [2,16,17]). Motivated by the rich results on Cayley graphs of groups, many authors studied Cayley graphs of semigroups (see, [7,8,13,15]). Let S be a semigroup and T be a nonempty subset of S. The Cayley graph Cay(S, T ) of S relative to T is defined as the graph with vertex set S and edge
B
Chunhua Li [email protected] Baogen Xu [email protected] Huawei Huang [email protected]
1
School of Science, East China Jiaotong University, Nanchang 330013, Jiangxi, China
2
School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, Guizhou, China
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set E(Cay(S, T )) consisting of those ordered pair (a, b), where a = b and as = b for some s ∈ T (see, [3,5,12]). As a generalization of Cayley graphs of groups, the Cayley graphs of semigroups are closely related to the finite state automata and have many applications (see, [3–7,9–11,14]). Note that some Cayley graphs of semigroups may be defined by left actions (see, [5,13]). However, for a given semigroup S and a fixed nonempty subset T of S, the Cayley graph Cay(S, T ) defined by right and left actions may not be same. In order to get more applications, this paper gives a new definition of the Cayley graph of a semigroup by using generalized Green relations. Recall from Fountain [1] that on a semigroup S the relation L∗ is defined by the rule (a, b) ∈ L∗ in S if and only if there is a semigroup T containing S as a subsemigroup, such that (a, b) ∈ L in T for all a, b ∈ S. The relation R∗ is defined dually. The intersection of L∗ and R∗ is denoted by H∗ and we denote the smallest equivalence relation containing both L∗ and R∗ by D∗ . A semigroup S is abundant if each L∗ class and each R∗ class of S contains an idempotent, and superabundant if each H∗ class of S contains an idempotent. As usual, L∗ a and R∗ a denote L∗ class and R∗ class containing a, respectively. In particular, a ∗ [a + ] denotes a typical idempotent in L∗ a [R∗ a ] if L∗ a [R∗ a ] contains an idempotent. A left ideal I of a semigroup S is a left ∗ ideal if L∗ a ⊆ I for all a ∈ I . Dually, we can define a right ∗ ideal. An ideal I of a semigroup S is called a ∗ ideal if it is both a left ∗ ideal and a right ∗ ideal. The smallest ∗ ideal containing the element a of the semigroup S will be denoted by J ∗ (a). Two elements a, b of a semigroup S have J ∗ − related if and only if J ∗ (a) = J ∗ (b). Generally, we call L∗ , R∗ , H∗ , D∗ and J ∗ Green ∗ relations. For an arbitrary semigroup S, we have D∗ ⊆ J ∗ . A semigroup S is J ∗ −simple if it has no proper ∗ ideal. Clearly a semigroup S is J ∗ −simple if and only if J ∗ is the universal relation on S. As usual, for a graph , V () and E() stand for its vertex set and edge set, respectively. Recall that a partial order set (X , ≤) is a linear order, if a ≤ b or b ≤ a for all a, b ∈ X . In particular, the relation graph of a linear order is a linear graph. It is easy to check that a graph is linear if it satisfies the following conditions (L1), (L2), (L3) and (L4) of Remark 1.1. Remark 1.1 (L1) has loops, not multiple edges; (L2) is transmissible (a graph is transmissible, if for all a, b, c ∈ V (), (a, b), (b, c) ∈ E() implies (a, c) ∈ E()); (L3) (a, b) ∈ E() or (b, a) ∈ E() for all a, b ∈ V (); (L4) (a, b) ∈ E() and (b, a) ∈ E() imply that a = b for all a, b ∈ V (). We proceed as follows: after providing some known results and notations, we first give the definition of Cayley graphs over Green ∗ relations of an abundant semigroup. Then we obtain some basic properties of Cayley graphs over Green ∗ relations. Finally, we prove necessary and sufficient conditions for a Cayley graph over Green ∗ relations of an abundant semigroup to be a linear graph and a complete graph (a graph is complete, if E() = {(x, y) | x, y ∈ V ()}), respectively.
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2 Preliminaries Throughout this paper, we follow the notions and notations in [1–3]. Lemma 2.1 [1] Let S be a semigroup and let a, b ∈ S, e = e2 ∈ S. Then 1. aL∗ b [aR∗ b] ⇐⇒ (∀x, y ∈ S 1 ) ax = ay [xa = ya] if and only if bx = by [xb = yb]; 2. aL∗ e [aR∗ e] ⇐⇒ (∀x, y ∈ S 1 ) ax = ay [xa = ya] implies ex = ey [xe = ye] and ae = a [ea = a]. From Lemma 2.1, it is easily seen that L∗ is a right congruence (an equivalence relation is said to be a right [resp., left] congruence if it is right [resp., left] compatible), while R∗ is a left congruence. Lemma 2.2 [1] Let S be a semigroup and let a, b ∈ S. Then b ∈ J ∗ (a) if and only if there are a0 , a1 , . . . , an ∈ S, x1 , . . . , xn , y1 , . . . , yn ∈ S 1 such that a = a0 , b = bn , and (ai , xi ai−1 yi ) ∈ D∗ for i = 1, . . . , n. Lemma 2.3 [1] Let S be a superabundant semigroup and let a, b ∈ S. Then J ∗ (a) ∩J ∗ (b) = J ∗ (ab) and J ∗ (ab) = J ∗ (ba). Recall a relation R on a semigroup S is inversely compatible, if (a, b), (c, d) ∈ R implies that (ca, bd) ∈ R for all a, b, c, d ∈ S. It is easily seen that the relation J ∗ on a superabundant semigroup is inversely compatible from Lemma 2.3.
3 Definitions and Properties The aim of this section is to give a definition of Cayley graphs over Green ∗ relations of abundant semigroups. Further, we also consider some properties of Cayley graphs over Green ∗ relations. Definition 3.1 Let S be an abundant semigroup and K be one of the relations L∗ , R∗ , H∗ , D∗ and J ∗ on S. The Cayley graph Cay(S, K) of S relative to K is defined as the graph with vertex set S and edge set E(Cay(S, K)) consisting of those ordered pairs (a, b) such that xay = b for some (x, y) ∈ K. We call the Cayley graphs defined in Definition 3.1 the Cayley graphs over Green ∗ relations. Example 3.1 Let S be the four-element set whose multiplication table is given below: Then S is a semigroup since the associativity is checked easily. On the other hand, it is readily verified that the L∗ −classes of S are {0}, {e} and { f , a}, and that the R∗ −classes of S are {0}, { f } and {e, a}. Since 0, e and f are idempotents, we have that S is abundant. In addition, it is not difficult to check that the H∗ −classes of S are
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0 e f a
0
e
f
a
0 0 0 0
0 e 0 0
0 a f a
0 a 0 0
{0}, {e}, { f } and {a}, while D∗ −classes of S are {0} and {e, f , a}. In fact, we have D∗ = J ∗ on S. Hence, we have E(Cay(S, L∗ )) = {(0, 0), (e, 0), ( f , 0), (a, 0), (e, e), ( f , f ), ( f , a)}, E(Cay(S, R∗ )) = {(0, 0), (e, 0), ( f , 0), (a, 0), (e, e), ( f , f ), (e, a)}, E(Cay(S, H∗ )) = {(0, 0), (e, 0), ( f , 0), (a, 0), (e, e), ( f , f )}, E(Cay(S, D∗ )) = {(0, 0), (e, 0), ( f , 0), (a, 0), (e, e), ( f , f ), (e, a), ( f , a), (a, a)} and E(Cay(S, J ∗ )) = E(Cay(S, D∗ )). Obviously, Cay(S, K) is transmissible, but it is not a complete graph. Proposition 3.1 Let S be an abundant semigroup and let a, b ∈ S. If (a, b) ∈ E(Cay(S, K)), then J ∗ (b) ⊆ J ∗ (a). Proof Let c ∈ S and c ∈ J ∗ (b). Then, by Lemma 2.2, there are b0 , b1 , . . . , bn ∈ S, and x1 , . . . , xn , y1 , . . . , yn ∈ S 1 such that b = b0 , c = bn , and (bi , xi bi−1 yi ) ∈ D∗ for i = 1, . . . , n. That is, (b1 , x1 b0 y1 ) ∈ D∗ , (b2 , x2 b1 y2 ) ∈ D∗ , . . . , (bn , xn bn−1 yn ) ∈ D∗ . If (a, b) ∈ E(Cay(S, K)), then, by Definition 3.1, there are x, y ∈ S such that xay = b and (x, y) ∈ K. Put a0 = a, a1 = b0 = b, a2 = b1 , . . . , an+1 = bn = c, s1 = x, s2 = x1 , . . . , sn+1 = xn , t1 = y, t2 = y1 , . . . , tn+1 = yn . Then (a1 , s1 a0 t1 ) ∈ D∗ , (a2 , s2 a1 t2 ) ∈ D∗ , . . . , (an+1 , sn+1 an tn+1 ) ∈ D∗ . By Lemma 2.2, c = an+1 ∈ J ∗ (a). That is, J ∗ (b) ⊆ J ∗ (a). Definition 3.2 Let S be an abundant semigroup and K be one of the relations L∗ , R∗ , H∗ , D∗ and J ∗ on S. Define the subsets of S as follows: 1 = {xay | (x, y) ∈ K} ∪ {a} K[a] = {xay | (x, y) ∈ K } and K[a]
Proposition 3.2 Let S be an abundant semigroup and let a, b ∈ S. Then the following statements are true: 1 ⊆ K1 and a = b, then (a, b) ∈ E(Cay(S, K)); 1. if K[b] [a] 2. if D∗ [b] ⊆ D∗ [a] , then (a, b) ∈ E(Cay(S, D∗ ));
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3. if J ∗ [b] ⊆ J ∗ [a] , then (a, b) ∈ E(Cay(S, J ∗ )). Proof 1. It is clear. 2. Let a ∈ S. Then, by Lemma 2.1(2), a = a + aa ∗ and a + R∗ aL∗ a ∗ since S is abundant. That is, (a + , a ∗ ) ∈ D∗ . Hence a ∈ D∗ [a] . If D∗ [b] ⊆ D∗ [a] , then b ∈ D∗ [b] ⊆ D∗ [a] . By Definition 3.2, there are x, y ∈ S such that xay = b and (x, y) ∈ D∗ . This implies that (a, b) ∈ E(Cay(S, D∗ )). 3. By (2), a ∈ D∗ [a] for all a ∈ S. Therefore, for all a ∈ S, a ∈ J ∗ [a] since D∗ [a] ⊆ J ∗ [a] . The rest of the proof is similar to the proof of (2). Remark 3.1 1. In Proposition 3.2(1), the condition a = b can not be removed. For example, in our Example 3.1, (a, a) ∈ / E(Cay(S, L∗ )). On the other hand, the 1 1 condition K[b] ⊆ K[a] can not be weakened to K[b] ⊆ K[a] . For example, in our / E(Cay(S, L∗ )). Example 3.1, L∗ [a] = {0} ⊆ {0, e} = L∗ [e] . However, (e, a) ∈ 1 2. In general, for an arbitrary element a of an abundant semigroup S, K[a] = K[a] from Definition 3.2. However, by the proofs of Proposition 3.2(2) and (3), it is easily seen that D∗ [a] = D∗ 1[a] and J ∗ [a] = J ∗ 1[a] for all a ∈ S. Corollary 3.3 Let S be an abundant semigroup. Then the following statements are true: 1. J ∗ [b] ⊆ J ∗ [a] implies that J ∗ (b) ⊆ J ∗ (a) for all a, b ∈ S; 2. (a, a) ∈ E(Cay(S, D∗ )) and (a, a) ∈ E(Cay(S, J ∗ )) for all a ∈ S. Proof 1. It follows from Propositions 3.2(3) and 3.1. 2. It follows from Proposition 3.2(2) and (3).
4 Main Results In this section, we consider some conditions for Cayley graphs over Green ∗ relations of abundant semigroups to be transmissible, linear and complete, respectively. We start with the following lemma. Lemma 4.1 Let S be an abundant semigroup. Then the following statements are true: 1. If S is K inversely compatible, then Cay(S, K) is transmissible; 2. If S is K inversely compatible, then (a, a) ∈ E(Cay(S, K)) for all a ∈ S; 1 for all a ∈ S. 3. If S is K inversely compatible, then K[a] = K[a] Proof 1. Let a, b, c ∈ S such that (a, b), (b, c) ∈ E(Cay(S, K)). Then there are x1 , y1 , x2 , y2 ∈ S with x1 ay1 = b, x2 by2 = c, (x1 , y1 ) ∈ K and (x2 , y2 ) ∈ K. Hence c = (x2 x1 )a(y1 y2 ). Note that S is K inversely compatible. We have (x2 x1 , y1 y2 ) ∈ K. This implies that (a, c) ∈ E(Cay(S, K)). That is, Cay(S, K) is transmissible. 2. Note that S is abundant. We have that aL∗ a ∗ and aR∗ a + for all a ∈ S. It means that a = a + a = aa ∗ from Lemma 2.1(2). Consider the following four possible cases:
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(i) If K = L∗ , then aL∗ a ∗ and a + L∗ a + imply a = a + aL∗ a ∗ a + for all a ∈ S since S is L∗ inversely compatible. Hence a ∗ L∗ a ∗ a + . By Lemma 2.1(2), a ∗ a + = a ∗ a + a ∗ . Thus a ∗ a + = a ∗ a + a + = a ∗ a + a ∗ a + = (a ∗ a + )2 . That is, a ∗ a + is the idempotent of S. But aL∗ a ∗ a + . By Lemma 2.1(2), we have a = a + a = a + (aa ∗ a + ) = a + aa + , where a + L∗ a + . Therefore (a, a) ∈ E(Cay(S, L∗ )) for all a ∈ S; (ii) If K = R∗ , then it is easily seen that (a, a) ∈ E(Cay(S, R∗ )) for all a ∈ S from the dual of (i); (iii) If K = H∗ , then a + H∗ a + and a ∗ H∗ a ∗ imply a ∗ a + H∗ a + a ∗ for all a ∈ S since S is H∗ inversely compatible. Hence a ∗ a + L∗ a + a ∗ since H∗ ⊆ L∗ . Similarly, aH∗ a and a + H∗ a + imply a = a + aH∗ aa + for all a ∈ S since S is H∗ inversely compatible. That is, aL∗ aa + since H∗ ⊆ L∗ . Note that L∗ is a right congruence. We have aL∗ a ∗ , which implies aa + L∗ a ∗ a + . Thus aL∗ a ∗ a + , and so a ∗ L∗ a ∗ a + . By the proof of (i), we can see that a ∗ a + is the idempotent of S and a = a + a = a + (aa ∗ a + ) = a + aa + , where a + H∗ a + . This gives that (a, a) ∈ E(Cay(S, H∗ )) for all a ∈ S; (iv) If K = D∗ or K = J ∗ , then the result is clear by Corollary 3.3(2). Summarizing the above cases, we conclude that (2) holds. 3. By the proof of (2), it is easy to see that for all a ∈ S there exist x, y ∈ S such that xay = a and (x, y) ∈ K. That is, a ∈ K[a] for all a ∈ S. Therefore (3) holds. Corollary 4.2 Let S be a superabundant semigroup. Then Cay(S, J ∗ ) is transmissible. Proof It follows from Lemma 4.1(1) since a superabundant semigroup S is J ∗ inversely compatible. Proposition 4.3 Let S be an abundant semigroup and let a, b ∈ S. If S is K inversely 1 ⊆ K1 . compatible, then (a, b) ∈ E(Cay(S, K)) if and only if K[b] [a] Proof The converse part follows from Proposition 3.2(1), Lemmas 4.1(2) and 4.1(3). We only prove the necessity part. Let a, b ∈ S and (a, b) ∈ E(Cay(S, K)). Then there are x, y ∈ S such that xay = b and (x, y) ∈ K. Let c ∈ K[b] . Then there are s, t ∈ S such that sbt = c and (s, t) ∈ K. Hence c = sxayt. Note that S is K inversely compatible. We have (sx, yt) ∈ K. This gives c ∈ K[a] . That is K[b] ⊆ K[a] . But 1 ⊆ K1 . b ∈ K[a] . Therefore K[b] [a] Note that an arbitrary superabundant semigroup S is J ∗ inversely compatible. Hence, the following corollary is now immediate by Proposition 4.3. Corollary 4.4 Let S be a superabundant semigroup and let a, b ∈ S. Then (a, b) ∈ E(Cay(S, J ∗ )) if and only if J ∗ 1[b] ⊆ J ∗ 1[a] . ∗
Next, we characterize some abundant semigroups whose Cayley graphs over Green relations are linear.
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Theorem 4.5 Let S be an abundant semigroup in which S is K inversely compatible. Then Cay(S, K) is a linear graph if and only if there is a linear order on S such 1 = {b ∈ S | b a} for all a ∈ S. that K[a] Proof “ ⇒ . Suppose that Cay(S, K) is linear. Then we define an order on S by 1 ⊆ K1 for a, b ∈ S. Obviously, is reflexive and the rule a b if and only if K[a] [b] transitive. On the contrary, suppose that there exist a, b ∈ S such that a b, b a 1 ⊆ K1 , K1 ⊆ K1 and a = b. By Proposition 3.2(1), and a = b. Therefore K[a] [a] [b] [b] (b, a) ∈ E(Cay(S, K)) and (a, b) ∈ E(Cay(S, K)). Note that Cay(S, K) is linear. We have a = b. This contradiction suggests that a b and b a imply a = b for all a, b ∈ S. Therefore is a partial order. Now, we prove that is a linear order. By hypothesis, (a, b) ∈ E(Cay(S, K)) or 1 ⊆ K1 or K1 ⊆ K1 for all (b, a) ∈ E(Cay(S, K)) for all a, b ∈ S. That is, K[b] [a] [a] [b] a, b ∈ S. This gives b a or a b for all a, b ∈ S, and so is linear. On the other hand, for all a ∈ S, we have 1 = K[a] = = =
K[a] ( By Lemma 4.1 (3) ) { b ∈ S | (a, b) ∈ E(Cay(S, K))} 1 ⊆ K1 } { b ∈ S | K[b] ( By Proposition 4.3 ) [a] { b ∈ S | b a}
This completes the proof of the direct part. “ ⇐ . Note that S is K inversely compatible. By Lemma 4.1(2), for all a ∈ S we have (a, a) ∈ E(Cay(S, K)). That is, the condition (L1) of Remark 1.1 holds. 1 = {b ∈ S | b a} Now, suppose that there is a linear order on S such that K[a] 1 for all a ∈ S. Then, by Lemma 4.1(3), K[a] = K[a] = {b ∈ S | b a} for all a ∈ S since S is K inversely compatible. Let a, b ∈ S. Then a b or b a since is linear. That is, a ∈ K[b] or b ∈ K[a] . Hence xby = a or sat = b, where x, y, s, t ∈ S, (x, y) ∈ K and (s, t) ∈ K. Thus (b, a) ∈ E(Cay(S, K)) or (a, b) ∈ E(Cay(S, K)). This shows that the condition (L3) of Remark 1.1 holds. Furthermore, if a, b ∈ S 1 such that (a, b) ∈ E(Cay(S, K)) and (b, a) ∈ E(Cay(S, K)), then b ∈ K[a] = K[a] 1 . So by the assumption, b a and a b. But is linear. We and a ∈ K[b] = K[b] have a = b. Therefore, the condition (L4) of Remark 1.1 holds. Finally, Cay(S, K) is transmissible from Lemma 4.1(1). That is, the condition (L2) of Remark 1.1 holds. Summarizing the above arguments, we conclude that Cay(S, K) is a linear graph. The following corollary is a special case of Theorem 4.5. Corollary 4.6 Let S be a superabundant semigroup. Then Cay(S, J ∗ ) is a linear graph if and only if there is a linear order on S such that J ∗ 1[a] = {b ∈ S | b a} for all a ∈ S. In the rest of the paper, we consider the necessary and sufficient conditions for a Cayley graph over Green ∗ relations of an abundant semigroup to be a complete graph with a loop incident to each vertex.
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Theorem 4.7 Let S be an abundant semigroup in which S is K inversely compatible. Then Cay(S, K) is a complete graph with a loop incident to each vertex if and only if 1 = K1 for all a, b ∈ S. K[a] [b] Proof “ ⇒ Suppose that Cay(S, K) is a complete graph with a loop incident to each vertex. Then (a, a) ∈ E(Cay(S, K)) for all a ∈ S. On the other hand, let a, b ∈ S such that a = b. Then (a, b) ∈ E(Cay(S, K)) and (b, a) ∈ E(Cay(S, K)). By 1 = K1 . Proposition 4.3, K[a] [b] 1 = K1 implies that (a, b) ∈ “ ⇐ Let a, b ∈ S such that a = b. Then K[a] [b] E(Cay(S, K)) and (b, a) ∈ E(Cay(S, K)) from Proposition 4.3. On the other hand, (a, a) ∈ E(Cay(S, K)) for all a ∈ S from Lemma 4.1(2). Therefore, Cay(S, K) is a complete graph with a loop incident to each vertex. This completes the proof. Corollary 4.8 Let S be a superabundant semigroup. Then the following statements are true: 1. Cay(S, J ∗ ) is a complete graph with a loop incident to each vertex if and only if J ∗ 1[a] = J ∗ 1[b] for all a, b ∈ S; 2. if Cay(S, J ∗ ) is a complete graph with a loop incident to each vertex, then S is J ∗ simple. Proof 1. It follows from Theorem 4.7. 2. It follows from (1) and Corollary 3.3(1).
Remark 4.1 In Corollary 4.8(2), the converse part is not true. For example, let S = {2n | n ∈ N ∪ {0}}, where N is the set of all positive integers. Then it is easily seen that S is a superabundant semigroup such that L∗ = R∗ = H∗ = D∗ = J ∗ = S × S on S, where 1 is the idempotent of S. That is, S is J ∗ simple. However, (2, 1) ∈ / E(Cay(S, J ∗ )) since x · 2 · y = 1 for all x, y ∈ S. Therefore, Cay(S, J ∗ ) is not complete. Acknowledgements The authors are very grateful to the referees for their valuable suggestions which lead to an improvement of this paper. This work is supported by the National Science Foundation of China (Nos. 11261018, 11961026), the Natural Science Foundation of Jiangxi Province (Nos. 20181BAB201002, 20171BAB201009).
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Cayley Graphs Over Green ∗ Relations of Abundant Semigroups Chunhua Li1 · Baogen Xu1 · Huawei Huang2 Received: 13 April 2019 / Revised: 21 September 2019 © Springer Japan KK, part of Springer Nature 2019
Abstract In this paper, we first introduce the concept of Cayley graphs over Green ∗ relations of abundant semigroups by using Green ∗ relations. After obtaining some basic properties, we get some conditions for Cayley graphs over Green ∗ relations of abundant semigroups to be transmissible. Finally, we give necessary and sufficient conditions for a Cayley graph over Green ∗ relations of an abundant semigroup to be linear and complete, respectively. Keywords Abundant semigroups · Cayley graphs over Green ∗ relations · Transmissible · Linear graphs · Complete graphs Mathematics Subject Classification 06F05 · 20M10
1 Introduction The Cayley graph of a group was introduced by Arthur Cayley in 1878 to interpret the notion of abstract groups. Since then, a lot of interesting results have been obtained by using the concept of Cayley graphs of groups (see, [2,16,17]). Motivated by the rich results on Cayley graphs of groups, many authors studied Cayley graphs of semigroups (see, [7,8,13,15]). Let S be a semigroup and T be a nonempty subset of S. The Cayley graph Cay(S, T ) of S relative to T is defined as the graph with vertex set S and edge
B
Chunhua Li [email protected] Baogen Xu [email protected] Huawei Huang [email protected]
1
School of Science, East China Jiaotong University, Nanchang 330013, Jiangxi, China
2
School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, Guizhou, China
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set E(Cay(S, T )) consisting of those ordered pair (a, b), where a = b and as = b for some s ∈ T (see, [3,5,12]). As a generalization of Cayley graphs of groups, the Cayley graphs of semigroups are closely related to the finite state automata and have many applications (see, [3–7,9–11,14]). Note that some Cayley graphs of semigroups may be defined by left actions (see, [5,13]). However, for a given semigroup S and a fixed nonempty subset T of S, the Cayley graph Cay(S, T ) defined by right and left actions may not be same. In order to get more applications, this paper gives a new definition of the Cayley graph of a semigroup by using generalized Green relations. Recall from Fountain [1] that on a semigroup S the relation L∗ is defined by the rule (a, b) ∈ L∗ in S if and only if there is a semigroup T containing S as a subsemigroup, such that (a, b) ∈ L in T for all a, b ∈ S. The relation R∗ is defined dually. The intersection of L∗ and R∗ is denoted by H∗ and we denote the smallest equivalence relation containing both L∗ and R∗ by D∗ . A semigroup S is abundant if each L∗ class and each R∗ class of S contains an idempotent, and superabundant if each H∗ class of S contains an idempotent. As usual, L∗ a and R∗ a denote L∗ class and R∗ class containing a, respectively. In particular, a ∗ [a + ] denotes a typical idempotent in L∗ a [R∗ a ] if L∗ a [R∗ a ] contains an idempotent. A left ideal I of a semigroup S is a left ∗ ideal if L∗ a ⊆ I for all a ∈ I . Dually, we can define a right ∗ ideal. An ideal I of a semigroup S is called a ∗ ideal if it is both a left ∗ ideal and a right ∗ ideal. The smallest ∗ ideal containing the element a of the semigroup S will be denoted by J ∗ (a). Two elements a, b of a semigroup S have J ∗ − related if and only if J ∗ (a) = J ∗ (b). Generally, we call L∗ , R∗ , H∗ , D∗ and J ∗ Green ∗ relations. For an arbitrary semigroup S, we have D∗ ⊆ J ∗ . A semigroup S is J ∗ −simple if it has no proper ∗ ideal. Clearly a semigroup S is J ∗ −simple if and only if J ∗ is the universal relation on S. As usual, for a graph , V () and E() stand for its vertex set and edge set, respectively. Recall that a partial order set (X , ≤) is a linear order, if a ≤ b or b ≤ a for all a, b ∈ X . In particular, the relation graph of a linear order is a linear graph. It is easy to check that a graph is linear if it satisfies the following conditions (L1), (L2), (L3) and (L4) of Remark 1.1. Remark 1.1 (L1) has loops, not multiple edges; (L2) is transmissible (a graph is transmissible, if for all a, b, c ∈ V (), (a, b), (b, c) ∈ E() implies (a, c) ∈ E()); (L3) (a, b) ∈ E() or (b, a) ∈ E() for all a, b ∈ V (); (L4) (a, b) ∈ E() and (b, a) ∈ E() imply that a = b for all a, b ∈ V (). We proceed as follows: after providing some known results and notations, we first give the definition of Cayley graphs over Green ∗ relations of an abundant semigroup. Then we obtain some basic properties of Cayley graphs over Green ∗ relations. Finally, we prove necessary and sufficient conditions for a Cayley graph over Green ∗ relations of an abundant semigroup to be a linear graph and a complete graph (a graph is complete, if E() = {(x, y) | x, y ∈ V ()}), respectively.
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2 Preliminaries Throughout this paper, we follow the notions and notations in [1–3]. Lemma 2.1 [1] Let S be a semigroup and let a, b ∈ S, e = e2 ∈ S. Then 1. aL∗ b [aR∗ b] ⇐⇒ (∀x, y ∈ S 1 ) ax = ay [xa = ya] if and only if bx = by [xb = yb]; 2. aL∗ e [aR∗ e] ⇐⇒ (∀x, y ∈ S 1 ) ax = ay [xa = ya] implies ex = ey [xe = ye] and ae = a [ea = a]. From Lemma 2.1, it is easily seen that L∗ is a right congruence (an equivalence relation is said to be a right [resp., left] congruence if it is right [resp., left] compatible), while R∗ is a left congruence. Lemma 2.2 [1] Let S be a semigroup and let a, b ∈ S. Then b ∈ J ∗ (a) if and only if there are a0 , a1 , . . . , an ∈ S, x1 , . . . , xn , y1 , . . . , yn ∈ S 1 such that a = a0 , b = bn , and (ai , xi ai−1 yi ) ∈ D∗ for i = 1, . . . , n. Lemma 2.3 [1] Let S be a superabundant semigroup and let a, b ∈ S. Then J ∗ (a) ∩J ∗ (b) = J ∗ (ab) and J ∗ (ab) = J ∗ (ba). Recall a relation R on a semigroup S is inversely compatible, if (a, b), (c, d) ∈ R implies that (ca, bd) ∈ R for all a, b, c, d ∈ S. It is easily seen that the relation J ∗ on a superabundant semigroup is inversely compatible from Lemma 2.3.
3 Definitions and Properties The aim of this section is to give a definition of Cayley graphs over Green ∗ relations of abundant semigroups. Further, we also consider some properties of Cayley graphs over Green ∗ relations. Definition 3.1 Let S be an abundant semigroup and K be one of the relations L∗ , R∗ , H∗ , D∗ and J ∗ on S. The Cayley graph Cay(S, K) of S relative to K is defined as the graph with vertex set S and edge set E(Cay(S, K)) consisting of those ordered pairs (a, b) such that xay = b for some (x, y) ∈ K. We call the Cayley graphs defined in Definition 3.1 the Cayley graphs over Green ∗ relations. Example 3.1 Let S be the four-element set whose multiplication table is given below: Then S is a semigroup since the associativity is checked easily. On the other hand, it is readily verified that the L∗ −classes of S are {0}, {e} and { f , a}, and that the R∗ −classes of S are {0}, { f } and {e, a}. Since 0, e and f are idempotents, we have that S is abundant. In addition, it is not difficult to check that the H∗ −classes of S are
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0 e f a
0
e
f
a
0 0 0 0
0 e 0 0
0 a f a
0 a 0 0
{0}, {e}, { f } and {a}, while D∗ −classes of S are {0} and {e, f , a}. In fact, we have D∗ = J ∗ on S. Hence, we have E(Cay(S, L∗ )) = {(0, 0), (e, 0), ( f , 0), (a, 0), (e, e), ( f , f ), ( f , a)}, E(Cay(S, R∗ )) = {(0, 0), (e, 0), ( f , 0), (a, 0), (e, e), ( f , f ), (e, a)}, E(Cay(S, H∗ )) = {(0, 0), (e, 0), ( f , 0), (a, 0), (e, e), ( f , f )}, E(Cay(S, D∗ )) = {(0, 0), (e, 0), ( f , 0), (a, 0), (e, e), ( f , f ), (e, a), ( f , a), (a, a)} and E(Cay(S, J ∗ )) = E(Cay(S, D∗ )). Obviously, Cay(S, K) is transmissible, but it is not a complete graph. Proposition 3.1 Let S be an abundant semigroup and let a, b ∈ S. If (a, b) ∈ E(Cay(S, K)), then J ∗ (b) ⊆ J ∗ (a). Proof Let c ∈ S and c ∈ J ∗ (b). Then, by Lemma 2.2, there are b0 , b1 , . . . , bn ∈ S, and x1 , . . . , xn , y1 , . . . , yn ∈ S 1 such that b = b0 , c = bn , and (bi , xi bi−1 yi ) ∈ D∗ for i = 1, . . . , n. That is, (b1 , x1 b0 y1 ) ∈ D∗ , (b2 , x2 b1 y2 ) ∈ D∗ , . . . , (bn , xn bn−1 yn ) ∈ D∗ . If (a, b) ∈ E(Cay(S, K)), then, by Definition 3.1, there are x, y ∈ S such that xay = b and (x, y) ∈ K. Put a0 = a, a1 = b0 = b, a2 = b1 , . . . , an+1 = bn = c, s1 = x, s2 = x1 , . . . , sn+1 = xn , t1 = y, t2 = y1 , . . . , tn+1 = yn . Then (a1 , s1 a0 t1 ) ∈ D∗ , (a2 , s2 a1 t2 ) ∈ D∗ , . . . , (an+1 , sn+1 an tn+1 ) ∈ D∗ . By Lemma 2.2, c = an+1 ∈ J ∗ (a). That is, J ∗ (b) ⊆ J ∗ (a). Definition 3.2 Let S be an abundant semigroup and K be one of the relations L∗ , R∗ , H∗ , D∗ and J ∗ on S. Define the subsets of S as follows: 1 = {xay | (x, y) ∈ K} ∪ {a} K[a] = {xay | (x, y) ∈ K } and K[a]
Proposition 3.2 Let S be an abundant semigroup and let a, b ∈ S. Then the following statements are true: 1 ⊆ K1 and a = b, then (a, b) ∈ E(Cay(S, K)); 1. if K[b] [a] 2. if D∗ [b] ⊆ D∗ [a] , then (a, b) ∈ E(Cay(S, D∗ ));
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3. if J ∗ [b] ⊆ J ∗ [a] , then (a, b) ∈ E(Cay(S, J ∗ )). Proof 1. It is clear. 2. Let a ∈ S. Then, by Lemma 2.1(2), a = a + aa ∗ and a + R∗ aL∗ a ∗ since S is abundant. That is, (a + , a ∗ ) ∈ D∗ . Hence a ∈ D∗ [a] . If D∗ [b] ⊆ D∗ [a] , then b ∈ D∗ [b] ⊆ D∗ [a] . By Definition 3.2, there are x, y ∈ S such that xay = b and (x, y) ∈ D∗ . This implies that (a, b) ∈ E(Cay(S, D∗ )). 3. By (2), a ∈ D∗ [a] for all a ∈ S. Therefore, for all a ∈ S, a ∈ J ∗ [a] since D∗ [a] ⊆ J ∗ [a] . The rest of the proof is similar to the proof of (2). Remark 3.1 1. In Proposition 3.2(1), the condition a = b can not be removed. For example, in our Example 3.1, (a, a) ∈ / E(Cay(S, L∗ )). On the other hand, the 1 1 condition K[b] ⊆ K[a] can not be weakened to K[b] ⊆ K[a] . For example, in our / E(Cay(S, L∗ )). Example 3.1, L∗ [a] = {0} ⊆ {0, e} = L∗ [e] . However, (e, a) ∈ 1 2. In general, for an arbitrary element a of an abundant semigroup S, K[a] = K[a] from Definition 3.2. However, by the proofs of Proposition 3.2(2) and (3), it is easily seen that D∗ [a] = D∗ 1[a] and J ∗ [a] = J ∗ 1[a] for all a ∈ S. Corollary 3.3 Let S be an abundant semigroup. Then the following statements are true: 1. J ∗ [b] ⊆ J ∗ [a] implies that J ∗ (b) ⊆ J ∗ (a) for all a, b ∈ S; 2. (a, a) ∈ E(Cay(S, D∗ )) and (a, a) ∈ E(Cay(S, J ∗ )) for all a ∈ S. Proof 1. It follows from Propositions 3.2(3) and 3.1. 2. It follows from Proposition 3.2(2) and (3).
4 Main Results In this section, we consider some conditions for Cayley graphs over Green ∗ relations of abundant semigroups to be transmissible, linear and complete, respectively. We start with the following lemma. Lemma 4.1 Let S be an abundant semigroup. Then the following statements are true: 1. If S is K inversely compatible, then Cay(S, K) is transmissible; 2. If S is K inversely compatible, then (a, a) ∈ E(Cay(S, K)) for all a ∈ S; 1 for all a ∈ S. 3. If S is K inversely compatible, then K[a] = K[a] Proof 1. Let a, b, c ∈ S such that (a, b), (b, c) ∈ E(Cay(S, K)). Then there are x1 , y1 , x2 , y2 ∈ S with x1 ay1 = b, x2 by2 = c, (x1 , y1 ) ∈ K and (x2 , y2 ) ∈ K. Hence c = (x2 x1 )a(y1 y2 ). Note that S is K inversely compatible. We have (x2 x1 , y1 y2 ) ∈ K. This implies that (a, c) ∈ E(Cay(S, K)). That is, Cay(S, K) is transmissible. 2. Note that S is abundant. We have that aL∗ a ∗ and aR∗ a + for all a ∈ S. It means that a = a + a = aa ∗ from Lemma 2.1(2). Consider the following four possible cases:
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(i) If K = L∗ , then aL∗ a ∗ and a + L∗ a + imply a = a + aL∗ a ∗ a + for all a ∈ S since S is L∗ inversely compatible. Hence a ∗ L∗ a ∗ a + . By Lemma 2.1(2), a ∗ a + = a ∗ a + a ∗ . Thus a ∗ a + = a ∗ a + a + = a ∗ a + a ∗ a + = (a ∗ a + )2 . That is, a ∗ a + is the idempotent of S. But aL∗ a ∗ a + . By Lemma 2.1(2), we have a = a + a = a + (aa ∗ a + ) = a + aa + , where a + L∗ a + . Therefore (a, a) ∈ E(Cay(S, L∗ )) for all a ∈ S; (ii) If K = R∗ , then it is easily seen that (a, a) ∈ E(Cay(S, R∗ )) for all a ∈ S from the dual of (i); (iii) If K = H∗ , then a + H∗ a + and a ∗ H∗ a ∗ imply a ∗ a + H∗ a + a ∗ for all a ∈ S since S is H∗ inversely compatible. Hence a ∗ a + L∗ a + a ∗ since H∗ ⊆ L∗ . Similarly, aH∗ a and a + H∗ a + imply a = a + aH∗ aa + for all a ∈ S since S is H∗ inversely compatible. That is, aL∗ aa + since H∗ ⊆ L∗ . Note that L∗ is a right congruence. We have aL∗ a ∗ , which implies aa + L∗ a ∗ a + . Thus aL∗ a ∗ a + , and so a ∗ L∗ a ∗ a + . By the proof of (i), we can see that a ∗ a + is the idempotent of S and a = a + a = a + (aa ∗ a + ) = a + aa + , where a + H∗ a + . This gives that (a, a) ∈ E(Cay(S, H∗ )) for all a ∈ S; (iv) If K = D∗ or K = J ∗ , then the result is clear by Corollary 3.3(2). Summarizing the above cases, we conclude that (2) holds. 3. By the proof of (2), it is easy to see that for all a ∈ S there exist x, y ∈ S such that xay = a and (x, y) ∈ K. That is, a ∈ K[a] for all a ∈ S. Therefore (3) holds. Corollary 4.2 Let S be a superabundant semigroup. Then Cay(S, J ∗ ) is transmissible. Proof It follows from Lemma 4.1(1) since a superabundant semigroup S is J ∗ inversely compatible. Proposition 4.3 Let S be an abundant semigroup and let a, b ∈ S. If S is K inversely 1 ⊆ K1 . compatible, then (a, b) ∈ E(Cay(S, K)) if and only if K[b] [a] Proof The converse part follows from Proposition 3.2(1), Lemmas 4.1(2) and 4.1(3). We only prove the necessity part. Let a, b ∈ S and (a, b) ∈ E(Cay(S, K)). Then there are x, y ∈ S such that xay = b and (x, y) ∈ K. Let c ∈ K[b] . Then there are s, t ∈ S such that sbt = c and (s, t) ∈ K. Hence c = sxayt. Note that S is K inversely compatible. We have (sx, yt) ∈ K. This gives c ∈ K[a] . That is K[b] ⊆ K[a] . But 1 ⊆ K1 . b ∈ K[a] . Therefore K[b] [a] Note that an arbitrary superabundant semigroup S is J ∗ inversely compatible. Hence, the following corollary is now immediate by Proposition 4.3. Corollary 4.4 Let S be a superabundant semigroup and let a, b ∈ S. Then (a, b) ∈ E(Cay(S, J ∗ )) if and only if J ∗ 1[b] ⊆ J ∗ 1[a] . ∗
Next, we characterize some abundant semigroups whose Cayley graphs over Green relations are linear.
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Theorem 4.5 Let S be an abundant semigroup in which S is K inversely compatible. Then Cay(S, K) is a linear graph if and only if there is a linear order on S such 1 = {b ∈ S | b a} for all a ∈ S. that K[a] Proof “ ⇒ . Suppose that Cay(S, K) is linear. Then we define an order on S by 1 ⊆ K1 for a, b ∈ S. Obviously, is reflexive and the rule a b if and only if K[a] [b] transitive. On the contrary, suppose that there exist a, b ∈ S such that a b, b a 1 ⊆ K1 , K1 ⊆ K1 and a = b. By Proposition 3.2(1), and a = b. Therefore K[a] [a] [b] [b] (b, a) ∈ E(Cay(S, K)) and (a, b) ∈ E(Cay(S, K)). Note that Cay(S, K) is linear. We have a = b. This contradiction suggests that a b and b a imply a = b for all a, b ∈ S. Therefore is a partial order. Now, we prove that is a linear order. By hypothesis, (a, b) ∈ E(Cay(S, K)) or 1 ⊆ K1 or K1 ⊆ K1 for all (b, a) ∈ E(Cay(S, K)) for all a, b ∈ S. That is, K[b] [a] [a] [b] a, b ∈ S. This gives b a or a b for all a, b ∈ S, and so is linear. On the other hand, for all a ∈ S, we have 1 = K[a] = = =
K[a] ( By Lemma 4.1 (3) ) { b ∈ S | (a, b) ∈ E(Cay(S, K))} 1 ⊆ K1 } { b ∈ S | K[b] ( By Proposition 4.3 ) [a] { b ∈ S | b a}
This completes the proof of the direct part. “ ⇐ . Note that S is K inversely compatible. By Lemma 4.1(2), for all a ∈ S we have (a, a) ∈ E(Cay(S, K)). That is, the condition (L1) of Remark 1.1 holds. 1 = {b ∈ S | b a} Now, suppose that there is a linear order on S such that K[a] 1 for all a ∈ S. Then, by Lemma 4.1(3), K[a] = K[a] = {b ∈ S | b a} for all a ∈ S since S is K inversely compatible. Let a, b ∈ S. Then a b or b a since is linear. That is, a ∈ K[b] or b ∈ K[a] . Hence xby = a or sat = b, where x, y, s, t ∈ S, (x, y) ∈ K and (s, t) ∈ K. Thus (b, a) ∈ E(Cay(S, K)) or (a, b) ∈ E(Cay(S, K)). This shows that the condition (L3) of Remark 1.1 holds. Furthermore, if a, b ∈ S 1 such that (a, b) ∈ E(Cay(S, K)) and (b, a) ∈ E(Cay(S, K)), then b ∈ K[a] = K[a] 1 . So by the assumption, b a and a b. But is linear. We and a ∈ K[b] = K[b] have a = b. Therefore, the condition (L4) of Remark 1.1 holds. Finally, Cay(S, K) is transmissible from Lemma 4.1(1). That is, the condition (L2) of Remark 1.1 holds. Summarizing the above arguments, we conclude that Cay(S, K) is a linear graph. The following corollary is a special case of Theorem 4.5. Corollary 4.6 Let S be a superabundant semigroup. Then Cay(S, J ∗ ) is a linear graph if and only if there is a linear order on S such that J ∗ 1[a] = {b ∈ S | b a} for all a ∈ S. In the rest of the paper, we consider the necessary and sufficient conditions for a Cayley graph over Green ∗ relations of an abundant semigroup to be a complete graph with a loop incident to each vertex.
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Theorem 4.7 Let S be an abundant semigroup in which S is K inversely compatible. Then Cay(S, K) is a complete graph with a loop incident to each vertex if and only if 1 = K1 for all a, b ∈ S. K[a] [b] Proof “ ⇒ Suppose that Cay(S, K) is a complete graph with a loop incident to each vertex. Then (a, a) ∈ E(Cay(S, K)) for all a ∈ S. On the other hand, let a, b ∈ S such that a = b. Then (a, b) ∈ E(Cay(S, K)) and (b, a) ∈ E(Cay(S, K)). By 1 = K1 . Proposition 4.3, K[a] [b] 1 = K1 implies that (a, b) ∈ “ ⇐ Let a, b ∈ S such that a = b. Then K[a] [b] E(Cay(S, K)) and (b, a) ∈ E(Cay(S, K)) from Proposition 4.3. On the other hand, (a, a) ∈ E(Cay(S, K)) for all a ∈ S from Lemma 4.1(2). Therefore, Cay(S, K) is a complete graph with a loop incident to each vertex. This completes the proof. Corollary 4.8 Let S be a superabundant semigroup. Then the following statements are true: 1. Cay(S, J ∗ ) is a complete graph with a loop incident to each vertex if and only if J ∗ 1[a] = J ∗ 1[b] for all a, b ∈ S; 2. if Cay(S, J ∗ ) is a complete graph with a loop incident to each vertex, then S is J ∗ simple. Proof 1. It follows from Theorem 4.7. 2. It follows from (1) and Corollary 3.3(1).
Remark 4.1 In Corollary 4.8(2), the converse part is not true. For example, let S = {2n | n ∈ N ∪ {0}}, where N is the set of all positive integers. Then it is easily seen that S is a superabundant semigroup such that L∗ = R∗ = H∗ = D∗ = J ∗ = S × S on S, where 1 is the idempotent of S. That is, S is J ∗ simple. However, (2, 1) ∈ / E(Cay(S, J ∗ )) since x · 2 · y = 1 for all x, y ∈ S. Therefore, Cay(S, J ∗ ) is not complete. Acknowledgements The authors are very grateful to the referees for their valuable suggestions which lead to an improvement of this paper. This work is supported by the National Science Foundation of China (Nos. 11261018, 11961026), the Natural Science Foundation of Jiangxi Province (Nos. 20181BAB201002, 20171BAB201009).
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