EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 580-588 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Left and right magnifying elements in generalized semigroups of transformations by using partitions of a set Ronnason Chinram1,4, Pattarawan Petchkaew2, Samruam Baupradist3,∗ 1 Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, 90110, Thailand 2 Mathematics and Statistics Program, Faculty of Science and Technology, Songkhla Rajabhat University, Songkhla, 90000, Thailand 3 Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand 4 Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand Abstract. An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = Ma]. Let X be a nonempty set and T (X) be the semigroup of all transformations from X into itself under the composition of functions. For a partition P = {Xα | α ∈ I} of the set X, let T (X,P ) = {f ∈ T (X) | (Xα)f ⊆ Xα for all α ∈ I}. Then T (X,P ) is a subsemigroup of T (X) and if P = {X}, T (X,P ) = T (X). Our aim in this paper is to give necessary and sufficient conditions for elements in T (X,P ) to be left or right magnifying. Moreover, we apply those conditions to give necessary and sufficient conditions for elements in some generalized linear transformation semigroups. 2010 Mathematics Subject Classifications: 20M10, 20M20 Key Words and Phrases: functions, transformation semigroups, partitions, left magnifying elements, right magnifying elements. 1. Introduction and Preliminaries The notions of left and right magnifying elements of semigroups were introduced by Ljapin [7]. An element a of a semigroup S is called left [right] magnifying if there exists a proper subset M of S such that S = aM [S = Ma]. Minimal subsets associated with the magnifying element, were introduced and studied by Migliorini in [9] and [10]. In [2], Catino and Migliorini gave necessary and sufficient conditions for any semigroup to contain left or right magnifying elements. In [8], Magill, Jr. gave necessary and sufficient ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3260 Email addresses: ronnason.c@psu.ac.th (R. Chinram), pattarawan.pe@gmail.com (P. Petchkaew), samruam.b@chula.ac.th (S. Baupradist) http://www.ejpam.com 580 c© 2018 EJPAM All rights reserved. R. Chinram, P. Petchkaew, S. Baupradist / Eur. J. Pure Appl. Math, 11 (3) (2018), 580-588 581 conditions for elements in transformation semigroups to be left or right magnifying and applied those conditions for elements in linear transformation semigroups and semigroups of all continuous selfmaps of topological spaces to be left or right magnifying. Gutan studied semigroups with strong and nonstrong magnifying elements in [3] and showed that every semigroup containing magnifying elements is factorizable in [4]. In [5], semigroups with magnifiers admitting minimal subsemigroups were studied by Gutan. Semigroups with good and bad magnifying were investigated by Gutan and Kisielewicz in [6]. Let X be a nonempty set and let T (X) be the set of all transformations from X into itself, that is, T (X) = {f : X → X | f is a function}. It is well-known that T (X) is a semigroup under the composition of functions and it is called the full transformation semigroup on X. Transformation semigroups play an important role in semigroup theory since it is well-known that every semigroup is isomorphic to a subsemigroup of a suitable full transformation semigroup. We will write functions from the right, (x)f rather than f(x) and compose from the left to the right, (x)(fg) rather than (g ◦f)(x), for f, g ∈ T (X) and x ∈ X. For a partition P = {Xα | α ∈ I} of a set X, consider the semigroup T (X,P ) = {f ∈ T (X) | (Xα)f ⊆ Xα for all α ∈ I}. We have that T (X,P ) is a subsemigroup of T (X) and if P = {X}, then T (X,P ) = T (X). In 2015, Araujo, Bentz, Mitchelll and Schneider [1] solved the problem of finding the minimum size of the generating sets of T (X,P ), when P is an arbitrary partition. Next, in 2016, Purisang and Rakbud investigated the regularity of transformation semigroups which defined by a partition in [11]. These are our motivation to do this research. Our aim in this paper is to give necessary and sufficient conditions for elements in T (X,P ) to be left or right magnifying. 2. Left magnifying elements of T (X,P ) Our purpose in this section is to give necessary and sufficient conditions for elements in T (X,P ) to be left magnifying. Lemma 1. If a function f is left magnifying of T (X,P ), then f is one-to-one. Proof. Assume that f is a left magnifying element of T (X,P ). Then there exists a proper subset M of T (X,P ) such that fM = T (X,P ). Let idX be an identity function on X. Clearly, idX ∈ T (X,P ). So there exists a function h ∈ M such that fh = idX . This implies that f is one-to-one. Lemma 2. If f ∈ T (X,P ) is bijective, then f is not left magnifying of T (X,P ). Proof. Suppose that f is left magnifying of T (X,P ). Then there exists a proper subset M of T (X,P ) such that fM = T (X,P ). This implies that fM = fT (X,P ). Since f is bijective, its inverse function f−1 exists and f−1 ∈ T (X,P ). So M = f−1fM = f−1fT (X,P ) = T (X,P ), a contradiction. Therefore, f is not left magnifying of T (X,P ). Lemma 3. If f ∈ T (X,P ) is one-to-one but not onto, then f is left magnifying of T (X,P ). R. Chinram, P. Petchkaew, S. Baupradist / Eur. J. Pure Appl. Math, 11 (3) (2018), 580-588 582 Proof. Assume that f is one-to-one but not onto. Let M = {h ∈ T (X,P ) | (x)h = x for all x /∈ ran f}. Then M is a proper subset of T (X,P ). Claim that fM = T (X,P ). Let g be any function in T (X,P ). Define a function h ∈ T (X,P ) by for all x ∈ X, (x)h = { (x′)g if x ∈ ran f and (x′)f = x, x if x /∈ ran f. Let x′, x ∈ X be such that (x′)f = x and x ∈ Xα for some α ∈ I. Clearly, x′ ∈ Xα. Therefore, (x)h = (x′)g ∈ Xα. Then h ∈M . For all x ∈ X, we have (x)fh = ((x)f)h = (x)g. Then fh = g, this implies that fM = T (X,P ). Hence f is left magnifying of T (X,P ). Example 1. Consider X = N and P = {{x | x is odd}, {x | x is even}}. Let f ∈ T (X,P ) by (x)f = { x+ 2 if x is even, x if x is odd, that is, f = ( 1 2 3 4 5 6 7 8 · · · 1 4 3 6 5 8 7 10 · · · ) . Then f ∈ T (X,P ) and f is one-to-one but not onto because 2 /∈ ran f . Let M = {h ∈ T (X,P ) | (2)h = 2}. Let g ∈ T (X,P ) be any function. Define a function h ∈ T (X,P ) by (x)h =  2 if x = 2, (x− 2)g if x is even and x > 2, (x)g if x is odd. So h ∈ M . If x is odd, we have (x)fh = ((x)f)h = (x)h = (x)g. If x is even, we have (x)fh = ((x)f)h = (x+ 2)h = (x)g. Then fh = g. For example, if g ∈ T (X,P ) such that (x)g = { 2x if x is even, x if x is odd, that is, g = ( 1 2 3 4 5 6 7 8 · · · 1 4 3 8 5 12 7 16 · · · ) . Define a function h ∈ T (X,P ) by (2)h = 2, (2x + 2)h = (2x)g = 4x and (2x − 1)h = (2x− 1)g = 2x− 1 for all x ∈ X, that is, h = ( 1 2 3 4 5 6 7 8 · · · 1 2 3 4 5 8 7 12 · · · ) . R. Chinram, P. Petchkaew, S. Baupradist / Eur. J. Pure Appl. Math, 11 (3) (2018), 580-588 583 So h ∈M and we have fh = ( 1 2 3 4 5 6 7 8 · · · 1 4 3 6 5 8 7 10 · · · )( 1 2 3 4 5 6 7 8 · · · 1 2 3 4 5 8 7 12 · · · ) = ( 1 2 3 4 5 6 7 8 · · · 1 4 3 8 5 12 7 16 · · · ) = g. The following theorem is the main result in this section. Theorem 1. Let P = {Xα | α ∈ I} be a partition of a set X. (1) A semigroup T (X,P ) has a left magnifying element if and only if Xα is infinite for some α ∈ I. (2) A function f is left magnifying of T (X,P ) if and only if f is one-to-one but not onto. Proof. This follows by Lemma 1, Lemma 2 and Lemma 3. Example 2. Let X = N and P = {{1, 2}, {3, 4, 5}, {x | x > 5}}. By Theorem 1(1), T (X,P ) has a left magnifying element. Let f ∈ T (X,P ) by (x)f = { x if x ≤ 6, x+ 1 if x > 6, that is, f = ( 1 2 3 4 5 6 7 8 · · · 1 2 3 4 5 6 8 9 · · · ) . Then f is one-to-one but not onto. By Theorem 1(2), f is left magnifying of T (X,P ). Corollary 1. The following statements hold for a semigroup T (X). (1) A semigroup T (X) has a left magnifying if and only if X is infinite. (2) A function f is left magnifying of T (X) if and only if f is one-to-one but not onto. Proof. This follows by Theorem 1 by using P = {X}. 3. Right magnifying elements of T (X,P ) In this section, we give necessary and sufficient conditions for elements in T (X,P ) to be right magnifying. Lemma 4. If f is a right magnifying element of T (X,P ), then f is onto. Proof. Assume that f is a right magnifying element of T (X,P ). Then there exists a proper subset M of T (X,P ) such that Mf = T (X,P ). Since idX ∈ T (X,P ), there exists a function h ∈M such that hf = idX . This implies that f is onto. R. Chinram, P. Petchkaew, S. Baupradist / Eur. J. Pure Appl. Math, 11 (3) (2018), 580-588 584 Lemma 5. If f ∈ T (X,P ) is bijective, then f is not right magnifying of T (X,P ). Proof. Assume that f is bijective. Then its inverse function f−1 exists and f−1 ∈ T (X,P ). Suppose that f is a right magnifying element of T (X,P ). Then there exists a proper subset M of T (X,P ) such that Mf = T (X,P ). Hence Mf = T (X,P )f and M = Mff−1 = T (X,P )ff−1 = T (X,P ), a contradiction. Therefore, f is not right magnifying of T (X,P ). Lemma 6. Let f ∈ T (X,P ) be onto but not one-to-one. Then f is right magnifying of T (X,P ). Proof. Assume that f is onto but not one-to-one. Let M = {h ∈ T (X,P ) | h is not onto}. Then M 6= T (X,P ). Let g be any function in T (X,P ). Since f is onto, there exists for each x ∈ Xα, an element yx ∈ Xα such that (yx)f = (x)g (if (x1)g = (x2)g, we must choose yx1 = yx2). Define a function h ∈ T (X,P ) by (x)h = yx for all x ∈ X. We claim that h is not onto. Since f is not one-to-one, there exist an element y′ ∈ X and distinct elements y1, y2 ∈ X such that (y1)f = (y2)f = y′. If y′ /∈ ran g, we have y1, y2 /∈ ranh. If y′ ∈ ran g, there is at most one between y1 and y2 in ranh. Then h is not onto. Hence h ∈M and for all x ∈ X, we have (x)hf = (yx)f = (x)g. Then hf = g, hence Mf = T (X,P ). Therefore, f is right magnifying of T (X,P ). Example 3. Consider X = N and P = {{x | x is odd}, {x | x is even}}. Let f ∈ T (X,P ) by f(1) = 1, f(2) = 2 and (x)f = x − 2 for all positive integer x > 2, that is, f = ( 1 2 3 4 5 6 7 8 · · · 1 2 1 2 3 4 5 6 · · · ) . Then f ∈ T (X,P ) and f is onto but not one-to-one. Let M = {h ∈ T (X,P ) | h is not onto}. Let g be any function in T (X,P ). By Lemma 6, there exists h ∈ M such that hf = g. For example, if g ∈ T (X,P ) is such that (x)g = x+ 2 for all x ∈ X, that is, g = ( 1 2 3 4 5 6 7 8 · · · 3 4 5 6 7 8 9 10 · · · ) . Define a function h ∈ T (X,P ) by (x)h = x+ 4 for all positive integer x, that is, h = ( 1 2 3 4 5 6 7 8 · · · 5 6 7 8 9 10 11 12 · · · ) . So h ∈M and we have hf = ( 1 2 3 4 5 6 7 8 · · · 5 6 7 8 9 10 11 12 · · · )( 1 2 3 4 5 6 7 8 · · · 1 2 1 2 3 4 5 6 · · · ) = ( 1 2 3 4 5 6 7 8 · · · 3 4 5 6 7 8 9 10 · · · ) = g. Our main result in this section is the following theorem. R. Chinram, P. Petchkaew, S. Baupradist / Eur. J. Pure Appl. Math, 11 (3) (2018), 580-588 585 Theorem 2. Let P = {Xα | α ∈ I} be a partition of a set X. (1) A semigroup T (X,P ) has a right magnifying element if and only if Xα is infinite for some α ∈ I. (2) A function f is right magnifying of T (X,P ) if and only if f is onto but not one-to- one. Proof. This follows by Lemma 4, Lemma 5 and Lemma 6. Example 4. Let X = N and P = {{1, 2}, {3, 4, 5}, {x | x > 5}}. By Theorem 2(1), T (X,P ) has a right magnifying element. Let f ∈ T (X,P ) by (x)f = { x if x ≤ 6, x− 1 if x > 6, that is, f = ( 1 2 3 4 5 6 7 8 · · · 1 2 3 4 5 6 6 7 · · · ) . Then f is onto but not one-to-one. By Theorem 2(2), f is right magnifying of T (X,P ). Corollary 2. The following statements hold for a semigroup T (X). (1) A semigroup T (X) has a right magnifying if and only if X is infinite. (2) A function f is right magnifying of T (X) if and only if f is onto but not one-to-one. Proof. This follows by Theorem 2 by using P = {X}. 4. Application to left and right magnifying elements of some generalized transformation semigroup Let V1 and V2 be subspaces of a vector space V over a field F such that V = V1 ⊕ V2. This mean that V = V1 + V2 and V1 ∩ V2 = {0}. Let L(V ) be the semigroup of all linear transformations from V into itself under the composition of functions and LP (V ) = {f ∈ L(V ) | (V1)f ⊆ V1 and (V2)f ⊆ V2}. Then LP (V ) is a subsemigroup of L(V ). If V = V1 and V2 = {0}, then LP (V ) = L(V ). Our purpose in this section is to give necessary and sufficient condition for elements in LP (V ) to be right or left magnifying. Lemma 7. If a function f is left magnifying of LP (V ), then f is one-to-one. Proof. This is similar to the proof of Lemma 1. Lemma 8. If f ∈ LP (V ) is bijective, then f is not left magnifying of LP (V ). Proof. This is similar to the proof of Lemma 2. R. Chinram, P. Petchkaew, S. Baupradist / Eur. J. Pure Appl. Math, 11 (3) (2018), 580-588 586 Lemma 9. If f ∈ LP (V ) is one-to-one but not onto, then f is left magnifying of LP (V ). Proof. Assume that f is one-to-one but not onto. Let M = {h ∈ LP (V ) | (v)h = 0 for all v /∈ ran f}. Claim that fM = LP (V ). Let g be any linear transformation in LP (V ). Let B1 and B2 be bases of V1 and V2, respectively. Clearly, B1 +B2 is a basis of V . Define a linear transformation h ∈ LP (V ) by for all v ∈ B1 ∪B2, (v)h = { (v′)g if v ∈ ran f and (v′)f = v, 0 if v /∈ ran f. Let v′, v ∈ V be such that (v′)f = v. Assume that v ∈ B1. Clearly, v′ ∈ V1. Therefore, (v)h = (v′)g ∈ V1. Similarly, if v ∈ B2, then (v)h = (v′)g ∈ V2. Thus h ∈ M and fh = g, this implies that fM = LP (V ). Hence f is left magnifying of LP (V ). Theorem 3. The following statements hold for a semigroup LP (V ). (1) A semigroup LP (V ) has a left magnifying element if and only if dimV1 is infinite or dimV2 is infinite. (2) A linear transformation f is left magnifying of LP (V ) if and only if f is one-to-one but not onto. Proof. This follows by Lemma 7, Lemma 8 and Lemma 9. Corollary 3. Let L(V ) be the linear transformation semigroup on a vector space V . (1) A semigroup L(V ) has a left magnifying if and only if dimV is infinite. (2) A linear transformation f is left magnifying of L(V ) if and only if f is one-to-one but not onto. Proof. This follows by Theorem 3 by using V = V1 and V2 = {0}. Lemma 10. If f is a right magnifying element of LP (V ). Then f is onto. Proof. This is similar to the proof of Lemma 4. Lemma 11. If f ∈ LP (V ) is bijective, then f is not right magnifying of LP (V ). Proof. This is similar to the proof of Lemma 5. Lemma 12. Let f ∈ LP (V ) be onto but not one-to-one, then f is right magnifying of LP (V ). REFERENCES 587 Proof. Assume that f is onto but not one-to-one. Let M = {f ∈ LP (V ) | f is not onto}. Then M 6= LP (V ). Let g be any linear transformation in LP (V ). Let B1 and B2 be bases of V1 and V2, respectively. Since f is onto, there exists for each v ∈ B1, an element uv ∈ B1 such that (uv)f = (v)g and there exists for each v ∈ B2, an element uv ∈ B2 such that (uv)f = (v)g. Define a linear transformation h ∈ LP (V ) by (v)h = uv for all v ∈ B1 ∪ B2. Then h ∈ LP (V ). Since f is not one-to-one, h is not onto, and so h ∈ M . Then hf = g, and hence Mf = LP (V ). Therefore, f is right magnifying of LP (V ). Theorem 4. The following statements hold for a semigroup LP (V ). (1) A semigroup LP (V ) has a right magnifying element if and only if dimV1 is infinite or dimV2 is infinite. (2) A linear transformation f is right magnifying of LP (V ) if and only if f is onto but not one-to-one. Proof. This follows by Lemma 10, Lemma 11 and Lemma 12. Corollary 4. Let L(V ) be the linear transformation semigroup on a vector space V . (1) A semigroup L(V ) has a right magnifying if and only if dimV is infinite. (2) A linear transformation f is right magnifying of L(V ) if and only if f is onto but not one-to-one. Proof. This follows by Theorem 4 by using V = V1 and V2 = {0}. Acknowledgements This paper was supported by Algebra and Applications Research Unit, Prince of Songkla University. References [1] J Araujo, W Bentz, J D Mitchelll and C Schneider. The rank of the semigroup of transformations stabilising a partition of a finite set, Mathematical Proceedings of the Cambridge Philosophical Society, 159:339–353, 2015. [2] F Catino and F Migliorini. Magnifying elements in semigroups. Semigroup Forum, 44:314–319, 1992. [3] M Gutan. Semigroups with strong and nonstrong magnifying elements. Semigroup Forum, 53:384–386, 1996. [4] M Gutan. Semigroups which contain magnifying elements are factorizable. Commu- nications in Algebra, 25:3953–3963, 1997. 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