EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5623 ISSN 1307-5543 – ejpam.com Published by New York Business Global Left and Right Regular Elements of Some Subsemigroups of the Linear Transformations Semigroups With Invariant Subspace Nares Sawatraksa1, Piyaporn Tantong1,∗ 1 Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Nakhon Sawan, Thailand Abstract. In this paper, we investigate the left regularity, right regularity, and complete regularity of elements in subsemigroups of the semigroups of linear transformations with invariant subspaces. We provide necessary and sufficient conditions for these subsemigroups to be left regular, right regular, and completely regular. Specifically, we examine semigroups of linear transformations with restricted range, invariant subspaces, and fixed subspaces. The results offer a comprehensive characterization of regular elements within these algebraic structures and extend existing work in this field. Our findings have potential applications in algebraic theory, particularly in the study of transformation semigroups and their subsemigroups. 2020 Mathematics Subject Classifications: 20M20, 15A04, 47A15, 54D15 Key Words and Phrases: Left regular, right regular, linear transformation, invariant subspace 1. Introduction and Preliminaries An element a of a semigroup S is called left regular if a = xa2 for some x ∈ S, right regular if a = a2x for some x ∈ S, and completely regular if a = axa and ax = xa for some x ∈ S. For a semigroup S, let LReg(S), RReg(S) and CReg(S) denote the set of all left regular elements, right regular elements, and completely regular elements of S, respectively. It is important to note that every completely regular element is also left and right regular. Additionally, Petrich and Reilly [10, Proposition 2.1.3] proved that an element a of a semigroup S is completely regular if and only if a is both a left and right regular element of S. A semigroup S is called left (right, completely) regular if all its elements are left (right, completely) regular, that is, LReg(S) = S (RReg(S) = S, CReg(S) = S). The characterizations of left regularity, right regularity, and complete regularity for semigroups have been studied in detail, as seen in [2, 6–9, 12, 15]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5623 Email addresses: nares.sa@nsru.ac.th (N. Sawatraksa), piyaporn.ta@nsru.ac.th (P. Tantong) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 2 of 15 Let V be a vector space over a field, and let L(V ) be the semigroup (under composition) consisting of all linear operators on V . It is well known that L(V ) is a regular semigroup [4, page 63]. Moreover, Tantong [15] characterized left, right and completely regularity for elements of L(V ) and provided necessary and sufficient conditions for L(V ) to be left regular, right regular and completely regular in terms of the dimension of V . Let W be a fixed subspace of V . In 2008, Sullivan [14] defined a subsemigroup of the linear transformation semigroup as: L(V,W ) = {α ∈ L(V ) : V α ⊆ W}. This semigroup is called the semigroup of linear transformations with restricted range. The author examined Green’s relations and ideals for the semigroup L(V,W ). Later, in 2019, Sangkhanan and Sanwong [11] proved certain isomorphism theorems and calculated the ranks of these semigroups for any proper subspace W of a finite dimensional vector space V over a finite field. Additionally, Sullivan [14] showed that: Q = {α ∈ L(V,W ) : V α ⊆ Wα} is the largest regular subsemigroup of L(V,W ). In 2015, Sangkhanan and Sommanee [13] described all the maximal regular subsemigroups of Q when W is a finite dimensional subspace of V over a finite field. Furthermore, they computed the rank and idempotent rank of Q where W is an n-dimensional subspace of an m-dimensional vector space V over a finite field. Let W be a subspace of a vector space V . The semigroup of linear transformations with invariant subspace are defined by: S(V,W ) = {α ∈ L(V ) : Wα ⊆ W}. In 2012, Huisheng [5] described the relations L∗ and R∗ on S(V,W ). In the same year, Honyam and Sanwong [3] presented the relations of Green and ideals of the semigroup and proved that it is never isomorphic to T (U) for any vector space U when W is a non- zero proper subspace of V . In 2019, Chaiya [1] characterized the natural partial order on S(V,W ) and determined the compatibility of their elements and found all maximal and minimal elements. Furthermore, she presented necessary and sufficient conditions for S(V,W ) to be factorizable, unit-regular, and directly finite. For a fixed subspace W of a vector space V , let Fix(V,W ) = {α ∈ L(V ) : wα = w for all w ∈ W}. Then, Fix(V,W ) is a subsemigroup of S(V,W ) and we call it the semigroup of linear transformations with fixed subspaces. In 2018, Chaiya et. al. [16] discussed the Green’s relations, regularity, and ideals of Fix(V,W ), and characterized when Fix(V,W ) is fac- torisable, unit-regular, and directly finite. The objective of this paper is to characterize the left, right, and complete regularity of elements within the semigroups L(V,W ), S(V,W ), Q, and Fix(V,W ). We also present a method to construct an element β in these semigroups such that it is left and right regular. Furthermore, we establish necessary and sufficient conditions for the semigroups L(V,W ), S(V,W ), Q, and Fix(V,W ) to be left regular, right regular, and completely regular. N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 3 of 15 2. Semigroups of Linear Transformations with Invariant Subspaces In this section, we assume that V is a vector space over a field F, andW is a subspace of V . Let S represent one of the semigroups S(V,W ), L(V,W ), and Q. We first characterize right regularity for elements of S. Theorem 1. Let α ∈ S. Then, α ∈ RReg(S) if and only if α|V α is a one-to-one trans- formations on V α. Proof. Assume that α is a right regular element of S. Thus, there exists an element β of S such that α = α2β. Let v1, v2 ∈ V α, and suppose v1α = v2α. Then, there exist v′1, v ′ 2 ∈ V such that v′1α = v1 and v′2α = v2. Therefore, v1 = v′1α = v′1α 2β = v1αβ = v2αβ = v′2α 2β = v′2α = v2. Hence, α|V α is one-to-one. Conversely, assume that α|V α is one-to-one. We will construct β ∈ S such that α = α2β. Let B be a basis for V α, and let B′ = {vα : v ∈ B}. To show that B′ is a linearly independent subset of V , let a1, a2, . . . , an ∈ F, and v1, v2, . . . , vn ∈ B be such that a1(v1α) + a2(v2α) + . . .+ an(vnα) = 0. Thus, (a1v1+a2v2+. . .+anvn)α = 0. Since a1v1+a2v2+. . .+anvn ∈ V α and α|V α is one-to- one, it follows that a1v1+a2v2+. . .+anvn = 0. Since v1, v2, . . . , vn are linearly independent, we conclude that ai = 0 for all i = 1, 2, . . . , n. Hence, B′ is linearly independent. We then construct a basis B′′ for V such that B′ ⊆ B′′. For each u ∈ B′, there exists a unique u′ ∈ B such that u′α = u by assumption. Define β : B′′ → V by vβ = { v′ if v ∈ B′, 0 otherwise. Thus, β is well-defined and can be extended to a linear transformation on V . Let v ∈ V . Since B′ ⊆ B′′, there exist positive integers k and n such that v1, v2, . . . , vk ∈ B′ and vk+1, vk+2, . . . , vn ∈ B′′ \B′ and a1, a2, . . . , an ∈ F with v = a1v1+ a2v2+ . . .+ anvn. This implies that vβ = (a1v1 + a2v2 + . . .+ akvk + ak+1vk+1 + ak+2vk+2 + . . .+ anvn)β = a1(v1β) + a2(v2β) + . . .+ ak(vkβ) + ak+1(vk+1β) + ak+2(vk+2β) + . . .+ an(vnβ) = a1v ′ 1 + a2v ′ 2 + . . .+ akv ′ k + ak+1(0) + ak+2(0) + . . .+ an(0) = a1v ′ 1 + a2v ′ 2 + . . .+ akv ′ k. Since v′1, v ′ 2, . . . , v ′ k are all elements in a basis B of V α, we conclude that a1v ′ 1 + a2v ′ 2 + . . . + akv ′ k ∈ V α. Therefore, β ∈ L(V,W ). This implies that Wβ ⊆ V β ⊆ W , and so N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 4 of 15 β ∈ S(V,W ). Moreover, we will now show that β ∈ Q. Let w = a1v1 + a2v2 + . . .+ akvk. Then, w ∈ W , and wβ = (a1v1 + a2v2 + . . .+ akvk)β = a1(v1β) + a2(v2β) + . . .+ ak(vkβ) = a1v ′ 1 + a2v ′ 2 + . . .+ akv ′ k = vβ. We can conclude that β ∈ Q. This shows that β ∈ S. Finally, we show that α = α2β. Let v ∈ V . Since vα ∈ V α, we can express vα = a1u ′ 1 + a2u ′ 2 + . . .+ anu ′ n where u′1, u ′ 2, . . . , u ′ n ∈ B with u′iα = ui for all i ∈ {1, 2, . . . , n}, and a1, a2, . . . , an ∈ F. Therefore, vα2β = (a1u ′ 1 + a2u ′ 2 + . . .+ anu ′ n)αβ = (a1u ′ 1α+ a2u ′ 2α+ . . .+ anu ′ nα)β = (a1u1 + a2u2 + . . .+ anun)β = a1u ′ 1 + a2u ′ 2 + . . .+ anu ′ n = vα. Hence, α is right regular, as required. The following next theorem characterizes when S is a right regular semigroup. Theorem 2. The following statements are equivalent: (i) S is a right regular semigroup. (ii) RReg(S) is a subsemigroup of S. (iii) dim(W ) ≤ 1. Proof. (i) ⇒ (ii) This is clear by definition. (ii) ⇒ (iii) We will prove the contrapositive. Assume that dim(W ) ≥ 2. Then, there exists a basis BW of W such that |BW | ≥ 2. Now, let B be a basis for V such that BW ⊆ B. Let a and b be distinct elements of BW . Define two mappings α and β from B into V as follows: xα =  a if x = b, b if x = a, 0 otherwise, and xβ = { b if x = b, 0 otherwise. It is easy to verify that α and β are well-defined and can be extended to linear transforma- tions on V . From their definitions, α, β ∈ S, and we will show that both are right regular N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 5 of 15 elements of S, but their product αβ is not right regular. To check that α is right regular, we use Theorem 1. Suppose u ∈ V α and uα = 0. From the definition of α, we know that V α = {k1a+ k2b : k1, k2 ∈ F}. Then, u = k1a + k2b where k1, k2 ∈ F. Thus 0 = uα = k1b + k2a. Since a, b are linearly independent, we must have k1 = k2 = 0, implying that u = 0. Thus ker(α|V α) = {0}, and hence, α|V α is one-to-one. By Theorem 1, α is right regular. Similarly, we can show that β is right regular by applying Theorem 1 to β|V β. Finally, we will show that αβ is not right regular. Note that 0αβ = 0 = aβ = bαβ. Since 0, b ∈ V αβ and 0 ̸= b, it follows that αβ|V αβ is not one-to-one. From Theorem 1, αβ is not right regular. Hence, RReg(S) is not a subsemigroup of S. (iii) ⇒ (i) Suppose that dim(W ) ≤ 1. Let α ∈ S. If α is the zero transformation, then it is trivially right regular. Suppose that α is not the zero transformation. Then, dim(V α) ≥ 1. Since dim(W ) ≤ 1, by the Dimension Theorem, dim(kerα) = 0, which implies that ker(α|V α) = {0}. Therefore, α|V α must be one-to-one, and by Theorem 1, α is right regular. Consequently, S = RReg(S), meaning that S is a right regular semigroup. Next, we give a characterization for left regular elements in S \Q and Q, respectively. Let S∗ be either L(V,W ) or S(V,W ). Theorem 3. Let α ∈ S∗. The following statements are equivalent: (i) α ∈ LReg(S∗). (ii) α|Wα is an onto transformation on V α. (iii) V α ⊆ Wα2. (iv) For every basis B for V , we have Bα ⊆ Wα2. (v) There exists a basis B for V such that Bα ⊆ Wα2. Proof. (i) ⇒ (ii) Assume that α is left regular in S∗. Then, α = βα2 for some β ∈ S∗. For any y ∈ V α, then there exists x ∈ V such that y = xα. Thus, xβ ∈ W , and y = xα = xβα2 = (xβα)α, which proves that α|Wα is onto. (ii) ⇒ (iii) If α|Wα : Wα → V α is onto, then Wα2 = (Wα)α = (Wα)α|Wα = V α, which proves that V α ⊆ Wα2. (iii) ⇒ (iv) and (iv) ⇒ (v) are clear by definition. (v) ⇒ (i) Suppose there is a basis B for V such that Bα ⊆ Wα2. For each v ∈ B, we choose and fix v′ ∈ W such that vα = v′α2. Define β : B → V by vβ = v′ for all v ∈ B. N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 6 of 15 By the uniqueness condition, β is well-defined and can be extended to a linear transfor- mation on V . We will now show that β ∈ S∗, and that α = βα2, proving that α is left regular. For any v ∈ V , there exist v1, v2, . . . , vn ∈ B and a1, a2, . . . , an ∈ F such that v = a1v1 + a2v2 + . . .+ anvn. Thus, vβ = (a1v1 + a2v2 + . . .+ anvn)β = a1(v1β) + a2(v2β) + . . .+ an(vnβ) = a1v ′ 1 + a2v ′ 2 + . . .+ anv ′ n ∈ W. Hence β ∈ L(V,W ) or β ∈ S(V,W ), depending on the semigroup. Finally, for each v ∈ B, we have vβα2 = vβαα = v′αα = vα, showing that βα2 = α, as required. The following result follows from Theorems 1 and 3. Corollary 1. Let α ∈ S∗. The following statements are equivalent: (i) α ∈ CReg(S∗). (ii) α|V α : V α → V α is one-to-one, and α|Wα : Wα → V α is onto. (iii) For every v ∈ V , there exists a unique v′ ∈ Wα such that vα = v′α. (iv) For every basis B of V , and for every v ∈ B, there exists a unique v′ ∈ Wα such that vα = v′α. (v) There exists a basis B of V , and for every v ∈ B, there exists a unique v′ ∈ Wα such that vα = v′α. Next, we give a necessary and sufficient condition when the semigroup S∗ to be left regular. Theorem 4. The following statements are equivalent: (i) S∗ is a left regular semigroup. (ii) LReg(S∗) is a subsemigroup of S∗. (iii) dim(W ) ≤ 1. Proof. (i) ⇒ (ii) This is clear by definition. (ii) ⇒ (iii) We will prove by contrapositive. Assume that dim(W ) ≥ 2. Then, there exists a basis BW of W such that |BW | ≥ 2. Let B be a basis for V such that BW ⊆ B. N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 7 of 15 Let a and b be distinct elements of BW . Define two transformations α and β on B as follows: xα =  a if x = b, b if x = a, 0 otherwise, and xβ = { b if x = b, 0 otherwise. Both α and β are well-defined and can be extended to the linear transformations on V . Since a, b, 0 ∈ W , it follows that α, β ∈ S∗. From the previous theorem (Theorem 3), we know that α and β are left regular because Bα ⊆ Wα2 and Bβ ⊆ Wβ2. However, their product αβ is not left regular. To see this, note that: aα = b, bα = a, aβ = 0, bβ = b. Thus, aαβ = b and bαβ = 0 = 0αβ, showing that αβ is not one-to-one on its image. Therefore, αβ is not left regular. This implies that LReg(S∗) is not a subsemigroup of S∗. (iii) ⇒ (i) Suppose that dim(W ) ≤ 1. Let α ∈ S∗. If α is the zero transformation, it is trivially left regular. Now, assume that α is not the zero transformation. Then, dim(V α) ̸= 0. Since dim(W ) ≤ 1, V α ⊆ W , and thus α is surjective onW . By assumption, we have dim(V α) = 1. Hence, LReg(S∗) = S∗, meaning that S∗ is a left regular semigroup. Corollary 2. The following statements are equivalent: (i) S∗ is a left regular semigroup. (ii) CReg(S∗) is a subsemigroup of S∗. (iii) dim(W ) ≤ 1. The following theorem gives a necessary and sufficient condition for an element of Q to be left regular. Theorem 5. Let α ∈ Q. The following statements are equivalent: (i) α ∈ LReg(Q). (ii) α|Wα is an onto transformation on V α. (iii) V α ⊆ Wα2. Proof. (i) ⇒ (ii) and (ii) ⇒ (iii) These follow directly from Theorem 3. (iii) ⇒ (i) Assume that V α ⊆ Wα2. Let BW be a basis for W , and let B be a basis for V such that BW ⊆ B. For each v ∈ B \ BW , since V α ⊆ Wα, we choose and fix N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 8 of 15 v′ ∈ W such that vα = v′α. For each w ∈ W , we let w′ = w. For each w′ ∈ W and by assumption, we choose and fix w′′ ∈ W such that w′α = w′′α2. Define β : B → V by vβ = v′′ for all v ∈ B It is easy to verify that β is well-defined, and it can be extended to a linear transformation on V . By the method of constructing β, it is easy to see that β ∈ L(V,W ). We now show that V β ⊆ Wβ. Let v ∈ V . Then, v = a1v1 + a2v2 + . . .+ anvn where v1, v2, . . . , vk ∈ B, and a1, a2, . . . , ak ∈ F. This implies that vβ = (a1v1 + a2v2 + . . .+ akvk)β = a1(v1β) + a2(v2β) + . . .+ ak(vkβ) = a1v ′′ 1 + a2v ′′ 2 + . . .+ akv ′′ k = a1v ′ 1β + a2v ′ 2β + . . .+ akv ′ kβ = (a1v ′ 1 + a2v ′ 2 + . . .+ akv ′ k)β. Since W is a subspace of V and v′1, v ′ 2, . . . , v ′ k ∈ W , we have that a1v ′ 1+a2v ′ 2+ . . .+akv ′ k ∈ W . This implies that β ∈ Q. Moreover, we will now show that α = βα2. For each v ∈ B, we have vβα2 = v′′αα = v′α = vα. Hence, βα2 = α, proving that α is left regular. The following corollary follows directly from Theorems 1 and 5. Corollary 3. Let α ∈ Q. The following statements are equivalent: (i) α ∈ CReg(Q). (ii) α|V α : V α → V α is one-to-one, and α|Wα : Wα → V α is onto. (iii) For every w ∈ W , there exists a unique w′ ∈ Wα such that wα = w′α. Lastly, we characterize the left regular semigroup structure of the semigroup Q. Theorem 6. The following statements are equivalent: (i) Q is a left regular semigroup. (ii) CReg(Q) is a subsemigroup of S∗. (iii) dim(W ) ≤ 1. Proof. The proof can be established in the same way as Theorem 4. The following corollary is an immediate consequence of Theorems 2 and 6. Corollary 4. The following statements are equivalent: (i) Q is a left regular semigroup. (ii) CReg(Q) is a subsemigroup of S∗. (iii) dim(W ) ≤ 1. N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 9 of 15 3. Semigroups of Linear Transformations with Fixed Subspaces In this section, we now explore the necessary and sufficient conditions for an element of Fix(V,W ) to be right regular. Theorem 7. Let α ∈ Fix(V,W ). Then, α ∈ RReg(Fix(V,W )) if and only if α|V α is a one-to-one transformation on V α. Proof. The necessity follows directly from Theorem 1. To prove the sufficiency, we suppose that α|V α is one-to-one. Since α ∈ Fix(V,W ), we get that W ⊆ V α. Let BW be a basis for W and B be a basis for V α such that BW ⊆ B. Define B′ = {uα |u ∈ B}. Then, BW ⊆ B′. By the same reasoning used in Theorem 1, B′ is linearly independent, and there exists a basis B′′ for V such that B′ ⊆ B′′. For each u ∈ B′, there exists a unique u′ ∈ B such that u′α = u by assumption. Define β : B′′ → V by vβ = { v′ if v ∈ B′, 0 otherwise. By the uniqueness, β is well-defined. Hence, β can be extended to a linear transformation on V . Let w ∈ W . Then, there are w1, w2, . . . , wk ∈ BW , and a1, a2, . . . , ak ∈ F such that w = a1w1+a2w2+ . . .+akwk. Since BW ⊆ B, and by the definition of α, we observe that w′ iα = wi = wiα. It follows from assumption that wi = w′ i. This implies that wβ = (a1w1 + a2w2 + . . .+ akwk)β = a1(w1β) + a2(w2β) + . . .+ ak(wkβ) = a1w1 + a2w2 + . . .+ akwk = w. It implies that β belong to Fix(V,W ). We now demonstrate that α = α2β. If v ∈ V , then we have vα ∈ V α, and we can write vα = a1u ′ 1+a2u ′ 2+. . .+anu ′ n where u′1, u ′ 2, . . . , u ′ n ∈ B with u′iα = ui for all i ∈ {1, 2, . . . , n} and a1, a2, . . . , an ∈ F. Therefore, vα2β = (a1u ′ 1 + a2u ′ 2 + . . .+ anu ′ n)αβ = (a1u ′ 1α+ a2u ′ 2α+ . . .+ anu ′ nα)β = (a1u1 + a2u2 + . . .+ anun)β = a1u ′ 1 + a2u ′ 2 + . . .+ anu ′ n = vα. Hence, α is right regular. This completes the proof of theorem. Next, we give a necessary and sufficient condition when the semigroup Fix(V,W ) to be left regular. Tantong [15], prove that L(V ) is a right regular semigroup if and only if dim(V ) ≤ 1. We will use this result in the proof of the next theorem. Theorem 8. The following statements are equivalent: N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 10 of 15 (i) Fix(V,W ) is a right regular semigroup. (ii) RReg(Fix(V,W )) is a subsemigroup of Fix(V,W ). (iii) V = W or dim(V ) ≤ 1. Proof. (i) ⇒ (ii) This is clear by definition. (ii) ⇒ (iii) We will prove the contrapositive. Assume that W ̸= V and dim(V ) > 1. If dim(W ) = 0, then Fix(V,W ) = L(V ) and hence RReg(Fix(V,W )) is not a subsemigroup of Fix(V,W ). Suppose that dim(W ) > 0. Let a be a non-zero element of W . Then, there is a basis BW for W such that a ∈ BW . Let B be a basis for V with BW ⊆ B. By assumption, let b ∈ B \BW . Define transformations α and β as follows: xα =  x if x ∈ BW , b if x = b, 0 otherwise, and xβ =  x if x ∈ BW , a if x = b, 0 otherwise. It is easy to verify that α and β are well-defined and can be extended to the linear transformations on V . By the definitions of α and β, we see that both α|W and β|W are the identity transformation on W , so that α, β ∈ Fix(V,W ). Next, we check that α and β are right regular elements of Fix(V,W ) by using Theorem 7. Let u ∈ V α be such that uα = 0. By the definition of α, we know that V α = {w + kb : w ∈ W and k ∈ F} . Then, u = w + kb where k ∈ F and w ∈ W . Thus 0 = uα = (w + kb)α = wα+ bα = w + kb = u Hence, ker(α|V α) = {0} and so α|V α is one-to-one. It follows from Theorem 7 that α is right regular of Fix(V,W ). Similarly, we can show that β|V β is one-to-one. It follows from Theorem 7 that β is right regular. Finally, we will show that αβ is not right regular. Note that bαβ = bβ = a = aβ = aαβ. Since a, b ∈ V αβ and a ̸= b, it follows that αβ|V αβ is not one-to-one. From Theorem 7, αβ is not right regular. Hence, RReg(Fix(V,W )) is not a subsemigroup of Fix(V,W ). (iii) ⇒ (i) Assume that W = V or dim(V ) ≤ 1. If W = V , then Fix(V,W ) consists only of the identity transformation, and hence, Fix(V,W ) is trivially a right regular N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 11 of 15 semigroup. If dim(V ) ≤ 1, then Fix(V,W ) = L(V ) and hence, it is a right regular semigroup. Secondly, we investigate the condition under which an element of Fix(V,W ) is left regular. Theorem 9. Let α ∈ Fix(V,W ). The following statements are equivalent: (i) α ∈ LReg(Fix(V,W )). (ii) α|V α is an onto transformation on V α. (iii) V α ⊆ V α2. (iv) For every basis B for V , we have Bα ⊆ V α2. (v) There exist two basses B for V and BW for W such that BW ⊆ B and Bα ⊆ V α2. Proof. (i) ⇒ (ii) Assume that α is left regular of Fix(V,W ). Then, α = βα2 for some β ∈ Fix(V,W ). If y ∈ V α, then y = xα for some x ∈ V . Thus, y = xα = xβα2 = (xβα)α. This prove that α|V α : V α → V α is onto. (ii) ⇒ (iii) Assume that α|V α : V α → V α is onto. Then, V α2 = (V α)α = (V α)α|V α = V α, which proves that V α ⊆ V α2. (iii) ⇒ (iv) This implication is clear by definition. If V α ⊆ V α2, then for every basis B of V , we will have Bα ⊆ V α2 (iv) ⇒ (v) Suppose that the condition (iv) holds. Let BW be a basis for W . Then, there exists a basis B for V such that BW ⊆ B. By the assumption, we have Bα ⊆ V α2, which implies that condition (v) holds. (v) ⇒ (i) Suppose that there is a basis B for V and a basis BW for W such that BW ⊆ B and Bα ⊆ V α2. For each u ∈ B \ BW , we choose and fix u′ ∈ V such that uα = u′α2. For each u ∈ BW , we set u′ = u. Then, uαα = uα = u′α. Define β : B → V by vβ = v′ for all v ∈ B. It is easy to verify that β is well-defined and can be extended to a linear transformation on V . We will now show that β ∈ Fix(V,W ), and that α = βα2. For each w ∈ W , we have w = a1w1+a2w2+. . .+anwn for some w1, w2, . . . , wn ∈ BW , and scalars a1, a2, . . . , an ∈ F. Therefore, wβ = (a1w1 + a2w2 + . . .+ anwn)β = a1(w1β) + a2(w2β) + . . .+ an(wnβ) N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 12 of 15 = a1w ′ 1 + a2w ′ 2 + . . .+ anw ′ n = a1w1 + a2w2 + . . .+ anwn = w. Thus, α ∈ Fix(V,W ). Finally, for every v ∈ B, we have vβα2 = vβαα = v′αα = vα. Therefore βα2 = α, which shows that α is left regular, as required. Next corollary is result from Theorems 7 and 9. Corollary 5. Let α ∈ Fix(V,W ). The following statements are equivalent: (i) α ∈ CReg(Fix(V,W )). (ii) α|V α : V α → V α is a bijective tranformation. (iii) For every v ∈ V , there exists a unique v′ ∈ V α such that vα = v′α. (iv) For every basis B for V , and for every v ∈ B, there exists a unique v′ ∈ V α such that vα = v′α. (v) There exist two basses B for V , and BW for W such that BW ⊆ B, and for every v ∈ B, there exists a unique v′ ∈ V α such that vα = v′α. Finally, we show that for the semigroup Fix(V,W ) to be left regular whenever V is a finite dimensional vector space. Theorem 10. Let V be a finite dimensional vector space. The following statements are equivalent: (i) Fix(V,W ) is a left regular semigroup. (ii) LReg(Fix(V,W )) is a subsemigroup of Fix(V,W ). (iii) V = W or dim(V ) = dim(W ) + 1. Proof. (i) ⇒ (ii) This is clear by definition. (ii) ⇒ (iii) Suppose that W ̸= V and dim(V ) ̸= dim(W ) + 1. Then, dim(V ) − dim(W ) > 1. Let BW be a basis for W and extend it to a basis B for V . Let a and b be distinct elements of B \BW . Define two mappings α and β from B into V as follows: xα =  a if x = b, b if x = a, x otherwise, and xβ = { x if x ∈ BW ∪ {a}, 0 otherwise. N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 13 of 15 Clearly, both α and β are well-defined and can be extended to the linear transformation on V . Let w ∈ W . Then, w = k1w1 + k2w2 + . . .+ knwn for some k1, k2, . . . , kn ∈ F and w1, w2, . . . , wn ∈ B. Therefore, wα = (k1w1 + k2w2 + . . .+ knwn)α = (k1w1)α+ (k2w2) + . . .+ (knwn) = k1w1α+ k2w2α+ . . .+ knwnα = k1w1 + k2w2 + . . .+ knwn = w. Hence, α ∈ Fix(V,W ). By the symmetry, we can show that β ∈ Fix(V,W ). It follows from B ⊆ V α and Theorem 9 that α is left regular. Let v ∈ B. If v ∈ BW ∪ {a}, then vβ = v = vβ2. Otherwise, vβ = 0 = (vβ)β. This implies that β is a left regular element of Fix(V,W ) by Theorem 9. Finally, we will show that βα is not left regular. Note that aβα = aα = b. Thus, b ∈ V βα. Claim that b ̸= vβα for all v ∈ V βα. Suppose that vβα = b for some v ∈ V βα. Then, v = v′βα for some v′ ∈ V . Thus, v′ = a1v1+a2v2+. . .+anvn where v1, v2, . . . , vn ∈ B and a1, a2, . . . , an ∈ F. If v′ ∈ W , then v = v′βα = v′, and so b = vβα = v ∈ W , which is a contradiction. Hence, v′ /∈ W . Since b ̸= 0 and vβα = b, we get that v ̸= 0, so that 0 ̸= v′βα = a1v1βα + a2v2βα + . . . + anvnβα. By the definition of β, there exists k ∈ {1, 2, . . . , n} such that vkβ = a and so v = akvkβα = akaα = akb. Therefore, b = vβα = akbβα = ak(0α) = 0, which is a contradiction. So we have the claim. Hence, βα|V βα is not onto. From Theorem 9, βα is not left regular. Hence, LReg(Fix(V,W )) is not a subsemigroup of Fix(V,W ). (iii) ⇒ (i) Assume that W = V or dim(W ) + 1 = dim(V ). If W = V , then Fix(V,W ) contains only the identity transformation and is trivially left regular. Sup- pose that dim(W ) + 1 = dim(V ). Let α ∈ Fix(V,W ). Then, by Theorem 9, we get that α|V α : V α → V α is onto. By the Dimension Theorem and assumptions, we have dim(W ) + 1 = dim(kerα) + dim(V α). Since V α contains the subspace W , clarifying this will help in explaining the two cases. Case 1. dim(W ) = dim(V α) and dim(kerα) = 1. Then, V α = W . Since α|W is the identity transformation on W , by Theorem 9, we have α ∈ LReg(Fix(V,W )). Case 2. dim(W ) + 1 = dim(V α) and dim(kerα) = 0. Then, V α = V . Therefore, α is onto. Hence, by Theorem 9, we have α ∈ LReg(Fix(V,W )). Hence, Fix(V,W ) is a left regular semigroup, as required. The next corollary is a result from Theorems 8 and 10. Corollary 6. The following statements are equivalent: N. Sawatraksa, P. Tantong / Eur. J. Pure Appl. Math, 18 (1) (2025), 5623 14 of 15 (i) Fix(V,W ) is a completely regular semigroup. (ii) CReg(Fix(V,W )) is a subsemigroup of Fix(V,W ). (iii) V = W or dim(V ) ≤ 1. 4. Conclusions In this paper, we have investigated the left, right, and complete regularity of elements in the semigroups of linear transformations with invariant subspaces. Specifically, we focused on the subsemigroups L(V,W ), S(V,W ), Q, and Fix(V,W ). We provided nec- essary and sufficient conditions for these semigroups to exhibit left, right, and complete regularity. Our results contribute to the characterization of regular elements within these mathematical structures. One of the key findings is the relationship between the subspaces V and W in deter- mining the regularity of the transformations. For instance, we have shown that an element of L(V,W ) is right regular if and only if its restriction to its image is a one-to-one transfor- mation. Similarly, left regularity is characterized by onto mappings in certain conditions. These results align with and extend previous work, offering a broader understanding of regular semigroups and their elements. The significance of this study lies in its application to algebraic structures such as semigroups, transformation semigroups, and subspaces. Specifically, Cayley’s Theorem states that every semigroup can be embedded in a full transformation semigroup. Our characterization of subsemigroups of transformation semigroups provides deeper insights into their structure, allowing for further exploration of their properties and potential applications. In future research, this framework can be expanded to explore other algebraic struc- tures, including transformation semigroups that preserve different types of equivalence relations. Additionally, studying the interaction between these regular elements and more complex algebraic operations could lead to new theoretical developments and applications in fields such as linear algebra, automata theory, and representation theory. Acknowledgements The authors would like to express their sincere gratitude to the Research and De- velopment Institute, Nakhon Sawan Rajabhat University, for their financial support in this research project. Their generous contribution has been essential to the successful completion of this study. References [1] Y Chaiya. 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