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Báo cáo toán học: The Translational Hull of a Strongly Right or Left Adequate Semigroup

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Chúng tôi chứng minh rằng thân tịnh của một nửa nhóm đủ mạnh mẽ bên phải hoặc trái là cùng loại. Kết quả của chúng tôi khuếch đại một kết quả nổi tiếng của Fountain và Lawson thân tịnh của một nửa nhóm đầy đủ được đưa ra vào năm 1985. 2000 Toán Phân loại Chủ đề...
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Báo cáo toán học: " The Translational Hull of a Strongly Right or Left Adequate Semigroup"Vietnam Journal of Mathematics 34:4 (2006) 441–447 9LHWQD P -RXUQDO RI 0$ 7+ (0$ 7, &6 ‹ 9$67 The Translational Hull of a Strongly Right or Left Adequate Semigroup X. M. Ren1 * and K. P. Shum2+ 1 Dept. of Mathematics, Xi’an University of Architecture and Technology Xi’an 710055, China2 Faculty of Science, The Chinese University of Hong Kong, Hong Kong, China Dedicated to Professor Do Long Van on the occasion of his 65th birthday Received May 10, 2005 Revised October 5, 2006Abstract. We prove that the translational hull of a strongly right or left adequatesemigroup is still of the same type. Our result amplifies a well known result of Fountainand Lawson on translational hull of an adequate semigroup given in 1985.2000 Mathematics Subject Classification: 20M10.Keywords: Translational hulls, right adequate semigroups, strongly right adequatesemigroups.1. IntroductionWe call a mapping λ from a semigroup S into itself a left translation of S ifλ(ab) = (λa)b for all a, b ∈ S . Similarly, we call a mapping ρ from S into itselfa right translation of S if (ab)ρ = a(bρ) for all a, b ∈ S . A left translation λ anda right translation ρ of S are said to be linked if a(λb) = (aρ)b for all a, b ∈ S.In this case, we call the pair (λ, ρ) a bitranslation of S . The set Λ(S ) of all left∗ Thisresearch is supported by the National Natural Science Foundation of China (Grant No.10671151); the NSF grant of Shaanxi Province, grant No. 2004A10 and the SF grant of Ed-ucation Commission of Shaanxi Province, grant No. 05JK240, P. R. China.+ This research is partially supported by a RGC (CUHK) direct grant No. 2060297 (2005/2006).442 X. M. Ren and K..P. Shumtranslations (and also the set P (S ) of all right translations) of the semigroupS forms a semigroup under the composition of mappings. By the translationalhull of S , we mean a subsemigroup Ω(S ) consisting of all bitranslations (λ, ρ)of S in the direct product Λ(S ) × P (S ). The concept of translational hull ofsemigroups and rings was first introduced by Petrich in 1970 (see [11]). Thetranslational hull of an inverse semigroup was first studied by Ault [1] in 1973.Later on, Fountain and Lawson [2] further studied the translational hulls of ad-equate semigroups. Recently, Guo and Shum [6] investigated the translationalhull of a type-A semigroup, in particular, the result obtained by Ault [1] was sub-stantially generalized and extended. Thus, the translational hull of a semigroupplays an important role in the general theory of semigroups. Recall that the generalized Green left relation L∗ is defined on a semigroupS by aL∗b when ax = ay if and only if bx = by, for all x, y ∈ S 1 (see, forexample, [4]). We now call a semigroup S an rpp semigroup if every L∗ -class ofS contains an idempotent of S . According to Fountain in [3], an rpp semigroupwhose idempotents commute is called a right adequate semigroup. By Guo,Shum and Zhu [7], an rpp semigroup S is called a strongly rpp semigroup iffor any a ∈ S, there is a unique idempotent e such that aL∗ e and a = ea.Thus, we naturally call a right adequate semigroup S a strongly right adequatesemigroup if S is a strongly rpp semigroup. Dually, we may define the Green starright relation R∗ on a semigroup S and define similarly a strongly left adequatesemigroup . In this paper, we shall show that the translational hull of a strongly right(left) adequate semigroup is still the same type. Thus, the result obtained byFountain and Lawson in [2] for the translational hull of an adequate semigroupwill be amplified. As a consequence, we also prove that the translational hull ofa C -rpp semigroup is still a C -rpp semigroup.2. PreliminariesThroughout this paper, we will use the notions and terminologies given in [3, 8,9]. We first call a semigroup S an idempotent balanced semigroup if for anya ∈ S , there exist idempotents e and f in S such that a = ea = af holds. The following lemmas will be useful in studying the translational hull of astrongly right (left) adequate semigroup.Lemma 2.1. Let S be an idempotent balanced semigroup. Then the followingstatements hold: (i) If λ1 and λ2 are left translations of S , then λ1 = λ2 if and only if λ1 e = λ2 e for all e ∈ E .(ii) If ρ1 and ρ2 are right translations of S , then ρ1 = ρ2 if and only if eρ1 = eρ2 for all e ∈ E .Proof ...

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