Semigroups on Semilattice and the Constructions of ...

Southeast Asian Bulletin of Mathematics (2014) 38: 719–730

Southeast Asian Bulletin of Mathematics c SEAMS. 2014 ⃝

Semigroups on Semilattice and the Constructions of Generalized Cryptogroups Dedicated to Prof. Polly Sy on her 60th birthday

K.P. Shum ∗ Institute of Mathematics, Yunnan University, Kunming 690091, P.R. China Email: [email protected]

Xueming Ren

†

Department of Mathematics, Xi’an University of Architecture and Technology, Xi’an 710055, P.R. China Email: [email protected]

Yanhui Wang

‡

College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, P.R. China Email: [email protected]

Received 10 April 2013 Accepted 9 May 2014 Communicated by Y.Q. Guo AMS Mathematics Subject Classification(2000): 20M20 Abstract. In this survey article, we introduce the systems of semigroups on a semilattice together with some modified Green’s relations on abundant semigroups. We demon∗ Part of the materials and content of this article have been presented and introduced by the first author in the invited talks in the recent international conferences of Mathematics held in Kazan University at Tatarstan and in Mandalay University at Myanmar in 2012 and 2013, respectively. † The author was supported by the National Natural Science Foundation of China (Grant No:11471255). ‡ The author was supported by the Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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˜ ˜ system of strate that the regular ℋ-cryptogroup can be constructed by using an ℋ𝐺 completely 𝒥˜ semigroups. Some other kind of generalized cryptogroups can also be similarly constructed. Thus, the systems of semigroups is a powerful tool for construct˜ ing the generalized cryptogroups. Our result concerning the regular ℋ-cryptogroups extends a well known result of M. Petrich on completely regular semigroups and some other known results of J. B. Fountain on superabundant semigroups. ˜ ˜ Keywords: Green ∼-relations; ℋ-cryptogroups; Homomorphisms of ℋ-abundant semigroups

1. Introduction It was first proved by A. H. Clifford [1] and M. Petrich and N.R. Reilly [14] that a regular semigroup can be expressed as a union of groups if and only if it is a semilattice of completely simple semigroups. Also, it is well-known that if all idempotents of a completely regular semigroup 𝑆 are central then the semigroup can be expressed as a strong semilattice of groups [1]. Thus, we call such kind of completely regular semigroups Clifford semigroups and such semigroups are usually regarded as a generalization of groups. Moreover, Petrich and Reilly called a completely regular semigroup a normal cryptogroup if the usual Green’s relation ℋ defined on the semigroup is a normal band congruence. In particular, a completely regular semigroup 𝑆 is said to be a normal cryptogroup if and only if 𝑆 can be expressed as a strong semilattice of completely simple semigroups (see [14] and [15]). This important result was further generalized to abundant semigroups by Fountain [4] by proving that an abundant semigroup 𝑆 is a superabundant semigroup if and only if 𝑆 is a semilattice of completely 𝒥 ∗ -simple semigroups. The structure of superabundant semigroups whose set of idempotents forms a subsemigroup has been further investigated by X.M. Ren and K.P. Shum in [17] and [18]. Recently, semigroups which are semillattices of left groups ˜ cryptogroups has have been studied in [11]. Further generalization of regular ℋbeen recently investigated by X.Z. Kong, Y. Ding and K.P. Shum in [12]. In generalizing the structure theorems of completely regular semigroups to abundant semigroups, the first step is to modify the usual Green’s relation on a given semigroup. For abundant semigroups, we use the so called Green ★relations which were first defined by F. Pastijn [13] and these kinds of generalized Green’s relations can be regarded as the usual Green’s relations on some oversemigroups of a semigroup 𝑆. For any element 𝑎, 𝑏 ∈ 𝑆, J.B. Fountain [4] defined the following notations ℒ∗ = {(𝑎, 𝑏) ∈ 𝑆 × 𝑆 : (∀𝑥, 𝑦 ∈ 𝑆 1 )𝑎𝑥 = 𝑎𝑦 ⇔ 𝑏𝑥 = 𝑏𝑦}, ℛ∗ = {(𝑎, 𝑏) ∈ 𝑆 × 𝑆 : (∀𝑥, 𝑦 ∈ 𝑆 1 )𝑥𝑎 = 𝑦𝑎 ⇔ 𝑥𝑏 = 𝑦𝑏}, ℋ∗ = ℒ∗ ∩ ℛ∗ , 𝒟∗ = ℒ∗ ∨ ℛ∗ . Later on, El-Qallali [3] further generalized the Green ∗-relations to Green ∼-

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relations by considering the construction of certain kinds of generalized abundant semigroups: ℒ˜ = {(𝑎, 𝑏) ∈ 𝑆 × 𝑆 : (∀𝑒 ∈ 𝐸(𝑆))𝑎𝑒 = 𝑎 ⇔ 𝑏𝑒 = 𝑏}, ˜ = {(𝑎, 𝑏) ∈ 𝑆 × 𝑆 : (∀𝑒 ∈ 𝐸(𝑆))𝑒𝑎 = 𝑎 ⇔ 𝑒𝑏 = 𝑏}, ℛ ˜ = ℒ˜ ∩ ℛ, ˜ 𝒟 ˜ = ℒ˜ ∨ ℛ. ˜ ℋ ˜ are equivalences on a semigroup 𝑆, howIt can be easily seen that ℒ˜ and ℛ ˜ ever, the ℒ relation is not necessary right compatible with the semigroup mul˜ relation is also not necessary left compatible with the tiplication and the ℛ ˜ semigroup multiplication. We now denote the ℒ-class containing the element 𝑎 ˜ ˜ of the semigroup 𝑆 by 𝐿𝑎 and observe that ℒ ⊆ ℒ∗ ⊆ ℒ. For the usual ℒ-relation or the generalized ℒ-relations, there always exists the corresponding dual relations, namely ℛ-relation or generalized ℛ-relations. In this article, we concentrate on the properties of the ℒ-relation and its generalized ℒ-relations, respectively. It is trivial that there exists at most one idempotent ˜𝑎 ∩ 𝐸(𝑆), for some 𝑎 ∈ 𝑆, then we of the semigroup 𝑆 in each ℋ class. If 𝑒 ∈ ℋ 0 ˜𝑎 . Clearly, for any 𝑥 ∈ ℋ ˜𝑎 simply denote the idempotent 𝑒 by 𝑥 , for any 𝑥 ∈ ℋ with 𝑎 ∈ 𝑆, 𝑥 = 𝑥𝑥0 = 𝑥0 𝑥. If a semigroup 𝑆 is regular, then every ℒ-class of 𝑆 contains at least one idempotent and so does every ℛ-class of 𝑆. If 𝑆 is a completely regular semigroup, then every ℋ-class of 𝑆 contains an idempotent. According to Fountain [4], a semigroup is called 𝑎𝑏𝑢𝑛𝑑𝑎𝑛𝑡 if every ℒ∗ -class and ℛ∗ -class of 𝑆 contains some idempotents. In other words, “abundant” means that the semigroup contains plenty of idempotents. It is obvious that ℒ∗ = ℒ on the regular elements of a semigroup. Thus, the regular semigroups are special abundant semigroups. In the literature, J.B. Fountain called such abundant semigroups superabundant [4] if every ℋ∗ -classes of an abundant semigroup contains one idempotent. Consequently, completely regular semigroups are special superabundant semigroups and superabundant semigroups can be regarded as a generalization of Clifford semigroups. Hence, a Clifford semigroup is a completely regular semigroup whose idempotents are central. As a further generalization of abundant ˜ semigroups, El-Qallali [3] called a semigroup 𝑆 semiabundant if every ℒ-class and ˜ every ℛ-class of 𝑆 contains at least one idempotent. A semigroup 𝑆 is hence ˜ ˜ called ℋ-abundant if every ℋ-class contains an idempotent of 𝑆. It is clear that ˜ ℋ-abundant semigroups are generalizations of superabundant semigroups in the class of semiabundant semigroups. One can easily see that ℒ˜ = ℒ on the regular ˜ elements in any ℋ-abundant semigroup. We now call a band 𝐵 regular (right quasi normal band) if 𝐵 satisfies the identity 𝑎𝑥𝑦𝑎 = 𝑎𝑥𝑎𝑦𝑎(𝑥𝑦𝑎 = 𝑥𝑎𝑦𝑎). According to [13], a completely regular semigroup 𝑆 is called a regular cryptogroup if the Green ℋ-relation on 𝑆 is a regular band congruence. The structure of regular cryptogroups was extensively investigated by Kong-Shum in [8] and [9]. In the class of abundant semigroups, X.J. Guo and K.P. Shum [5] called an abundant semigroup whose set of idempotents forms a regular band a cyber group. The semilattice structure of regular

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cyber-groups were further investigated by K.P. Shum and X.Z. Kong in [9]. The following questions naturally arise: (i) can we establish an analogous result of superabundant semigroups [4] in the class of semiabundant semigroups? or (ii) whether there exists an analogous result of cryptogroups [13] in the class of ˜ ℋ-abundant semigroups? In order to answer the above questions, we establish a ˜ theorem for ℋ-cryptogroups by using the Green ∼-relations and the so called 𝒦𝐺 ˜ semigroup system as described in [10]. We shall show that an ℋ-cryptogroup 𝑆 ˜ ˜ is a regular ℋ-cryptogroup if and only if 𝑆 can be expressed as an ℋ𝐺-semigroup system of simple semigroups. Our results generalize the corresponding results given in [1], [4], [7], [8] and [14].

2. 퓚𝑮-systems of semigroups We first restate the concept of 𝐺-strong semilattice decomposition of a semigroup 𝑆 given by X.Z. Kong and K.P. Shum in [8] and [9]. Let 𝑆 = (𝑌 ; 𝑆𝛼 ) be a semilattice of the semigroups 𝑆𝛼 , where each 𝑆𝛼 is a subsemigroup of the semigroup 𝑆 and 𝑌 itself is a semilattice. Define the 𝐺-strong semilattice of semigroups which is a generalization of the well-known concept of refined strong semilattice of semigroups stated in Kong-Shum [9]. Definition 2.1. Let 𝑆 = (𝑌 ; 𝑆𝛼 ) be a semilattice of semigroups. Suppose that the following conditions on 𝑆 are satisfied : (1) (∀𝛼, 𝛽 ∈ 𝑌, 𝛼 ⩾ 𝛽), there exists a family of homomorphisms 𝜑𝑑(𝛼,𝛽) : 𝑆𝛼 −→ 𝑆𝛽 , where 𝑑(𝛼, 𝛽) ∈ 𝐷(𝛼, 𝛽) and 𝐷(𝛼, 𝛽) is a non-empty index set. (2) (∀𝛼 ∈ 𝑌 ), 𝐷(𝛼, 𝛼) is a singleton. Denote the element in 𝐷(𝛼, 𝛼) by 𝑑(𝛼, 𝛼). In this case, the homomorphism 𝜑𝑑(𝛼,𝛼) : 𝑆𝛼 −→ 𝑆𝛼 is the identity automorphism of the semigroup 𝑆𝛼 .


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(3) (∀𝛼, 𝛽, 𝛾 ∈ 𝑌, 𝛼 ⩾ 𝛽 ⩾ 𝛾), if we write Φ𝛼,𝛽 = {𝜑𝑑(𝛼,𝛽) : 𝑑(𝛼, 𝛽) ∈ 𝐷(𝛼, 𝛽)} then Φ𝛼,𝛽 Φ𝛽,𝛾 ⊆ Φ𝛼,𝛾 , where Φ𝛼,𝛽 Φ𝛽,𝛾 = {𝜑𝑑(𝛼,𝛽) 𝜑𝑑(𝛽,𝛾) : ∀𝑑(𝛼, 𝛽) ∈ 𝐷(𝛼, 𝛽), 𝑑(𝛽, 𝛾) ∈ 𝐷(𝛽, 𝛾)}. (4) (∀𝑎 ∈ 𝑆𝛼 )(∀𝛽 ∈ 𝑌 ), there is a unique 𝜑𝑎𝑑(𝛽,𝛼𝛽) ∈ Φ𝛽,𝛼𝛽 such that for all 𝑏 ∈ 𝑆𝛽 , 𝑎𝑏 = (𝑎𝜑𝑏𝑑(𝛼,𝛼𝛽) )(𝑏𝜑𝑎𝑑(𝛽,𝛼𝛽) ). Then the above semilattice of semigroups is still a semigroup (see L. Zhang and K.P. Shum [21]). We now call this semigroup a generalized strong semilattice of semigroups 𝑆𝛼 and in brevity, we just call it the “𝐺-strong semilattice” of semigroups 𝑆𝛼 and denote it by 𝑆 = 𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ]. We now extend the above “𝐺-strong semilattice of semigroups” to a more general setting and call it the “system of semigroups”. Definition 2.2. Suppose that 𝒦 is an equivalence on a 𝐺-strong semilattice of semigroups 𝑆 = 𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ]. Then 𝑆 is partitioned into disjoint union of subsets by the 𝒦-relation. We now call 𝑆 a system of “𝒦𝐺-semigroups 𝑆𝛼 ” if for every 𝛼, 𝛽 ∈ 𝑌, the mapping 𝑎 7−→ 𝜑𝑎𝑑(𝛽,𝛼𝛽) has the property that 𝜑𝑎𝑑(𝛽,𝛼𝛽) = 𝜑𝑏𝑑(𝛽,𝛼𝛽) whenever the elements 𝑎, 𝑏 ∈ 𝑆𝛼 are in the same 𝒦-class of 𝑆. It is clear that the 𝐺-strong semilattice of semigroups 𝑆 can be determined by an equivalent relation 𝒦. We therefore call the above generalized strong semilattice of semigroups 𝑆𝛼 a system of “𝒦𝐺- semigroups 𝑆𝛼 ” and this system is denoted by 𝑆 = 𝒦𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ], where 𝒦 is any one of the usual Green’s relations ℒ, ℛ, 𝒟 and ℋ, defined on the semigroup 𝑆 respectively. Remark 2.3. (i) It is clear that the 𝒦𝐺-system of semigroups is a stronger concept than the concept of 𝐺-strong semilattice of semigroups but on the other hand, the concept of 𝒦𝐺-system of semigroups is weaker than the concept of the usual strong semilattice of semigroups. In fact, if we let 𝜌 and 𝛿 be two equivalences on the semigroup 𝑆 = (𝑌 ; 𝑆𝛼 ) with 𝜌 ⊆ 𝛿, then one can easily observe that 𝛿𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ] is “stronger” than 𝜌𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ]. Remark 2.3. (ii) As a special case, 1𝑆 𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ] is clearly the “weakest” 𝒦𝐺system of semigroups since 1𝑆 is the “smallest” equivalence on the semigroup 𝑆 and also 𝜂𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ] is the strongest 𝒦𝐺-system of semigroups since 𝜂 is the “greatest” equivalence on the semigroup 𝑆, where 1𝑆 is the identity relation on 𝑆 and 𝜂 is the semilattice congruence on 𝑆 which partitions 𝑆 into disjoint union of subsets 𝑆𝛼 (where 𝛼 ∈ 𝑌 ) of 𝑆. By the definition of “system of semigroups”, we can verify that above 𝒦-partitioned subsets of 𝑆 are subsemigroups of 𝑆. Hence, we can observe that 𝜂𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ] is the usual strong semilattice of semigroups since in this case, every index set 𝐷(𝛼, 𝛽) is only a singleton for 𝛼 ⩾ 𝛽 on 𝑌

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and hence there exists one and only one structure homomorphism in the set of structure homomorphisms 𝜑𝛼,𝛽 . ˜ ℛ, ˜ ℋ ˜ and 𝒟 ˜ have been defined on a semigroup After the Green ∼-relations ℒ, 𝑆, we can define the Green ∼-relation 𝒥˜ on 𝑆. For this purpose, consider the left ∼-ideal 𝐿 of a semigroup 𝑆. Definition 2.4. A left (right) ideal 𝐿 (𝑅) of a semigroup 𝑆 is called a left ∼-ideal ˜ 𝑎 ⊆ 𝐿(𝑅 ˜𝑎 ⊆ 𝑅) holds, for all 𝑎 ∈ 𝐿(𝑎 ∈ 𝑅). We call a subset 𝐼 of a of 𝑆 if 𝐿 semigroup 𝑆 a ∼-ideal of 𝑆 if it is both a left ∼-ideal and a right ∼-ideal. It can be easily observed that if 𝑆 is a regular semigroup, then every left (right, two-sided) ideal of 𝑆 is a left (right, two-sided) ∼-ideal. We also observe that for any idempotent 𝑒 in a semigroup 𝑆 , the left (right) ideal 𝑆𝑒(𝑒𝑆) is a left(right) ∼-ideal. For if 𝑎 ∈ 𝑆𝑒, then 𝑎 = 𝑎𝑒, and hence for every element 𝑏 in ˜ 𝑎 , we always have 𝑏 = 𝑏𝑒 ∈ 𝑆𝑒. 𝐿 By Definition 2.4, the semigroup 𝑆 is always a ∼-ideal of itself, and we ˜ denote the smallest ∼-ideal containing the element 𝑎 of 𝑆 by 𝐽(𝑎). Now, we ˜ ˜ ˜ define 𝒥 = {(𝑎, 𝑏) ∈ 𝑆 × 𝑆 : 𝐽(𝑎) = 𝐽(𝑏)}. ˜ Definition 2.5. An ℋ-abundant semigroup 𝑆 is called completely 𝒥˜-simple if 𝑆 does not contain any non-trivial proper ∼-ideal of 𝑆. ˜ We now state some properties of the ℋ-abundant semigroups. Some of their properties may have already been known or can be easily derived. ˜ Lemma 2.6. Let 𝑆 be an ℋ-abundant semigroup. Then we have the following properties: ˜ is a congruence on 𝑆 if and only if for any 𝑎, 𝑏 ∈ 𝑆, (1) The Green ∼-relation ℋ 0 0 0 0 (𝑎𝑏) = (𝑎 𝑏 ) . ˜ (2) If 𝑒, 𝑓 are 𝒟-related idempotents of 𝑆, then 𝑒𝒟𝑓. ˜ = ℒ˜ ∘ ℛ ˜=ℛ ˜ ∘ ℒ. ˜ (3) 𝒟 (4) If 𝑒, 𝑓 are idempotents in 𝑆 such that 𝑒𝒥˜𝑓 , then 𝑒𝒟𝑓 . Similar to the definition of cyber-group given in [5], we formulate the following ˜ definition of an ℋ-cryptogroup. ˜ ˜ Definition 2.7. An ℋ-abundant semigroup 𝑆 is called an ℋ-cryptogroup if the ˜ ˜ Green ∼-relation ℋ is a congruence on 𝑆. Also, we call an ℋ-abundant semi˜ ˜ is a congruence on 𝑆 such that 𝑆/ℋ ˜ is a group 𝑆 a regular ℋ-cryptogroup if ℋ regular band. ˜ ˜ Thus, an ℋ-cryptogroup is an analogy of cryptogroups in the class of ℋ˜ abundant semigroups. Also, by [5], an ℋ-cryptogroup is a generalized cyber


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group. ˜ The ℋ-cryptogroup 𝑆 has the following properties: Lemma 2.8. ˜ ˜ = 𝑆𝑎0 𝑆. (1) For any element 𝑎 of the ℋ-cryptogroup 𝑆, 𝐽(𝑎) ˜ ˜ (2) For the ℋ-cryptogroup 𝑆, 𝒥˜ = 𝒟. ˜ (3) If the ℋ-cryptogroup 𝑆 is completely 𝒥˜-simple, then the idempotents of 𝑆 are primitive. ˜ (4) If the ℋ-cryptogroup 𝑆 is completely 𝒥˜-simple, then the regular elements of 𝑆 generate a regular subsemigroup of 𝑆. ˜ We now establish the following theorem for ℋ-cryptogroups. ˜ Theorem 2.9. Let 𝑆 be an ℋ-cryptogroup. Then 𝑆 is a semilattice 𝑌 of completely 𝒥˜-simple semigroups 𝑆𝛼 (𝛼 ∈ 𝑌 ) such that for every 𝛼 ∈ 𝑌 and 𝑎 ∈ 𝑆𝛼 , ˜ 𝑎 (𝑆) = 𝐿 ˜ 𝑎 (𝑆𝛼 ) and 𝑅 ˜𝑎 (𝑆) = 𝐿 ˜ 𝑎 (𝑆𝛼 ). we have 𝐿 ˜ 2 and so, 𝐽(𝑎) ˜ ˜ 2 ). Now for 𝑎, 𝑏 ∈ 𝑆, we have Proof. If 𝑎 ∈ 𝑆, then 𝑎ℋ𝑎 = 𝐽(𝑎 (𝑎𝑏)2 ∈ 𝑆𝑏𝑎𝑆, and hence, 2 ˜ ˜ ˜ 𝐽(𝑎𝑏) = 𝐽((𝑎𝑏) ) ⊆ 𝐽(𝑏𝑎).

˜ ˜ Now, by symmetry, we have 𝐽(𝑎𝑏) = 𝐽(𝑏𝑎). Since, by Lemma 2.8 (i), we have 0 0 ˜ ˜ ˜ ∩ 𝐽(𝑏), ˜ 𝐽(𝑎) = 𝑆𝑎 𝑆 and 𝐽(𝑏) = 𝑆𝑏 𝑆 so that if 𝑐 ∈ 𝐽(𝑎) then 𝑐 = 𝑥𝑎0 𝑦 = 𝑧𝑏0 𝑡 2 0 0 ˜ 0 𝑡𝑥𝑎0 ) for some 𝑥, 𝑦, 𝑧, 𝑡 ∈ 𝑆. Now we have 𝑐 = 𝑧𝑏 𝑡𝑥𝑎 𝑦 ∈ 𝑆𝑏0 𝑡𝑥𝑎0 𝑆 ⊆ 𝐽(𝑏 0 0 0 0 ˜ 𝑡𝑥𝑎 ) = 𝐽(𝑎 ˜ 𝑏 𝑡𝑥) by using our previous arguments. Thus, and hence, 𝐽(𝑏 2 0 0 ˜ ˜ ˜ 0 𝑏0 ). Since 𝑎ℋ𝑎 ˜ 0 , 𝑏ℋ𝑏 ˜ 0 and ℋ ˜ 𝑐 ∈ 𝐽(𝑎 𝑏 ) and since 𝑐ℋ𝑐2 , we have 𝑐 ∈ 𝐽(𝑎 0 0 ˜ ˜ is a congruence on 𝑆, we have 𝑎𝑏ℋ𝑎 𝑏 . Consequently, 𝑐 ∈ 𝐽(𝑎𝑏), and thereby ˜ ∩ 𝐽˜(𝑏) ⊆ 𝐽(𝑎𝑏). ˜ ˜ ∩ 𝐽(𝑎) The converse containment is clear so that we obtain 𝐽(𝑎) ˜ ˜ 𝐽(𝑏) = 𝐽(𝑎𝑏). ˜ We can easily see that the set 𝑌 of all ∼-ideals 𝐽(𝑎)(𝑎 ∈ 𝑆) forms a semilattice ˜ is a homomorphism from 𝑆 under set intersection and that the mapping 𝑎 7→ 𝐽(𝑎) ˜ is just the 𝒥˜-class 𝐽˜𝑎 which is a subsemigroup onto 𝑌 . The inverse image of 𝐽(𝑎) of 𝑆. Hence 𝑆 is a semilattice 𝑌 of the semigroups 𝐽˜𝑎 . ˜ 𝐽). ˜ Then, Now let 𝑎, 𝑏 be elements of 𝒥˜-class 𝐽˜ and suppose that (𝑎, 𝑏) ∈ ℒ( 0 0 0 0 0 0 0 0 0 0 ˜ ˜ ˜ 𝑎 , 𝑏 ∈ 𝐽 so that (𝑎 , 𝑏 ) ∈ ℒ(𝐽), that is, 𝑎 𝑏 = 𝑎 , 𝑏 𝑎 = 𝑏 and (𝑎0 , 𝑏0 ) ∈ ˜ ˜ ˜ 𝑎 (𝑆) ⊆ 𝐽, ˜ we have ℒ(𝑆). It follows that (𝑎, 𝑏) ∈ ℒ(𝑆) and consequently, by 𝐿 ˜ 𝑎 (𝑆) = 𝐿 ˜ 𝑎 (𝐽). ˜ By using similar arguments, we also have 𝑅 ˜𝑎 (𝑆) = 𝑅 ˜𝑎 (𝐽˜). 𝐿 ˜ ˜ ˜ 𝑎 (𝑆) and From the above discussion, we can easily deduce that 𝐻𝑎 (𝐽) = 𝐻 ˜ ˜ ˜ so 𝐽 is indeed an ℋ-abundant semigroup. Furthermore, if 𝑎, 𝑏 ∈ 𝐽, then by ˜ Lemma 2.8 (i), (𝑎, 𝑏) ∈ 𝒟(𝑆) and hence, by Lemma 2.6 (iii), there exists an

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˜ 𝑎 (𝑆) ∩ 𝑅 ˜𝑏 (𝑆) = 𝐿 ˜ 𝑎 (𝐽) ˜ ∩𝑅 ˜𝑏 (𝐽). ˜ Thus 𝑎, 𝑏 are 𝒟-related ˜ element 𝑐 in 𝐿 in 𝐽˜ and ˜ ˜ so 𝐽 is 𝒥 -simple. ˜ For ℋ-cryptogroups, we also have the following theorem. ˜ Theorem 2.10. Let 𝑆 be an ℋ-cryptogroup which is expressed by the semilattice of semigroups 𝑆 = (𝑌 ; 𝑆𝛼 ). Then the following statements hold: (1) For 𝛼 and 𝛽 in the semilattice 𝑌 with 𝛼 ⩾ 𝛽, if 𝑎 ∈ 𝑆𝛼 then there exists 𝑏 ∈ 𝑆𝛽 with 𝑎 ⩾ 𝑏; ˜ if 𝑎 ⩾ 𝑏 and 𝑎 ⩾ 𝑐 then 𝑏 = 𝑐; (2) For 𝑎, 𝑏, 𝑐 ∈ 𝑆 with 𝑏ℋ𝑐, (3) For 𝑎 ∈ 𝐸(𝑆) and 𝑏 ∈ 𝑆, if 𝑎 ⩾ 𝑏 then 𝑏 ∈ 𝐸(𝑆); ˜ (4) If 𝜑 is a homomorphism which maps an ℋ-cryptogroup 𝑆 into another 0 0 ˜ ℋ-cryptogroup 𝑇 , then (𝑎𝜑) = 𝑎 𝜑.

˜ 3. Regular 퓗-cryptogroups ˜ ˜ ˜ We call an ℋ-abundant semigroup 𝑆 a regular ℋ-cryptogroup if the relation ℋ ˜ ˜ is a congruence on the ℋ-abundant semigroup 𝑆 such that 𝑆/ℋ forms a regular band. By using the above concept, we formulate the following crucial lemma. Lemma 3.1. Let 𝑆 be a semigroup. For every element 𝑎 ∈ 𝑆, we define a relation 𝜌𝑎 on 𝑆 by (𝑏1 , 𝑏2 ) ∈ 𝜌𝑎 if and only if (𝑎𝑏1 𝑎)0 = (𝑎𝑏2 𝑎)0 , (𝑏1 , 𝑏2 ∈ 𝑆). Then the following properties hold on 𝑆: (1) 𝜌𝑎 is a band congruence on 𝑆; (2) (∀𝑎, 𝑎1 ∈ 𝑆𝛼 ), 𝜌𝑎 = 𝜌𝑎1 , that is, 𝜌𝑎 depends only on the component 𝑆𝛼 containing the element 𝑎 rather than on the element itself, hence we can write 𝜌𝛼 = 𝜌𝑎 , for all 𝑎 ∈ 𝑆𝛼 ; (3) (∀𝛼, 𝛽 ∈ 𝑌 with 𝛼 ⩾ 𝛽), 𝜌𝛼 ⊆ 𝜌𝛽 and 𝜌𝛽 ∣𝑆𝛼 = 𝜔𝑆𝛼 , where 𝜔𝑆𝛼 is the universal relation on 𝑆𝛼 . Proof. (1) Clearly, 𝜌𝑎 is an equivalence on 𝑆, for every 𝑎 ∈ 𝑆 . We now claim that 𝜌𝑎 is left compatible with the semigroup multiplication. For this purpose, let (𝑥, 𝑦) ∈ 𝜌𝑎 and 𝑐 ∈ 𝑆. Then, by the definition of 𝜌𝑎 , (𝑎𝑥𝑎)0 = (𝑎𝑦𝑎)0 . Since ˜ 𝑆 is a regular ℋ-cryptogroup, by Lemma 2.6 (i) and the regularity of the band ˜ 𝑆/ℋ, we obtain (𝑎𝑐𝑥𝑎)0 = (𝑎𝑐(𝑎𝑥𝑎))0 = ((𝑎𝑐)0 (𝑎𝑥𝑎)0 )0 = ((𝑎𝑐)0 (𝑎𝑦𝑎)0 )0 = (𝑎𝑐𝑦𝑎)0 . Hence, (𝑐𝑥, 𝑐𝑦) ∈ 𝜌𝑎 . Dually, we can prove that 𝜌𝑎 is right compatible with the ˜ ⊆ 𝜌𝑎 and semigroup multiplication. Thus 𝜌𝑎 is a congruence on 𝑆. Obviously, ℋ so 𝜌𝑎 is a band congruence on 𝑆. (2) Let (𝑥, 𝑦) ∈ 𝜌𝑎 . Then, by the definition of 𝜌𝑎 , it follows that (𝑎𝑥𝑎)0 = (𝑎𝑦𝑎)0 and so 𝑎01 (𝑎𝑥𝑎)0 𝑎01 = 𝑎01 (𝑎𝑦𝑎)0 𝑎01 . This leads to (𝑎01 (𝑎𝑥𝑎)0 𝑎01 )0 =


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˜ = (𝑌 ; 𝑆𝛼 /ℋ) ˜ is a regular band and by Lemma 2.6 (i), (𝑎01 (𝑎𝑦𝑎)0 𝑎01 )0 . Since 𝑆/ℋ we obtain (𝑎1 𝑎𝑎1 𝑥𝑎1 𝑎𝑎1 )0 = (𝑎1 𝑎𝑎1 𝑦𝑎1 𝑎𝑎1 )0 . However, since 𝑎, 𝑎1 are elements of the completely 𝒥˜-simple semigroup 𝑆𝛼 , (𝑎1 𝑎𝑎1 )0 = 𝑎01 . By Lemma 2.6 (i) again, we deduce that (𝑎1 𝑥𝑎1 )0 = (𝑎1 𝑦𝑎1 )0 , that is, (𝑥, 𝑦) ∈ 𝜌𝑎1 . This shows that 𝜌𝑎 ⊆ 𝜌𝑎1 . Similarly, 𝜌𝑎1 ⊆ 𝜌𝑎 . Thus, 𝜌𝑎 = 𝜌𝑎1 . Since this relation holds for all 𝑎 ∈ 𝑆𝛼 , we write 𝜌𝑎 = 𝜌𝛼 . (3) Let 𝑎 ∈ 𝑆𝛼 , 𝑏 ∈ 𝑆𝛽 and 𝛼 ⩾ 𝛽. We need to prove that 𝜌𝛼 ⊆ 𝜌𝛽 . For this purpose, let (𝑥, 𝑦) ∈ 𝜌𝛼 = 𝜌𝑎 . Then, by the definition of 𝜌𝑎 , (𝑎𝑥𝑎)0 = (𝑎𝑦𝑎)0 and hence 𝑏(𝑎𝑥𝑎)0 𝑏 = 𝑏(𝑎𝑦𝑎)0 𝑏. By Lemma 2.6 (i) and the regularity of the band, we have (𝑏𝑎𝑏𝑥𝑏𝑎𝑏)0 = (𝑏𝑎𝑏𝑦𝑏𝑎𝑏)0. Since 𝛼 ⩾ 𝛽 in 𝑌 and 𝑎 ∈ 𝑆𝛼 , 𝑏 ∈ 𝑆𝛽 , we have (𝑏𝑎𝑏)0 = 𝑏0 . By Lemma 2.6 (i) again, we can show that (𝑏𝑥𝑏)0 = (𝑏𝑦𝑏)0 , that is, (𝑥, 𝑦) ∈ 𝜌𝑏 = 𝜌𝛽 . Thus, 𝜌𝛼 ⊆ 𝜌𝛽 as required. Furthermore, it is trivial that 𝜌𝛽 ∣𝑆𝛼 = 𝜔𝑆𝛼 , which is the universal relation on the semigroup 𝑆𝛼 . We now use the band congruence 𝜌𝛼 defined in Lemma 3.1 to describe the ˜ structure homomorphisms for the ℋ-cryptogroup 𝑆 = (𝑌 ; 𝑆𝛼 ), where each 𝑆𝛼 is a completely 𝒥˜-simple semigroup. We first consider the congruence 𝜌𝛼,𝛽 = 𝜌𝛼 ∣𝑆𝛽 for 𝛼, 𝛽 ∈ 𝑌 , which is a band congruence on the semigroup 𝑆𝛽 . Now,we denote all the 𝜌𝛼,𝛽 -classes of 𝑆𝛽 by {𝑆𝑑(𝛼,𝛽) : 𝑑(𝛼, 𝛽) ∈ 𝐷(𝛼, 𝛽)}, where 𝐷(𝛼, 𝛽) is a non-empty index set. In particular, the set 𝐷(𝛼, 𝛼) is clearly a singleton and we therefore write 𝑑(𝛼, 𝛼) = 𝐷(𝛼, 𝛼). Thus we have the following lemma. The proof of the following lemma is similar to the proof given in [12] and so we omit the details. ˜ Lemma 3.2. Let 𝑆 = (𝑌 ; 𝑆𝛼 ) be a regular ℋ-cryptogroup. Then, for all 𝛼, 𝛽 ∈ 𝑌 with 𝛼 ⩾ 𝛽,the following statements hold for all 𝑑(𝛼, 𝛽) ∈ 𝐷(𝛼, 𝛽). (1) For all 𝑎 ∈ 𝑆𝛼 , there exists a unique 𝑎𝑑(𝛼,𝛽) ∈ 𝑆𝑑(𝛼,𝛽) satisfying 𝑎 ⩾ 𝑎𝑑(𝛼,𝛽) ; (2) For all 𝑎 ∈ 𝑆𝛼 and 𝑥 ∈ 𝑆𝑑(𝛼,𝛽) , if 𝑎0 ⩾ 𝑒 for some idempotent 𝑒 ∈ 𝑆𝑑(𝛼,𝛽) then 𝑒𝑎𝑥 = 𝑎𝑥, 𝑥𝑎𝑒 = 𝑥𝑎, 𝑒𝑎 = 𝑎𝑒 and (𝑒𝑎)0 = 𝑒; (3) Let 𝑎 ∈ 𝑆𝛼 . Then, we define 𝜑𝑑(𝛼,𝛽) : 𝑆𝛼 −→ 𝑆𝑑(𝛼,𝛽) by 𝑎𝜑𝑑(𝛼,𝛽) = 𝑎𝑑(𝛼,𝛽) , where 𝑎𝑑(𝛼,𝛽) ∈ 𝑆𝑑(𝛼,𝛽) and 𝑎 ⩾ 𝑎𝑑(𝛼,𝛽) . Thus we can verify that 𝜑𝑑(𝛼,𝛽) is a homomorphism and 𝑎𝑑(𝛼,𝛽) = 𝑎(𝑎𝑏𝑎)0 = (𝑎𝑏𝑎)0 𝑎 for any 𝑏 ∈ 𝑆𝑑(𝛼,𝛽) . ˜ Now, for a given ℋ-abundant semigroup 𝑆, we establish a construction the˜ orem for a regular ℋ-cryptogroup. Our first step is to show that the homomorphisms given in Lemma 3.2 (iii) are the structure homomorphisms for the 𝐺-strong semilattice 𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ] induced by the semigroup 𝑆 = (𝑌 ; 𝑆𝛼 ) under the band congruence 𝜌𝛼 on the semigroup 𝑆𝛼 . ˜ Step I Consider 𝑆 = (𝑌 ; 𝑆𝛼 ) as an ℋ-cryptogroup and 𝜑𝛼,𝛽 = {𝜑𝑑(𝛼,𝛽) ∣ 𝑑(𝛼, 𝛽) ∈ 𝐷(𝛼, 𝛽)} for 𝛼 ⩾ 𝛽 on 𝑌 , where 𝐷(𝛼, 𝛽) is a non-empty index set. Then by straightforward verification, we can prove that

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(1) 𝜑𝛼,𝛽 𝜑𝛽,𝛾 ⊆ 𝜑𝛼,𝛾 for 𝛼 ⩾ 𝛽 ⩾ 𝛾 on 𝑌 ; (2) For 𝑎 ∈ 𝑆𝛼 and 𝛽 ∈ 𝑌 , 𝑎𝜑𝛼,𝛼𝛽 = {𝑎𝜑𝑑(𝛼,𝛼𝛽) ∣∀𝑑(𝛼, 𝛼𝛽) ∈ 𝐷(𝛼, 𝛼𝛽)} ⊆ 𝑆𝑑(𝛽,𝛼𝛽) , for some 𝜌𝛽,𝛼𝛽 -class 𝑆𝑑(𝛽,𝛼𝛽) ; ˜ ˜ Step II We prove that ℋ-cryptogroup 𝑆 is a regular ℋ-cryptogroup if and ˜ system of completely 𝒥˜-simple semigroups, that is, 𝑆 = only if 𝑆 is an ℋ𝐺 ˜ ℋ𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ]. Proof. By the definition of 𝒦𝐺- system of semigroups and the results obtained ˜ 𝑆 ⊆ 𝜌𝛼,𝛽 in § 2, we have already proved the necessity part of Step II since ℋ∣ 𝛽 for 𝛼 ⩾ 𝛽 on 𝑌 . We now prove the sufficiency part of Step II. ˜ is a regular band, we use a result in [15]. What we need To prove that 𝑆/ℋ is to verify that the usual Green’s relations ℒ and ℛ on 𝑆 are congruences on ˜ In other words, we have to verify that ℒ is a left congruence on 𝑆/ℋ ˜ 𝑆/ℋ. ˜ can be proved in a similar fashion. Since since ℛ is a right congruence on 𝑆/ℋ ˜ ˜ 𝑓ℋ ˜ and 𝑔 ℋ ˜ ∈ 𝑆/ℋ, ˜ where 𝑆 = (𝑌 ; 𝑆𝛼 ) is an ℋ-cryptogroup, we can let 𝑒ℋ, ˜ 𝑒, 𝑓 ∈ 𝑆𝛼 ∩ 𝐸(𝑆), 𝑔 ∈ 𝑆𝛽 ∩ 𝐸(𝑆) with (𝑒, 𝑓 ) ∈ ℒ. Then, 𝑒𝑓 = 𝑒 and 𝑓 𝑒 = 𝑓 . ˜ ˜ By the definition of ℋ𝐺-strong semilattice ℋ𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ], we can find the 𝑒𝑓 𝑓 𝑔 homomorphisms 𝜑𝑑(𝛽,𝛼𝛽) and 𝜑𝑑(𝛽,𝛼𝛽) ∈ 𝜑𝛽,𝛼𝛽 , 𝜑𝑑(𝛼,𝛼𝛽) ∈ 𝜑𝛼,𝛼𝛽 such that the following equalities hold. ˜ = {[𝑔(𝑒𝑓 )](𝑔𝑓 )}ℋ ˜ (𝑔𝑒𝑔𝑓 )ℋ 𝑔 𝑓 𝑔 ˜ = {[(𝑔𝜑𝑒𝑓 𝑑(𝛽,𝛼𝛽) )((𝑒𝑓 )𝜑𝑑(𝛼,𝛼𝛽) )][(𝑔𝜑𝑑(𝛽,𝛼𝛽) )(𝑓 𝜑𝑑(𝛼,𝛼𝛽) )]}ℋ 𝑔 ˜ = [(𝑔𝜑𝑒𝑓 𝑑(𝛽,𝛼𝛽) )(𝑓 𝜑𝑑(𝛼,𝛼𝛽) )]ℋ

and ˜ = [𝑔(𝑒𝑓 )]ℋ ˜ (𝑔𝑒)ℋ 𝑔 ˜ = [(𝑔𝜑𝑒𝑓 𝑑(𝛽,𝛼𝛽) )((𝑒𝑓 )𝜑𝑑(𝛼,𝛼𝛽) )]ℋ 𝑔 ˜ = [(𝑔𝜑𝑒𝑓 𝑑(𝛽,𝛼𝛽) )(𝑓 𝜑𝑑(𝛼,𝛼𝛽) )]ℋ.

˜ = (𝑔𝑒)ℋ. ˜ Analogously, we can also prove that (𝑔𝑓 𝑔𝑒)ℋ ˜= Thereby, (𝑔𝑒𝑔𝑓 )ℋ ˜ This proves that ℒ is left compatible with the semigroup multiplication of (𝑔𝑓 )ℋ. ˜ ˜ as required. 𝑆/ℋ. Since ℒ is always right congruence, ℒ is a congruence on 𝑆/ℋ, ˜ Dually, ℛ is also a congruence on 𝑆/ℋ. Thus by [15] (see II. 3.6 Proposition ), ˜ forms a regular band and hence 𝑆 is a regular ℋ-cryptogroup. ˜ 𝑆/ℋ Recall that a right quasi-normal band is a band satisfying the identity 𝑦𝑥𝑎 = 𝑦𝑎𝑥𝑎 [6]. Also, a left quasi-normal band is a band satisfying the identity 𝑎𝑥𝑦 = 𝑎𝑥𝑎𝑦. Thus, we can easily observe that both the right quasi-normal bands and


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the left quasi-normal bands are special cases of the regular bands. Also, a normal band (that is, a band satisfies the identity 𝑎𝑥𝑦𝑎 = 𝑎𝑦𝑥𝑎) is a special right quasinormal band and a left quasi-normal band. Based on the above observation, we ˜ now establish the following theorem for right quasi-normal ℋ-cryptogroups. We omit the proof because the arguments are similar to the previous construction.

4. Main theorems ˜ ˜ Theorem 4.1. An ℋ-abundant semigroup 𝑆 is a right quasi-normal ℋ˜ ˜ cryptogroup if and only if 𝑆 is an ℒ𝐺-strong semilattice of completely 𝒥 -simple ˜ semigroups, that is, 𝑆 = ℒ𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ]. Since a band 𝐵 is normal if for all elements 𝑒, 𝑓, 𝑔 in 𝐵, the identity 𝑒𝑓 𝑔𝑒 = 𝑒𝑔𝑓 𝑒 holds in 𝐵( see [6]). By using similar arguments as before , we formulate a modified version of a theorem of Petrich and Reilly in [12] on normal cryptogroups [12] and also the theorem of Fountain on superabundant semigroups in [4]. This theorem is an extended version of the above theorems in the the class of quasiabundant semigroups. ˜ ˜ Theorem 4.2. An ℋ-abundant semigroup 𝑆 is a normal ℋ-cryptogroup if and ˜ ˜ only if 𝑆 is a 𝒟𝐺-strong semilattice of completely 𝒥 -simple semigroups, that is, ˜ 𝑆 = 𝒟𝐺[𝑌 ; 𝑆𝛼 , 𝜑𝛼,𝛽 ].

5. Related problems In studying abundant semigroups, apart from using Green ★-relations, we can also modify the Green relations by using some other methods of generalizations, for example, we can generalize the Green’s relations to Green ♯-relations on a semigroup 𝑆 by using the following definitions: ℒ♯ = {(𝑎, 𝑏) ∈ 𝑆 × 𝑆 : (∀𝑥, 𝑦 ∈ 𝑆 1 )(𝑎𝑥, 𝑎𝑦) ∈ ℒ ⇔ (𝑏𝑥, 𝑏𝑦) ∈ ℒ}, ℛ♯ = {(𝑎, 𝑏) ∈ 𝑆 × 𝑆 : (∀𝑥, 𝑦 ∈ 𝑆 1 )(𝑥𝑎, 𝑦𝑎) ∈ ℛ ⇔ (𝑥𝑏, 𝑦𝑏) ∈ ℛ, ℋ♯ = ℒ♯ ∩ ℛ♯ , 𝒟♯ = ℒ♯ ∨ ℛ♯ . It is not difficult to verify that the newly defined Green ♯-relations have most of the nice properties as the Green ★-relations. Therefore, we can naturally define ℋ♯ -abundant semigroups and hence ask whether can we provide a constructive theorem for a regular a normal ℋ♯ -abundant cryptogroups by using semigroup system of semigroups? Since the Green ♯-relations have many similar properties as the Green ★-relations on abundant semigroups, we may not need to use the powerful tool “semigroup system of semigroups” to give the structure of the ℋ♯ -abundant semigroups. In closing this paper, we raise the following open

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question: can we describe the structure of regular ℋ♯ -abundant cryptogroups by using some other kind of modified strong semilattices of semigroups?

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