Southeast Asian Bulletin of Mathematics (2011) 35: 1009–1014
Southeast Asian Bulletin of Mathematics c SEAMS. 2011 ⃝
Coextensions of Abundant Semigroups P.G. Romeo Department of Mathematics, Cochin University of Science and Technology, Thrissur, Kerala, India Email: romeo
[email protected]
Received 11 April 2009 Accepted 10 October 2009 Communicated by Yuqi Guo AMS Mathematics Subject Classification(2000): 20M10 Abstract. Fundamental semigroups are those semigroups which cannot be shrunk homomorphically without collapsing idempotents together. A semigroup 𝑆 in which each ℒ∗ [ℛ∗ ]- class contains idempotent is an abundant semigroup. In this paper we describe a congruence 𝜇 on an abundant semigroup 𝑆 with biordered set of idempotents 𝐸 such that 𝑆/𝜇 is fundamental. Keywords: Abundant semigroup; Fundamental semigroup; Coextensions.
1. Introduction A set 𝑆 is a groupoid with respect to a binary operation if for every pair of elements 𝑎, 𝑏 ∈ 𝑆 there is an element 𝑎 ⋅ 𝑏 ∈ 𝑆 which is the product of 𝑎 by 𝑏. A groupoid 𝑆 is a semigroup if the binary operation on 𝑆 is associative. A semigroup 𝑆 is regular if for any 𝑎 ∈ 𝑆 there exists an 𝑥 ∈ 𝑆 such that 𝑎𝑥𝑎 = 𝑎. Regular semigroups are the most important class of semigroups, and Green’s equivalences are significant in their study. For an element 𝑎 in a semigroup 𝑆 the smallest left ideal containing 𝑎 is 𝑆𝑎 ∪ {𝑎} and is called the principal left ideal generated by 𝑎, written as 𝑆 1 𝑎. An equivalence relation ℒ on 𝑆 is defined by 𝑎ℒ𝑏 if and only if 𝑆 1 𝑎 = 𝑆 1 𝑏. Similarly we can define the relation ℛ by 𝑎ℛ𝑏 if and only if 𝑎𝑆 1 = 𝑏𝑆 1 . The intersection of ℛ and ℒ is of great importance and is denoted by ℋ and their join by 𝒥 . These equivalence relations are termed as Green’s equivalences. One approach to study semigroups which are not regular is to consider the
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generalized Green’s relations on semigroups. This idea was first introduced by McAlister [4] and Pastjin [6]. The generalized Green’s relations ℛ∗ [ℒ∗ ] on a semigroup 𝑆 is defined by 𝑎ℛ∗ 𝑏 [𝑎ℒ∗ 𝑏] if and only if 𝑎ℛ𝑏 [𝑎ℒ𝑏] in some semigroup 𝑆¯ containing 𝑆. Also we can define the join of ℛ∗ and ℒ∗ by 𝒥 ∗ and ℋ∗ = ℛ∗ ∩ ℒ∗ . Definition 1.1. A semigroup is abundant if every ℛ∗ - class and ℒ∗ -class contains an idempotent (cf. [8]). Lemma 1.2. The following are equivalent for elements 𝑎, 𝑏 ∈ 𝑆: (i) 𝑎ℛ∗ 𝑏 (ii) for all 𝑥, 𝑦 ∈ 𝑆 1 , 𝑥𝑎 = 𝑦𝑎 if and only if xb=yb. This condition is simplified when one of the elements is an idempotent. Corollary 1.3. Let a be an element of a semigroup S, and let 𝑒 ∈ 𝐸(𝑆). Then the following are equivalent: (i) 𝑎ℛ∗ 𝑒 (ii) ea=a and for all 𝑥, 𝑦 ∈ 𝑆 1 , 𝑥𝑎 = 𝑦𝑎 implies xe=ye. A semigroup 𝑆 is called fundamental if it cannot be shrunk homomorphicaly without collapsing idempotens together, that is, if 𝜑 is a homomorphism from 𝑆 then 𝜑∣𝐸 one-one ⇒ 𝜑 one-one this is equivalent to 𝑆 having no nontrivial idempotent separating congruence. A mapping 𝜙 : 𝑆 → 𝑇 between two semigroups S and T is a semigroup homomorphism if it preserves the semigroup multiplication. 𝑖𝑚𝜙 = {𝑎𝜙 ∣ 𝑎 ∈ 𝑆} is a subsemigroup and 𝑘𝑒𝑟𝜙 = {(𝑎, 𝑏) ∣ 𝑎𝜙 = 𝑏𝜙} is a congruence. If 𝑆 = 𝑇 then 𝜙 is an endomorphism. 𝑆 is a semigroup and 𝜈 is any congruence on 𝑆 then the natural mapping 𝜑 : 𝑆 → 𝑆/𝜈, 𝑎 7→ 𝑎𝜈 is a semigroup homomorphism. The fundamental homomorphism theorem states that if 𝜙 : 𝑆 → 𝑇 is a homomorphism then 𝑆/𝑘𝑒𝑟𝜙 ∼ = 𝑖𝑚𝜙. Definition 1.4. Let 𝑆 and 𝑇 be any semigroups and 𝜎 any congruence on 𝑆. Then 𝑆 is a coextension of 𝑇 by 𝜎 if there is an epimorphism 𝜙 : 𝑆 → 𝑇 with 𝑘𝑒𝑟𝜎 By fundamental homomorphism theorem we have 𝑆/𝜎 ∼ = 𝑇 . In case 𝑆 is a group the congruence 𝜎 corresponds to a normal subgroup 𝑁 and 𝑆 is an extension of 𝑁 by 𝑇 in the usual sense in group theory.
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2. Coextensions of Abundant Semigroups In the following we proceed to construct a homomorphism of an abundant semigroup, which in turn provides us a fundamental semigroup and the desired coextension. Let 𝑆 be an abundant semigroup with biordered set of idempotents 𝐸. Denote the 𝐿∗ -class [𝑅∗ -class] of 𝐸 which contains 𝑒 ∈ 𝐸 by 𝐿∗𝑒 [𝑅𝑒∗ ]. The ℒ∗ −class [ℛ∗ − class] of 𝑆 which contains 𝑎 ∈ 𝑆 will be denoted by ℒ∗𝑎 [ℛ∗𝑎 ]. For each 𝑎 ∈ 𝑆 we define two partial transformations on the sets 𝐸/𝐿 and 𝐸/𝑅, to disguise the partial transformation as full transformation the element ∞ which is not in 𝐸/𝐿 or 𝐸/𝑅 will be introduced. For any 𝑎 ∈ 𝑆, let 𝜌𝑎 ∈ 𝑇𝐸/𝐿∪{∞} be such that 𝐿∗𝑥 7→ ℒ∗𝑥𝑎 ∩ 𝐸 if 𝑥ℛ∗ 𝑥𝑎 ∞ 7→ ∞ and 𝜆𝑎 ∈ 𝑇𝐸/𝑅∪{∞} 𝑅𝑥∗ 7→ ℛ∗𝑎𝑥 ∩ 𝐸 if 𝑥ℒ∗ 𝑎𝑥 ∞ 7→ ∞. Observe that 𝜆𝑎 is the dual of 𝜌𝑎 , the definition of 𝜆𝑎 can be obtained from the definition of 𝜌𝑎 by interchanging 𝐿∗ - classes with 𝑅∗ - classes and ℒ∗ - classes with ℛ∗ - classes and reversing all products. Also note that 𝜌𝑎 and 𝜆𝑎 are well defined since every ℒ∗ - class and ℛ∗ - class contains idempotens. Let 𝜙 = (𝜌𝑎 , 𝜆∗𝑎 ) and define mappings ∗ 𝜙 : 𝑆 → 𝑇𝐸/𝐿∪{∞} × 𝑇𝐸/𝑅∪{∞} , 𝑎 7→ 𝜙𝑎 .
Lemma 2.1.Let 𝑆 be an abundant semigroup. Then 𝜙 is a semigroup homomorphism. Proof. It is sufficient to show that 𝜌𝑎 𝜌𝑏 = 𝜌𝑎𝑏 . Let 𝑥 ∈ 𝐸, first suppose that 𝐿∗𝑥 𝜌𝑎 𝜌𝑏 ∕= ∞. Then 𝐿∗𝑥 𝜌𝑎 ∕= ∞ so 𝑥ℛ∗ 𝑥𝑎 and 𝐿∗𝑥 𝜌𝑎 = ℒ∗𝑥𝑎 ∩ 𝐸 = 𝐿∗𝑦 for some 𝑦 ∈ 𝐸. Now 𝐿∗𝑦 𝜌𝑏 = 𝐿∗𝑥 𝜌𝑎 𝜌𝑏 ∕= ∞ so 𝑦ℛ∗ 𝑦𝑏 and 𝐿∗𝑦 𝜌𝑏 = ℒ∗𝑦𝑏 ∩ 𝐸. Hence 𝐿∗𝑥 𝜌𝑎 𝜌𝑏 = ℒ∗𝑦𝑏 ∩ 𝐸 = ℒ∗𝑥𝑎𝑏 ∩ 𝐸 = 𝐿∗𝑥 𝜌𝑎𝑏 . Conversely, suppose that 𝐿∗𝑥 𝜌𝑎𝑏 ∕= ∞, then 𝑥ℛ∗ 𝑥𝑎𝑏 and 𝐿∗𝑥 𝜌𝑎𝑏 = ℒ∗𝑥𝑎𝑏 ∩𝐸. Since 𝑥ℛ∗ 𝑥𝑎𝑏 we have for all 𝑠, 𝑡 ∈ 𝑆 1 , 𝑠(𝑥𝑎) = 𝑡(𝑥𝑎) ⇒ 𝑠(𝑥𝑎𝑏) = 𝑡(𝑥𝑎𝑏) ⇒ 𝑠𝑥 = 𝑡𝑥 ie., 𝑥ℛ∗ 𝑥𝑎 thus 𝐿∗𝑥 𝜌𝑎 = ℒ∗𝑥𝑎 ∩ 𝐸 and so there exists a 𝑦 ∈ ℒ∗𝑥𝑎 ∩ 𝐸 such that 𝐿∗𝑥 𝜌𝑎 = 𝐿∗𝑦 . Clearly 𝑦(𝑦𝑏) = 𝑦𝑏 and for all 𝑠, 𝑡 ∈ 𝑆 1 , 𝑠(𝑦𝑏) = 𝑡(𝑦𝑏) ⇒ 𝑠𝑦 = 𝑡𝑦 thus 𝑦ℛ∗ 𝑦𝑏 and hence 𝐿∗𝑦 𝜌𝑏 = ℒ∗𝑦𝑏 ∩ 𝐸. But 𝑦ℒ∗ 𝑥𝑎 ⇒ 𝑦𝑏ℒ∗ 𝑥𝑎𝑏, so 𝐿∗𝑦 𝜌𝑏 = ℒ∗𝑥𝑎𝑏 ∩ 𝐸 ⇒ 𝐿∗𝑥 𝜌𝑎 𝜌𝑏 = 𝐿∗𝑥 𝜌𝑎𝑏 . Thus we have 𝜌𝑎 𝜌𝑏 = 𝜌𝑎𝑏 .
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Let 𝜇 = 𝑘𝑒𝑟𝜙. Then 𝜇 = {(𝑎, 𝑏)∣∀𝑥 ∈ 𝐸, 𝑥ℛ∗ 𝑥𝑎, 𝑥ℛ∗ 𝑥𝑏 ⇒ 𝑥𝑎ℋ∗ 𝑥𝑏 and 𝑥ℒ∗ 𝑎𝑥, 𝑥ℒ∗ 𝑏𝑥 ⇒ 𝑎𝑥ℋ∗ 𝑏𝑥} Here it should be noted that congruence 𝜇 above is analogous to the congruence defined by Edwards [2]. Proposition 2.2. Let 𝑆 be an abundant semigroup and 𝜎 any congruence on 𝑆. Then 𝜎 ⊂ 𝜇 if and only if 𝜎 is a ℋ∗ - congruence. Proof. Suppose 𝜎 ⊂ 𝜇. Let 𝑒𝜎𝑎 for 𝑒 ∈ 𝐸 and 𝑎 ∈ 𝑆. Then 𝑒𝜇𝑎 ⇒ (𝑎, 𝑒) ∈ 𝑘𝑒𝑟𝜙 ⇒ 𝜙𝑎 = 𝜙𝑒 . Thus 𝐿∗𝑒 = 𝐿∗𝑒 𝜌𝑒 = 𝐿∗𝑒 𝜌𝑎 = ℒ∗𝑒𝑎 ∩ 𝐸 and 𝑅𝑒∗ = 𝑅𝑒∗ 𝜆𝑒 = 𝑅𝑒∗ 𝜆𝑎 = ℛ∗𝑒𝑎 ∩ 𝐸 so ℒ∗𝑒 = ℒ∗𝑒𝑎 ∩ 𝐸 ≤ ℒ∗𝑎 and ℛ∗𝑒 = ℛ∗𝑒𝑎 ∩ 𝐸 ≤ ℛ∗𝑎 thus ℋ𝑒∗ ≤ ℋ𝑎∗ and so 𝜎 is an ℋ∗ congruence. Conversely, let 𝜎 be an ℋ∗ congruence. It will suffices to show that for 𝑎, 𝑏 ∈ 𝑆 with 𝑎𝜎𝑏 implies 𝜌𝑎 = 𝜌𝑏 . Let 𝑥 ∈ 𝐸 be such that 𝐿∗𝑥 𝜌𝑎 ∕= ∞ so 𝑥ℛ∗ 𝑥𝑎 and 𝐿∗𝑥 𝜌𝑎 = ℒ∗𝑥𝑎 ∩ 𝐸. Since 𝑎ℋ∗ 𝑏 ⇒ 𝑎ℛ∗ 𝑏 ⇒ 𝑥𝑎ℛ∗ 𝑥𝑏 and 𝐿∗𝑥 𝜌𝑏 ∕= ∞ and so 𝐿∗𝑥 𝜌𝑏 = ℒ∗𝑥𝑏 ∩ 𝐸. But it is seen that for every 𝑥 ∈ 𝐸, ℒ∗𝑥𝑎 = ℒ∗𝑥𝑏 which implies 𝐿∗𝑥 𝜌𝑎 = 𝐿∗𝑥 𝜌𝑏 . Thus 𝜌𝑎 = 𝜌𝑏 and so (𝑎, 𝑏) ∈ 𝜇. It should be noted that if 𝑆 is a regular semigroup the generalized Green’s relations coincides with the Green’s relations and so 𝜇 is the maximum congruence contained in ℋ. It is also well known that in this case 𝜇 is the maximum idempotent separating congruence. Corollary 2.3. Let 𝑆 be an abundant semigroup. Then 𝜇 is the maximum ℋ∗ congruence. Lemma 2.4. Let 𝑆 be an abundant semigroup. Then 𝜇 is a biorder preserving congruence on 𝑆. Proof. For 𝑒, 𝑓 ∈ 𝐸 it is sufficient to prove 𝑒𝜔 𝑙 𝑓 ⇔ 𝑒𝜇𝜔 𝑙 𝑓 𝜇. We have 𝑒𝜔 𝑙 𝑓 ⇒ 𝑒𝑓 = 𝑒 then (𝑒𝜇)(𝑓 𝜇) = (𝑒𝑓 )𝜇 = 𝑒𝜇 so 𝑒𝜇𝜔 𝑙 𝑓 𝜇. Conversely, suppose 𝑒𝜇𝜔 𝑙 𝑓 𝜇, then 𝑒𝜇 = (𝑒𝜇)(𝑓 𝜇) ⇒ (𝑒𝜇) = (𝑒𝑓 )𝜇 that is (𝑒, 𝑒𝑓 ) ∈ 𝜇 so 𝜙𝑒 = 𝜙𝑒𝑓 ⇒ 𝜌𝑒 = 𝜌𝑒𝑓 . Also 𝐿∗𝑒 = 𝐿∗𝑒 𝜌𝑒 = 𝐿∗𝑒 𝜌𝑒𝑓 = ℒ∗𝑒(𝑒𝑓 ) ∩ 𝐸 = ℒ∗𝑒𝑓 ∩ 𝐸
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thus 𝑒ℛ∗ 𝑒𝑓 . But 𝐿∗𝑒 = ℒ∗𝑒𝑓 ∩ 𝐸 ⇒ 𝑒ℒ∗ 𝑒𝑓 so (𝑒𝑓 )𝑒 = 𝑒𝑓 . Hence we have (𝑒𝑓 )(𝑒𝑓 ) = (𝑒𝑓 𝑒)𝑓 = 𝑒𝑓 𝑓 = 𝑒𝑓 . Also since 𝑒ℛ∗ 𝑒𝑓 we have for every 𝑠, 𝑡 ∈ 𝑆 1 , 𝑠(𝑒𝑓 ) = 𝑡(𝑒𝑓 ) ⇒ 𝑠𝑒 = 𝑡𝑒 thus (𝑒𝑓 )(𝑒𝑓 ) = 𝑒𝑓 ⇒ (𝑒𝑓 )𝑒 = 𝑒 ⇒ 𝑒𝑓 = 𝑒 that is 𝑒𝜔 𝑙 𝑓 . Thus 𝜇 is biorder preserving. Lemma 2.5. Let 𝑆 be an abundant semigroup. Then 𝜇(𝑆/𝜇) = 1. Proof. Denote the congruence 𝜇 on 𝜇(𝑆/𝜇) by 𝜇′ . Let 𝑎, 𝑏 ∈ 𝑆 be such that (𝑎𝜇, 𝑏𝜇) ∈ 𝜇′ , it will suffices to show that 𝑎𝜇𝑏 so that 𝜇′ is the trivial congruence on 𝑆/𝜇. For that it is enough to show that 𝜌𝑎 = 𝜌𝑏 because by duality we have 𝜆𝑎 = 𝜆𝑏 and hence 𝜙𝑎 = 𝜙𝑏 ie., 𝑎𝜇𝑏. Let 𝑥 ∈ 𝐸 such that 𝐿∗𝑥 𝜌𝑎 ∕= ∞. Then 𝑥ℛ∗ 𝑥𝑎 since 𝜇 is a congruence 𝑥𝜇 ∈ 𝐸(𝑆/𝜇) and 𝑥𝜇ℛ∗ (𝑥𝑎)𝜇. Now ℒ∗(𝑥𝑎)𝜇 ∩ 𝐸(𝑆/𝜇) = 𝐿∗𝑥𝜇 𝜌𝑎𝜇 = 𝐿∗𝑥𝜇 𝜌𝑏𝜇 = ℒ∗(𝑥𝑏)𝜇 ∩ 𝐸(𝑆/𝜇) and so (𝑥𝑎)𝜇ℛ∗ 𝑥𝜇ℛ∗ (𝑥𝑏)𝜇. Thus (𝑥𝑎)𝜇ℋ∗ (𝑥𝑏)𝜇 ⇒ 𝑥𝑎ℋ∗ 𝑥𝑏 so we have 𝐿∗𝑥 𝜌𝑏 = ℒ∗𝑥𝑏 ∩ 𝐸 = ℒ∗𝑥𝑎 ∩ 𝐸 = 𝐿∗𝑥 𝜌𝑎 . Hence the proof. Lemma 2.6. Let 𝑆 be an abundant semigroup. Then 𝑆 is fundamental if and only if 𝜇 = 1 Proof. Suppose 𝑆 is fundamental, then 𝜇 = 1 since 𝜇 is idempotent separating. Conversely, if 𝜇 = 1 and let 𝜎 be an idempotent separating congruence then 𝜎 is an ℋ∗ −congruence and 𝜎 ⊂ 𝜇 so 𝜎 = 1 ie., 𝑆 is fundamental. In the light of the above Lemmas and Proposition, now we summarize our main result as follows. Theorem 2.7. Let 𝑆 be an abundant semigroup with biordered set of idempotents 𝐸. Then 𝑆/𝜇 is fundamental and 𝐸 is isomorphic to a sub biordered set of 𝐸(𝑆/𝜇). Thus an abundant semigroup is a biorder preserving coextension of a fundamental semigroup.
References [1] D. Easdown, Biorder preserving coextensions of fundamental semigroups, Pro. Edinburgh Math. Soc 2 (3) (1988) 463–467. [2] P.M. Edwards, Fundamental semigroups, Proc. Roy. Soc. Edinburg Sect. A. 99 (3–4) (1985) 313–317. [3] J.M. Howie, An Introduction to Semigroup Theory, Academic Press, London, 1976. [4] D.B. McAlister, One-to-one partial right translations of a right cancellative semigroups, J. Algebra 43 (1976) 231–251. [5] K. S. S. Nambooripad, Structure of regular semigroups-I, Memoirs Amer. Math. Soc. 22 (224)(1979) 1–117. [6] F. Pastjin, A representation of a semigroup by a semigroup of matrices over a group with zero, Semigroup Forum 10 (1975) 238–249.
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[7] P.G. Romeo, Universal regular semigroup of a concordant semigroup, Bull. Calcutta Math. Society 88 (1996) 17–36. [8] P.G. Romeo, Concordant semigroups and balanced categories, Southeast Asian Bull. Math. 31 (2007) 949–961. [9] A. Sheena, Structure of concordant semigroups, J. Algebra 118 (1) (1988) 205–260.
