Green s Relations and Partial Orders on Semigroups of Partial Linear Transformations with Restricted Range

Thai Journal of Mathematics Volume 12 2014 Number 1 : 81 93 http://thaijmathincmuacth ISSN 1686-0209 Green s Relations and Partial Orders on Semigroups of Partial Linear Transformations with Restricted Range Kritsada Sangkhanan and Jintana Sanwong 1 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand e-mail : kritsada kst@hotmailcom K Sangkhanan jintanas@cmuacth J Sanwong Abstract : Let V be any vector space and P V the set of all partial linear transformations defined on V, that is, all linear transformations α : S T where S, T are subspaces of V Then P V is a semigroup under composition Let W be a subspace of V We define P T V, W = {α P V : V α W } So P T V, W is a subsemigroup of P V In this paper, we present the largest regular subsemigroup and determine Green s relations on P T V, W Furthermore, we study the natural partial order on P T V, W in terms of domains and images and find elements of P T V, W which are compatible Keywords : regular elements; Green s relations; partial linear transformation semigroups; natural order; compatibility 2010 Mathematics Subject Classification : 20M20 1 Introduction A partial transformation semigroup is the collection of functions from a subset of X into X with composition which is denoted by P X In addition, the 1 Corresponding author Copyright c 2014 by the Mathematical Association of Thailand All rights reserved

82 Thai J M ath 12 2014/ K Sangkhanan and J Sanwong semigroup T X and IX are defined by T X = {α P X : dom X} and IX = {α P X : α is injective} We note that if we let α P X and Z X, the notation Zα means {zα : z Z dom α} It is clear that, X im α In [2], Fernandes and Sanwong introduced the partial transformation semigroup with restricted range They considered the semigroups P T X, Y and IX, Y defined by P T X, Y = {α P X : Xα Y } and IX, Y = {α IX : Xα Y } where Y is a subset of X They proved that P F = {α P T X, Y : X Y α} is the largest regular subsemigroup of P T X, Y Moreover, they determined Green s relations on P T X, Y and IX, Y In 2008, Sanwong and Sommanee [9] studied the subsemigroup T X, Y = T X P T X, Y of T X where Y is a subset of X They gave a necessary and sufficient condition for T X, Y to be regular In the case when T X, Y is not regular, the largest regular subsemigroup was obtained and this subsemigroup was shown to determine the Green s relations on T X, Y Also, a class of maximal inverse subsemigroups of T X, Y was obtained Analogously to P X, we can define a partial linear transformation on some vector spaces Let V be any vector space, P V the set of all linear transformations α : S T where S and T are subspaces of V, that is, every element α P V, the domain and range of α are subspaces of V Then we have P V under composition is a semigroup and it is called the partial linear transformation semigroup of V The subsemigroups T V and IV are defined by T V = {α P V : dom V } and IV = {α P V : α is injective} Similarly, the linear transformation semigroups with restricted range can be defined as follows For any vector space V and a subspace W of V, P T V, W = {α P V : V α W }, T V, W = {α T V : V α W } and IV, W = {α IV : V α W } Obviously, P T V, V = P V, T V, V = T V and IV, V = IV Hence we may regard P T V, W, T V, W and IV, W as generalizations of P V, T V and IV, respectively It is known that Green s relations on T V are as follows see [3], page 63 Let α, β T V Then

Green s relations and partial orders on semigroups 83 αlβ if and only if V V β; αrβ if and only if ker ker β; αdβ if and only if dimv dimv β; D = J In 2007, Droms [1] gave a complete description of Green s relations on P V and IV We have for α, β P V : And for α, β IV : αlβ if and only if V V β; αrβ if and only if ker ker β and dom dom β; αdβ if and only if dimv dimv β; D = J αlβ if and only if V V β; αrβ if and only if dom dom β; αdβ if and only if dimv dimv β; D = J Later in 2008, Sullivan [11] described Green s relations and ideals for the semigroup T V, W And its Green s relations are as follows Let Q = {α T V, W : V α W α} For α, β T V, W : αlβ if and only if β or α, β Q and V V β; αrβ if and only if ker ker β; αdβ if and only if ker ker β or α, β Q and dimv dimv β; αj β if and only if ker ker β or dimv dimw dimw β = dimv β Now, we deal with a natural partial order or Mitsch order [6] on any semigroup S defined by for a, b S a b if and only if a = xb = by, xa = a for some x, y S 1 In 2005, Sullivan [10] studied the natural partial order on P V The author found all elements of P V which are compatible with respect to In 2012, Sangkhanan and Sanwong [7] characterized the natural partial order on P T X, Y and found elements of P T X, Y which are compatible with Recently, they presented the largest regular subsemigroup of IV, W and determined its Green s relations in [8] Furthermore, the authors studied the natural partial order on IV, W in terms of domains and images Finally, they also found elements of IV, W which are compatible In this paper, we describe the largest regular subsemigroup of P T V, W and characterized its Green s relations Furthermore, we study the natural partial order on P T V, W in terms of domains and images Moreover, we characterize elements of P T V, W which are compatible

84 Thai J M ath 12 2014/ K Sangkhanan and J Sanwong 2 Regularity and Green s relations on P T V, W Since P T V, W = {α P V : V α W }, we have the following simple result on P T V, W which will be used throughout the paper Lemma 21 If S and T are subspaces of V with S T, then Sα T α for all α P T V, W For convenience, we adopt the convention: namely, if α P X then we write Xi and take as understood that the subscript i belongs to some unmentioned index set I, the abbreviation {a i } denotes {a i : i I}, and that X {a i } and a i α 1 = X i Similarly, we can use this notation for elements in P V To construct a map α P V, we first choose a basis {e i } for a subspace of V and a subset {a i } of V, and then let e i a i for each i I and extend this map linearly to V To shorten this process, we simply say, given {e i } and {a i } within the context, then for each α P V, we can write ei A subspace U of V generated by a linearly independent subset {e i } of V is denoted by e i and when we write U = e i, we mean that the set {e i } is a basis of U, and we have dim U = I For each α P V, the kernel and the range of α denoted by ker α and V α respectively, and the rank of α is dimv α Let V be a vector space and {u i } a subset of V The notation a i u i means the linear combination: a i a i a i1 u i1 + a i2 u i2 + + a in u in for some n N, u i1, u i2,, u in {u i } and scalars a i1, a i2,, a in Suppose that α P T V, W and U is a subspace of V If we write U u i α, it means that u i U dom α for all i In addition, we can show that {u i } is linearly independent Let P Q = {α P T V, W : V α W α} For α P Q and β P T V, W, we obtain V α W α which implies that V αβ W αβ So αβ P Q Therefore, P Q is a right ideal of P T V, W Lemma 22 The set P Q is a right ideal of P T V, W Theorem 23 Let α P T V, W Then α is regular if and only if α P Q Consequently, P Q is the largest regular subsemigroup of P T V, W Proof From Lemma 22, we see that P Q is a subsemigroup of P T V, W Let α P Q Then V α W w j α So {w j } is linearly independent If v dom α,

Green s relations and partial orders on semigroups 85 then v x j w j x j w j α for some scalars x j So v x j w j 0 implies v x j w j ker α Hence v ker α + w j Let u ker α w j Then u 0 and u = y j w j for some scalars y j, so 0 = u y j w j α implies y j = 0 for all j since {w j α} is linearly independent Hence ker α w j = 0 which follows that dom ker α w j If ker u i and W = W α v k, we can write ui w j 0 w j α and define β = vk w j α 0 w j We can see that V β = w j W, so β P T V, W and αβα Hence α is regular Now, let α be any regular element in P T V, W Then αβα for some β P T V, W, so V V αβ V αβα W α Therefore, α P Q By the above theorem, we have the following corollary Corollary 24 Let W be a non-zero subspace of a vector space V Then P T V, W is a regular semigroup if and only if V = W Proof It is clear that if V = W, then P T V, W = P V and P T V, W is regular Conversely, if W is a proper subspace of V, then we can write W = w i and V = w i v j Since W is a non-zero subspace, we choose w i1 {w i } and v j1 {v j } Define wi v j1 0 w i1 Hence W 0 w i1 = V α and then α is not regular by Theorem 23 By the above corollary, we note that if W is a non-zero proper subspace of a vector space V, then P T V, W is not a regular semigroup It is concluded that, in this case, P T V, W is not isomorphic to P U for any vector space U since P U is regular This shows that P T V, W is almost never isomorphic to P U Lemma 25 Let α, β P T V, W Then γβ for some γ P T V, W if and only if V α W β Proof If γβ for some γ P T V, W, then V V γβ W β Now, assume that V α W β and write V v i α Hence {v i } is linearly independent For each i, there is w i W such that v i w i β Thus {w i β} is linearly independent Now, let V β = w i β v j β, ker u r and ker β = u s Then {u r } {v i } and {u s } {w i } {v j } are linearly independent Since dom ker α v i and dom β = ker β w i v j, by the same proof as given for [11, Lemma 2] then is γ P T V, W such that γβ, as required By the above lemma, we get the following result immediately

86 Thai J M ath 12 2014/ K Sangkhanan and J Sanwong Lemma 26 Let α, β P T V, W P T V, W if and only if V α V β If β P Q, then γβ for some γ Theorem 27 Let α, β P T V, W Then αlβ if and only if α, β P Q and V V β or α, β P T V, W \ P Q and β Proof Assume that αlβ Then λβ and β = µα for some λ, µ P T V, W 1 Suppose that α P Q If λ = 1 or µ = 1, then β = α P Q and V V β On the other hand, if λ, µ P T V, W then V β = V µ V µλβ W β since V µλ W Thus β P Q From λβ and β = µα, we have V V β by Lemma 26 Now, suppose that α P T V, W \ P Q If λ, µ P T V, W, then V V λβ = V λµα W α which contradicts α P T V, W \ P Q Thus λ = 1 or µ = 1 and so β = α P T V, W \ P Q The converse is a direct consequence of Lemma 26 Theorem 28 If α, β P T V, W, then βγ for some γ P T V, W if and only if dom α dom β and ker β ker α Consequently, αrβ if and only if dom dom β and ker ker β Proof It is clear that if βγ for some γ P T V, W, then dom α dom β Let v ker β Then vβ = 0 implies v vβγ = 0, so ker β ker α Conversely, suppose dom α dom β and ker β ker α Write ker β = u i, ker u i, u j and dom ker α v k Since dom α dom β, we have dom β = dom α v s Then ui u j v k ui u 0 0 w k, β = j v k v s 0 w j w k w s for some w k, w j, w k, w s W We can see that {w j, w k } is linearly independent Define γ P T V, W by wj w γ = k 0 w k Then βγ, as required Lemma 29 Let α, β P T V, W If dom dom β and ker ker β then either both α and β are in P Q, or neither is in P Q Consequently, αrβ if and only if α, β P Q, dom dom β and ker ker β or α, β P T V, W \P Q, dom dom β and ker ker β Proof Assume that dom dom β and ker ker β and suppose that α, β P Q is false So one of α or β is not in P Q, we suppose that α / P Q Thus V \ W α W α, so there is v 0 V \ W such that v 0 α wα for all w W Thus v 0 w / ker α for all w W If β P Q, then V β = W β, so v 0 β = wβ for some w W v 0 dom dom β which implies that v 0 w ker β = ker α which is a contradiction Therefore β / P Q

Green s relations and partial orders on semigroups 87 As a direct consequence of Theorem 27, Theorem 28 and Lemma 29, we have the following corollary Corollary 210 Let α, β P T V, W Then αhβ if and only if α, β P Q, V V β, dom dom β and ker ker β or α, β P T V, W \ P Q and β Theorem 211 Let α, β P T V, W Then αdβ if and only if α, β P Q and dimv dimv β or α, β P T V, W \ P Q, dom dom β and ker ker β Proof Let α, β P T V, W be such that αdβ Then αlγ and γrβ for some γ P T V, W If α P Q, then since αlγ, we must have γ P Q and V V γ From γrβ, we get β P Q, dom γ = dom β and ker γ = ker β So we obtain dimv dimv γ = dimdom γ/ ker γ = dimdom β/ ker β = dimv β If α P T V, W \ P Q, then γ = α since αlγ and thus αrβ which implies that dom dom β and ker ker β So by Lemma 29, we must have β P T V, W \ P Q Conversely, assume that the conditions hold Clearly, if α, β P T V, W \P Q, dom dom β and ker ker β then αrβ, and so αdβ since R D If α, β P Q and dimv dimv β, then V W w j α and V β = W β = w j β Let ker u r and ker β = u s, so we can write ur w j 0 w j α, β = us w j 0 w j β where w j, w j W If γ P T V, W is defined by ur w γ = j 0 w j β, then dom γ = dom α, ker γ = ker α, V γ = V β and γ P Q, so αrγlβ Lemma 212 Let α, β P T V, W If λβµ for some λ P T V, W and µ P T V, W 1, then dimv α dimw β Proof Since V V λβµ W βµ, we have dimv α dimw βµ Let W βµ = w i µ where {w i } W β and {w i } is linearly independent Then w i W β which implies that Therefore, dimv α dimw β, dimw βµ = dim w i µ = dim w i dimw β

88 Thai J M ath 12 2014/ K Sangkhanan and J Sanwong Theorem 213 Let α, β P T V, W Then αj β if and only if dom dom β and ker ker β or dimv dimw dimw β = dimv β Proof Assume that αj β Then λβµ and β = λ αµ for some λ, λ, µ, µ P T V, W 1 If λ = 1 = λ, then βµ and β = αµ which imply that αrβ Thus dom dom β and ker ker β If either λ or λ is in P T V, W, then we can write σβδ and β = σ αδ for some σ, σ P T V, W and δ, δ P T V, W 1 For example, if λ = 1 and λ P T V, W, then βµ and β = λ αµ imply βµ = λ αµ µ = λ αµ µ = λ βµµ µ = λ βµµ µ Thus, by Lemma 212, it follows that dimw β dimv α dimw α dimv β dimw β, whence dimv dimw dimw β = dimv β Conversely, assume that the conditions hold If dom dom β and ker ker β, then αrβ and so αj β Now, suppose that dimv dimw dimw β = dimv β Let ker u i, ker β = u j and V v k α Then dom ker α v k Since dimv dimw β, we let W β = w k β and dom β = ker β w k v l Then we write ui v k 0 w k, β = uj w k v l 0 w k β w l Let V = w k β v m and define λ, µ P T V, W by ui v λ = k vm w 0 w k, µ = k β 0 w k Then λβµ, as required Similarly, we can show that β = λ αµ for some λ, µ P T V, W by using the equality dimv β = dimw α Corollary 214 If α, β P Q, then αj β on P T V, W if and only if αdβ on P T V, W Proof In general, we have D J Let α, β P Q and αj β on P T V, W Then dom dom β and ker ker β or dimv dimw dimw β = dimv β If dom dom β and ker ker β, then dimv dimdom α/ ker dimdom β/ ker β = dimv β Thus, both cases imply dimv dimv β and αdβ on P T V, W by Theorem 211 Theorem 215 D = J on P T V, W if and only if dim W is finite or V = W Proof It is clear that if V = W, then P T V, W = P V = P Q which follows that D = J by Corollary 214 Suppose that dim W is finite Let α, β P T V, W with αj β If dom dom β and ker ker β, then αrβ and hence αdβ

Green s relations and partial orders on semigroups 89 Now, assume that dimv dimw dimw β = dimv β Since dim W is finite, we have dimv α, dimv β are finite which implies that V W α and V β = W β Thus α, β P Q and dimv dimv β Therefore, αdβ Conversely, suppose that dim W is infinite and W V Let {v i } be a basis of W and {v j } a basis of V such that I J Then there is an infinite countable subset {u n } of {v i } where n N Let v {v j } \ {v i } and define α, β by v un u 1 u 2n u2n 1 v u, β = 2n 0 u 1 u 4n Then α, β P T V, W \ P Q and dimv dimw ℵ 0 = dimw β = dimv β, so αj β Since ker 0 = u 2n 1 = ker β, we have α and β are not D-related on P T V, W 3 Partial orders Recall that the natural partial order on any semigroup S is defined by or equivalently a b if and only if a = xb = by, xa = a for some x, y S 1, a b if and only if a = wb = bz, az = a for some w, z S 1 31 In this paper, we use 31 to define the partial order on the semigroup P T V, W, that is for each α, β P T V, W α β if and only if γβ = βµ, αµ for some γ, µ P T V, W 1 We note that if W V, then P T V, W has no identity elements So, in this case P T V, W 1 P T V, W In addintion, on P T V, W does not coincide with the restriction of on P V For example, let V = v 1, v 2, v 3 and W = v 1, v 2 Define If we let v1 v 2 v 3 v 1 v 1 v 1 γ = and β = v1 v 2 v 3 v 2 v 2 v 1 v1 v 2 v 3 v1 v and µ = 2 v 3, v 3 v 3 v 3 v 1 v 1 v 3 then γβ = βµ, αµ which implies that α β in P V but we cannot find γ P T V, W 1 such that γβ Hence α β in P T V, W In [4], Kowol and Mitsch characterized on T X as follows If α, β T X, then the following statements are equivalent 1 α β 2 Xα Xβ and βµ for some idempotent µ T X

90 Thai J M ath 12 2014/ K Sangkhanan and J Sanwong 3 ββ 1 αα 1 and λβ for some idempotent λ T X 4 Xα Xβ, ββ 1 αα 1 and x xβ for all x X with xβ Xα In [5], Marques-Smith and Sullivan extended the above result to P X as follows If α, β P X, then α β if and only if Xα Xβ, dom α dom β, αβ 1 αα 1 and ββ 1 dom β dom α αα 1 Later in [10], Sullivan proved an analogue for P V as follows If α, β P V, then α β if and only if V α V β, dom α dom β, ker β ker α and V αβ 1 Eα, β where Eα, β = {u V : u uβ} Recently, we extended the result for P X to P T X, Y see [7] For α, β P T X, Y, α β if and only if β or the following statements hold 1 Xα Y β 2 dom α dom β and ker β dom β dom α ker α 3 For each x dom β, if xβ Xα, then x dom α and x xβ Now, we aim to prove an analogue result for P T V, W and this result extends a similar result on P V Theorem 31 Let α, β P T V, W Then α β if and only if β or the following statements hold 1 V α W β 2 dom α dom β 3 V αβ 1 Eα, β Proof Suppose that α β Then there exist γ, µ P T V, W 1 such that γβ = βµ and αµ If γ = 1 or µ = 1, then β If γ, µ P T V, W, then 1 and 2 hold by Lemma 25 and Theorem 28 If v V αβ 1, then vβ V α which implies that vβ = wα for some w V, thus vβ = w wαµ = vβµ = vα Hence v dom α and v vβ So v Eα, β Conversely, assume that the conditions 1-3 hold To show that ker β ker α, let v ker β Then v dom β and vβ = 0 V α which implies that v V αβ 1 Eα, β We obtain v dom α and v vβ = 0 So, v ker α Again by Lemma 25 and Theorem 28, there exist γ, µ P T V, W such that γβ = βµ Now, we prove that V α dom µ, by letting w V α Then there is v dom α such that v w Since γβ, we have w = v vγβ from which it follows that vγ V αβ 1 Eα, β That is vγ dom α and vγ vγβ Thus vγβ = vγ vγβµ = wµ which implies that w dom µ So, V α dom µ Hence dom αµ = im α dom µα 1 = im αα 1 = dom α

Green s relations and partial orders on semigroups 91 For each v dom α, v vγβ We obtain vγ V αβ 1 Eα, β which implies that vγ dom α and vγ vγβ Thus Therefore, αµ v vγβ = vγ vγβµ = vαµ Let be a partial order on a semigroup S An element c S is said to be left [right] compatible if ca cb [ac bc] for each a, b S such that a b Now, we characterize all elements in P T V, W which are compatible with respect to We first prove the following lemma We note that a zero partial linear transformation is a zero map having domain as a subspace of V Lemma 32 Let dim W = 1 and α, β P T V, W If α β, then β or α is a zero partial linear transformation Proof Suppose that α β and α is not a zero partial linear transformation So 1 dim V α dim W = 1, and then V W For each v dom β, vβ V β W = V α which implies that v V αβ 1 Eα, β by Theorem 313 Hence v dom α and v vβ Thus dom β dom α Since α β, we have dom α dom β Therefore, dom dom β and β Theorem 52 in [7] showed that if Y > 1 and = γ P T X, Y, then 1 γ is left compatible with if and only if Y γ = Y ; 2 γ is right compatible with if and only if Y dom γ and γ Y is injective or Y dom γ = And Theorem 31 in [10] proved that if γ P V has non-zero rank and dim V > 1, then 1 γ P V is left compatible with if and only if γ is surjective; 2 γ P V is right compatible with if and only if γ T V and γ is injective For the semigroup P T V, W, we have the following result Theorem 33 Let γ P T V, W The following statements hold 1 If dim W = 1, then every element in P T V, W is always left compatible with 2 If dim W > 1, then γ is left compatible with if and only if W γ = W or γ is a zero partial linear transformation 3 If dim W 1, then γ P T V, W is right compatible with if and only if W dom γ and γ W is injective Proof 1 Assume that dim W = 1 Let α, β P T V, W with α β, and let λ P T V, W By the above lemma, we obtain β or α is a zero partial linear transformation If β, then λ λβ Now, we consider the case α is a zero partial linear transformation We obtain V λ 0 W λβ and dom λ im λ dom αλ 1 im λ dom βλ 1 = dom λβ Let v V λαλβ 1 We

92 Thai J M ath 12 2014/ K Sangkhanan and J Sanwong obtain vλβ V λα V α which implies that vλ V αβ 1 Eα, β since α β Thus vλβ = vλα So v Eλα, λβ Therefore, λα λβ 2 Assume that dim W > 1 Suppose that W γ W and γ is not a zero partial linear transformation If W γ = 0, then there is v V γ \ W γ V γ since γ is not a zero partial linear transformation From dim W > 1, there exists v w W \ W γ where {v, w} is linearly independent If W γ 0, there are 0 v W γ V γ and w W \ W γ where {v, w} is linearly independent since W γ W It is concluded that we can choose 0 w W \ W γ and 0 v V γ where {v, w} is linearly independent Define α, β P T V, W by v w v w w w, β = Then α β It is clear that v V γβ but v / V γα, so γα γβ Since w V γα but w / W γβ, we conclude that γα γβ Conversely, it is clear that γ γ for each α P T V, W if γ is a zero partial linear transformation In this case, we obtain γ is left compatible Assume that W γ = W Let α, β P T V, W be such that α β We have V γα V α W β = W γβ and v w dom γ imγ dom αγ 1 imγ dom βγ 1 = dom γβ Let v V γαγβ 1 Then vγβ V γα V α which implies that vγ V αβ 1 Eα, β Hence vγ dom α and vγ vγβ, so v Eγα, γβ Therefore, γα γβ 3 Suppose that dim W 1 Assume that W dom γ and γ W is injective Let α, β P T V, W be such that α β So V α W β which implies that V αγ W βγ Since W dom γ, we obtain dom αγ = im α dom γα 1 W dom γα 1 = W α 1 = dom α dom β = im β W β 1 im β dom γβ 1 = dom βγ For each v V αγβγ 1, we have vβγ = wαγ for some w V Since γ W is injective, we have vβ = wα V α, thus v dom α and v vβ Hence v dom im α W α 1 im α dom γα 1 = dom αγ and vαγ = vβγ from which it follows that v Eαγ, βγ Therefore, αγ βγ Conversely, if γ W is not injective, then ker γ W 0 Let 0 w ker γ W Define α, β P T V, W by 0 0 and β = w w So, we obtain α β by Theorem 31 and αγ βγ since w dom βγ but w / dom αγ We also have αγ βγ since wβγ = wγ = 0 V αγ which implies that w V αγβγ 1 but w / dom αγ If W dom γ, then there is w W \dom γ Define α, β P T V, W by w 0 and β = w w Thus α β and αγ βγ And αγ βγ since dom αγ = w 0 = dom βγ Therefore, γ is not right compatible Acknowledgements : The first author thanks the Development and Promotion of Science and Technology Talents Project, Thailand, for its financial support He also thanks the Graduate School, Chiang Mai University, Chiang Mai, Thailand, for its financial support that he received during the preparation of this paper The second author thanks Chiang Mai University for its financial support

Green s relations and partial orders on semigroups 93 References [1] S V Droms, Partial Linear Transformations of a Vector Space, MS Thesis, directed by Janusz Konieczny, University of Mary Washington, Virginia, USA, 2007 [2] V H Fernandes, J Sanwong, On the ranks of semigroups of transformations on a finite set with restricted range, Algebra Colloq, to appear [3] J M Howie, Fundamentals of Semigroup Theory, Oxford University Press, USA, 1995 [4] G Kowol, H Mitsch, Naturally ordered transformation semigroups, Monatsh Math 102 1986 115 138 [5] M Paula, O Marques-Smith and R P Sullivan, Partial orders on transformation semigroups, Monatsh Math 140 2003 103 118 [6] H Mitsch, A natural partial order for semigroups, Proc Amer Soc 97 3 1986 384 388 [7] K Sangkhanan, J Sanwong, Partial orders on semigroups of partial transformations with restricted range, Bull Aust Math Soc 86 1 2012, 100 118 [8] K Sangkhanan, J Sanwong, Semigroups of injective partial linear transformations with restricted range: Green s relations and partial orders, Int J Pure Appl Math 80 4 2012, 597 608 [9] J Sanwong, W Sommanee, Regularity and Green s relations on a semigroup of transformation with restricted range, Int J Math Math Sci 2008 2008, Art ID 794013, 11pp [10] R P Sullivan, Partial orders on linear transformation semigroups, P Roy Soc Edinb 135A 2005 413 437 [11] R P Sullivan, Semigroups of linear transformations with restricted range, Bull Aust Math Soc 77 2008 441 453 Received 1 October 2013 Accepted 6 November 2013 Thai J Math Online @ http://thaijmathincmuacth