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A partial order on transformation semigroups with restricted range that preserve double direction equivalence

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Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

Let T ( X ) be the full transformation semigroup on a set X . For an equivalence E on X , let

T E ∗ ( X ) = { α ∈ T ( X ) : ∀ x , y ∈ X , ( x , y ) ∈ E ⇔ ( x α , y α ) ∈ E } .

For each nonempty subset Y of X , we denote the restriction of E to Y by E Y . Let T E ∗ ( X , Y ) be the intersection of the semigroup T E ∗ ( X ) with the semigroup of all transformations with restricted range Y under the condition that ∣ X / E ∣ = ∣ Y / E Y ∣ . Equivalently, T E ∗ ( X , Y ) = { α ∈ T E ∗ ( X ) : X α ⊆ Y } , where ∣ X / E ∣ = ∣ Y / E Y ∣ . Then T E ∗ ( X , Y ) is a subsemigroup of T E ∗ ( X ) . In this paper, we characterize the natural partial order on T E ∗ ( X , Y ) . Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of T E ∗ ( X , Y ) lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.

Keywords: transformation semigroup; equivalence; partial order; compatibility; maximal and minimal
MSC 2010: 20M20

1 Introduction and preliminaries

Let T ( X ) be the set of all functions from a set X into itself. Then T ( X ) under the composition of functions is a semigroup which is called the full transformation semigroup on X . In 1975, Symons [1] studied the semigroup T ( X , Y ) defined by

T ( X , Y ) = { α ∈ T ( X ) : X α ⊆ Y } ,

where Y is a nonempty subset of X . Note that if we let α ∈ T ( X ) and Z ⊆ X , the notation Z α means { z α : z ∈ Z } . Clearly, X α = im α . In [1], the author studied the automorphism of T ( X , Y ) and the isomorphism between two semigroups T ( X 1 , Y 1 ) and T ( X 2 , Y 2 ) . In 2008, Sanwong and Sommanee [2] found a necessary and sufficient condition for T ( X , Y ) to be regular. Moreover, the largest regular subsemigroup was obtained. Later on, Sangkhanan and Sanwong [3] and, independently, Sun and Sun [4] endowed T ( X , Y ) with ≤ , the natural partial order, and determined when two elements of T ( X , Y ) related under this order, then found out elements of T ( X , Y ) which are compatible with ≤ on T ( X , Y ) . They also described its maximal and minimal elements and proved that every element in T ( X , Y ) must lie between maximal and minimal elements.

Let E be an equivalence on X . In 2005, Pei [5] defined a subsemigroup T E ( X ) of T ( X ) by

T E ( X ) = { α ∈ T ( X ) : ∀ x , y ∈ X , ( x , y ) ∈ E ⇒ ( x α , y α ) ∈ E } .

Then T E ( X ) is exactly S ( X ) , the semigroup of all continuous self-maps of the topological space X for which all E -classes form a basis. In [5], the author investigated regularity of elements and Green’s relations for T E ( X ) . In 2008, Sun et al. [6] described T E ( X ) with the natural partial order and gave a characterization for two elements of T E ( X ) related under this order. They found out elements of T E ( X ) which are compatible with ≤ on T E ( X ) . In addition, the maximal, minimal and covering elements were described.

Recently, Sangkhanan and Sanwong [7] defined a subsemigroup T E ( X , Y ) of T E ( X ) where Y ⊆ X by

T E ( X , Y ) = { α ∈ T ( X , Y ) : ∀ ( x , y ) ∈ E , ( x α , y α ) ∈ E } .

Then T E ( X , Y ) is the semigroup of all continuous self-maps of the topological space X for which all E -classes form a basis carrying X into a subspace Y . In [7], they gave a necessary and sufficient condition for T E ( X , Y ) to be regular and characterized Green’s relations on T E ( X , Y ) .

In 2010, Deng et al. [8] introduced a subsemigroup T E ∗ ( X ) of T ( X ) by

T E ∗ ( X ) = { α ∈ T ( X ) : ∀ x , y ∈ X , ( x , y ) ∈ E ⇔ ( x α , y α ) ∈ E } .

Then T E ∗ ( X ) is a semigroup of continuous self-maps of the topological space X for which all E -classes form a basis. The authors studied regularity of elements and Green’s relations for T E ∗ ( X ) . In 2013, Sun and Sun [9] endowed T E ∗ ( X ) with the natural partial order. They determined when two elements are related under this order and found the elements which are compatible. Moreover, the maximal and minimal elements were described. Finally, they studied the existence of the greatest lower bound of two elements.

Now, we define a subset T E ∗ ( X , Y ) of T E ∗ ( X ) by

T E ∗ ( X , Y ) = { α ∈ T E ∗ ( X ) : X α ⊆ Y } = T E ∗ ( X ) ∩ T ( X , Y )

under the condition that ∣ X / E ∣ = ∣ Y / E Y ∣ where E Y = E ∩ ( Y × Y ) . In 2019, Chaichompoo and Sangkhanan [10] showed that T E ∗ ( X , Y ) is a subsemigroup of T E ∗ ( X ) under composition of functions and studied the regularity of T E ∗ ( X , Y ) . Furthermore, they characterized Green’s relations on this semigroup. We see that if X = Y , then T E ∗ ( X , Y ) = T E ∗ ( X ) . Hence, T E ∗ ( X ) is a special case of T E ∗ ( X , Y ) . Furthermore, if E is the universal relation, E = X × X , then T E ∗ ( X , Y ) becomes T ( X , Y ) . Moreover, it is not difficult to check that T E ∗ ( X , Y ) is a semigroup of continuous self-maps of the topological space X for which all E classes form a basis carrying X into a subspace Y , and is referred to as a semigroup of continuous functions (see [11] for details).

In this paper, we aim to generalize the results of [3,4,9]. Actually, we study the natural partial order on T E ∗ ( X , Y ) which extends the results on T E ∗ ( X ) and T ( X , Y ) and improve some results in these semigroups. For example, in Section 4, we characterize the left and right compatible elements instead of the strictly compatibility which was studied in [9]. Moreover, in Section 5, we show that every element in T E ∗ ( X , Y ) lies between maximal and minimal elements which have never been studied before in T E ∗ ( X ) .

We first state some notations and results that will be used later. For each α ∈ T E ∗ ( X , Y ) and A ⊆ X , the restriction of α to A is denoted by α ∣ A . We adopt the notation introduced in [12], namely, if α ∈ T E ∗ ( X , Y ) , then we write

α = X i a i

and the subscript i belongs to some unmentioned index set I , the abbreviation { a i } denotes { a i : i ∈ I } and that X α = { a i } and a i α − 1 = X i .

Let X / E be the quotient set, where E is an equivalence on X , the set of all equivalence classes on X . For each α ∈ T E ∗ ( X , Y ) , the symbol π ( α ) will denote the decomposition of X induced by the map α , namely,

π ( α ) = { x α − 1 : x ∈ X α } .

Then π ( α ) = X / ker ( α ) where ker ( α ) = { ( x , y ) ∈ X × X : x α = y α } . For a subset A of X , we put

π A ( α ) = { M ∈ π ( α ) : M ∩ A ≠ ∅ } .

We also define π ˜ A ( α ) = π A ∩ Y ( α ) . It is clear that π ˜ A ( α ) is an appropriate extension of π A ( α ) in the sense that if Y = X , then π ˜ A ( α ) = π A ( α ) . Obviously, π ˜ A ( α ) ⊆ π A ( α ) .

We define the restriction of the equivalence E on a subset M of X by

E M = { ( x , y ) ∈ M × M : ( x , y ) ∈ E } = E ∩ ( M × M ) .

It is easy to see that E M is also an equivalence on M and

M / E M = { A ∩ M : A ∈ X / E , A ∩ M ≠ ∅ } .

Let A and B be two collections of subsets of X . If for each A ∈ A , there exists B ∈ B such that A ⊆ B , then A is said to refine B .

Note that, by the definition of T E ∗ ( X , Y ) studied in [10], we will assume that Y is a subset of X such that ∣ Y / E Y ∣ = ∣ X / E ∣ in the remaining of this paper.

Now, we deal with the natural partial order or Mitsch’s order [13] on any semigroup S defined, for a , b ∈ S , by

a ≤ b if and only if a = x b = b y , x a = a for some x , y ∈ S 1 .

Equivalently,

(1) a ≤ b if and only if a = w b = b z , a z = a for some w , z ∈ S 1 ,

where the notation S 1 denotes a monoid obtained from S by adjoining an identity 1 if necessary ( S 1 = S for a monoid S ). In this paper, we use (1) to define the partial order on the semigroup T E ∗ ( X , Y ) , that is, for each α , β ∈ T E ∗ ( X , Y ) ,

α ≤ β if and only if α = γ β = β μ , α = α μ for some γ , μ ∈ T E ∗ ( X , Y ) 1 .

Note that if Y ⊊ X , then T E ∗ ( X , Y ) has no identity elements. Hence, in this case, T E ∗ ( X , Y ) 1 ≠ T E ∗ ( X , Y ) . It is worth studying the natural partial order on T E ∗ ( X , Y ) in its own right since, in general, ≤ on T E ∗ ( X , Y ) does not coincide with the restriction of ≤ on T E ∗ ( X ) . In other words, for each α , β ∈ T E ∗ ( X , Y ) , if α ≤ β on T E ∗ ( X ) , then it is not necessary that α ≤ β on T E ∗ ( X , Y ) . For example, let X = { 1 , 2 , 3 , 4 } , Y = { 1 , 2 , 3 } and X / E = { { 1 , 2 } , { 3 , 4 } } . Define

α = 1 2 3 4 3 3 1 1 and β = 1 2 3 4 3 3 2 1 .

If we let

γ = 1 2 3 4 1 1 4 4 and μ = 1 2 3 4 1 1 3 3 ,

then α = γ β = β μ , α = α μ , which implies that α ≤ β in T E ∗ ( X ) but we cannot find γ ′ ∈ T E ∗ ( X , Y ) 1 such that α = γ ′ β . Hence, α ≰ β in T E ∗ ( X , Y ) .

Let us refer to the following corollary which will prove useful.

Corollary 1.1

[9, Corollary 2.2] Let α , β ∈ T E ∗ ( X ) and α ≤ β . Then the following statements hold.

  1. If X α = X β , then α = β .

  2. For each P ∈ π ( α ) , there exists some Q ∈ π ( β ) such that Q ⊆ P and P α = Q β .

2 Characterization

Now, we start this section with the characterization of ≤ on T E ∗ ( X , Y ) , which extends Theorem 2.1 of [9].

Theorem 2.1

Let α , β ∈ T E ∗ ( X , Y ) . Then α ≤ β if and only if either α = β or the following statements hold.

  1. π ( β ) refines π ( α ) .

  2. For each x ∈ X , if x β ∈ X α , then x α = x β .

  3. For each E -class A , A α ⊆ ( A ∩ Y ) β .

Proof

( ⇒ ) Suppose that α ≤ β . Then α = γ β = β μ , α = α μ for some γ , μ ∈ T E ∗ ( X , Y ) 1 . If γ = 1 or μ = 1 , then α = β . Now, we assume that γ , μ ∈ T E ∗ ( X , Y ) . Then (1) and (2) follow from Theorem 2.1 of [9]. To prove (3), let A ∈ X / E . Then A α = A γ β ⊆ ( B ∩ Y ) β for some class B . We claim that A = B . Indeed, let a ∈ A . Then a α = b β for some b ∈ B ∩ Y . Hence, a α = a α μ = b β μ = b α and so ( a α , b α ) ∈ E implies ( a , b ) ∈ E . Thus, b ∈ A ∩ B ≠ ∅ and so A = B .

( ⇐ ) The proof is an appropriate modification of the proof of Theorem 2.1 of [9]. In fact, assume that (1), (2) and (3) hold. Note that for each A ∈ X / E , A α is nonempty. Hence, by (3), ( A ∩ Y ) β ≠ ∅ and so A ∩ Y is nonempty for each A ∈ X / E . Define a function μ on each class A as follows. If A ∩ X β = ∅ , fix y 0 ∈ A ∩ Y and define x μ = y 0 for each x ∈ A . If A ∩ X β ≠ ∅ , we first define μ on A ∩ X β . For each z ∈ A ∩ X β , we have z = x β for some x ∈ X , so define z μ = x α . To define μ on A ⧹ X β , we fix z 0 ∈ A ∩ X β and define z μ = z 0 μ for each z ∈ A ⧹ X β . By using the same method as in the proof of Theorem 2.1 in [9], we can see that μ ∈ T E ∗ ( X , Y ) is well-defined and α = α μ , α = β μ .

Finally, we define a function γ . For each A ∈ X / E , A α ⊆ ( A ∩ Y ) β . Thus, for each x ∈ A , we choose y ∈ A ∩ Y such that x α = y β and define x γ = y . It is clear that X γ ⊆ Y . Similarly, we can show that γ ∈ T E ∗ ( X , Y ) and α = γ β . Therefore, α ≤ β .□

We note that if X = Y , then A ∩ Y = A for each E -class A . We can see that Theorem 2.1 becomes Theorem 2.1 of [9]. Moreover, if E = X × X , then X / E = { X } . Hence, this theorem also becomes Theorem 2.1 of [4].

Proposition 2.2

Let Y be a subset of X such that Y ∩ A = ∅ for some class A ∈ X / E and let α , β ∈ T E ∗ ( X , Y ) . Then α ≤ β if and only if α = β .

Proof

Let α ≤ β and suppose that α < β . Then for each A ∈ X / E , A α ⊆ ( A ∩ Y ) β by Theorem 2.1 (3). Hence, ( A ∩ Y ) β is nonempty and so A ∩ Y is also nonempty for all classes A . It leads to a contradiction. Therefore, α = β .□

By the above proposition, we can see that if Y ∩ A = ∅ for some class A ∈ X / E , then the natural partial order ≤ becomes the equality relation. From now on, we assume that Y ∩ A is nonempty for all E -classes A , equivalently, Y / E Y = { A ∩ Y : A ∈ X / E } .

3 Compatibility

Recall that an element γ ∈ T E ∗ ( X , Y ) is said to be strictly left compatible if γ α < γ β whenever α < β . Strictly right compatibility is defined dually. We note that, in Section 2 of [9], the authors studied the strictly compatibility on T E ∗ ( X ) but, in this paper, we remove the term “strictly,” that is, an element γ ∈ T E ∗ ( X , Y ) is said to be left compatible if γ α ≤ γ β whenever α ≤ β . Right compatibility is defined dually.

Theorem 3.1

γ ∈ T E ∗ ( X , Y ) is left compatible if and only if for each A ∈ X / E , ( A ∩ Y ) γ ∈ Y / E Y .

Proof

( ⇒ ) We prove by contrapositive. Assume that there is A ∈ X / E such that ( A ∩ Y ) γ ⊊ B ∩ Y ∈ Y / E Y for some E -class B . Then there exists y ∈ ( B ∩ Y ) ⧹ ( A ∩ Y ) γ . Since Y / E Y = { A ∩ Y : A ∈ X / E } , we can write ( X / E ) ⧹ { B } = { B i : i ∈ I } and ( Y / E Y ) ⧹ { B ∩ Y } = { B i ∩ Y : i ∈ I } . Now, we choose z ∈ ( A ∩ Y ) γ and define α , β ∈ T E ∗ ( X , Y ) by

α = B B i y b i and β = B ⧹ { y } { y } B i z y b i ,

where b i ∈ B i ∩ Y for all i ∈ I . It is routine to verify that α < β by using Theorem 2.1. We see that z ∉ A γ α but z ∈ A γ β , which implies that γ α ≠ γ β . Moreover, we obtain A γ α ⊆ ( B ∩ Y ) α = { y } ⊈ { z } = ( A ∩ Y ) γ β . Hence, γ α ≰ γ β . Therefore, γ is not left compatible.

( ⇐ ) Suppose that ( A ∩ Y ) γ ∈ Y / E Y for each A ∈ X / E . Let α , β ∈ T E ∗ ( X , Y ) be such that α ≤ β with α ≠ β . We show that γ α ≤ γ β by using Theorem 2.1. Assume that γ α ≠ γ β .

  1. Let U = y ( γ β ) − 1 ∈ π ( γ β ) . Then U γ β = { y } and so U γ ⊆ y β − 1 . Hence, y β − 1 ⊆ z α − 1 ∈ π ( α ) since π ( β ) refines π ( α ) . We obtain U γ ⊆ z α − 1 , which implies that U γ α = { z } . Therefore, U ⊆ z ( γ α ) − 1 ∈ π ( γ α ) and so π ( γ β ) refines π ( γ α ) .

  2. Let x γ β ∈ X γ α . Then x γ β ∈ X α , which implies that x γ β = x γ α since α ≤ β .

  3. Let A ∈ X / E . Then ( A ∩ Y ) γ = B ∩ Y for some B ∈ X / E by the assumption and so A γ = B ∩ Y . We see that B α ⊆ ( B ∩ Y ) β since α ≤ β and hence A γ α = ( B ∩ Y ) α ⊆ B α ⊆ ( B ∩ Y ) β = ( A ∩ Y ) γ β .

Therefore, γ α ≤ γ β .□

Theorem 3.2

Let γ ∈ T E ∗ ( X , Y ) . Then γ is right compatible if and only if for each A ∈ X / E , γ ∣ A ∩ Y is injective or ∣ ( A ∩ Y ) γ ∣ = 1 .

Proof

Assume that γ is right compatible and suppose to the contrary that there is A ∈ X / E such that γ ∣ A ∩ Y is not injective and ∣ ( A ∩ Y ) γ ∣ > 1 . Then there exist s , t ∈ A ∩ Y with s ≠ t and s γ = t γ = w for some w ∈ Y . Moreover, there is u ∈ A ∩ Y such that s ≠ u ≠ t and u γ = v for some v ≠ w in Y . Since Y / E Y = { C ∩ Y : C ∈ X / E } , we can write ( X / E ) ⧹ { A } = { A i : i ∈ I } and ( Y / E Y ) ⧹ { A ∩ Y } = { A i ∩ Y : i ∈ I } . Define α , β ∈ T E ∗ ( X , Y ) by

α = { s } A ⧹ { s } A i s u a i and β = { s } { t } A ⧹ { s , t } A i s t u a i ,

where a i ∈ A i ∩ Y for each i ∈ I . It is straightforward to show that α < β . We can see that s α γ = s γ = w = t γ = t β γ , which implies that t β γ ∈ X α γ but t α γ = u γ = v ≠ w = t γ = t β γ . Thus, α γ ≰ β γ , which contradicts to the right compatibility of γ . Therefore, γ ∣ A ∩ Y is injective or ∣ ( A ∩ Y ) γ ∣ = 1 .

Conversely, assume that for each A ∈ X / E , γ ∣ A ∩ Y is injective or ∣ ( A ∩ Y ) γ ∣ = 1 . Let α , β ∈ T E ∗ ( X , Y ) be such that α ≤ β with α ≠ β . We show that α γ ≤ β γ by using Theorem 2.1.

(1) Let U = y ( β γ ) − 1 ∈ π ( β γ ) . We can see that U β ⊆ A ∩ Y for some class A ∈ X / E . Then ( U β ) γ ∣ A ∩ Y = U β γ = { y } and so U β ⊆ y ( γ ∣ A ∩ Y ) − 1 . We consider the following two cases.

Case 1: γ ∣ A ∩ Y is injective. Then y ( γ ∣ A ∩ Y ) − 1 = { x } for some x ∈ A ∩ Y and hence U β = { x } . Thus, U ⊆ x β − 1 ∈ π ( β ) and then there is z α − 1 ∈ π ( α ) such that U ⊆ x β − 1 ⊆ z α − 1 since π ( β ) refines π ( α ) . We obtain U α = { z } , which implies that U α γ = { z γ } and so U ⊆ ( z γ ) ( α γ ) − 1 ∈ π ( α γ ) .

Case 2: ∣ ( A ∩ Y ) γ ∣ = 1 . Then ( A ∩ Y ) γ = { y } . We claim that U ∈ X / E . Clearly, U ⊆ ( A ∩ Y ) β − 1 = B for some class B ∈ X / E since β ∈ T E ∗ ( X , Y ) . Let b ∈ B . Then b β ∈ A ∩ Y from which it follows that b β γ ∈ ( A ∩ Y ) γ = { y } . Thus, b ∈ y ( β γ ) − 1 = U and so B ⊆ U . We conclude that U = B ∈ X / E . By Theorem 2.1 (3), U α ⊆ ( U ∩ Y ) β ⊆ U β , which implies that U α γ ⊆ U β γ = { y } . Hence, U ⊆ y ( α γ ) − 1 ∈ π ( α γ ) .

Therefore, π ( β γ ) refines π ( α γ ) .

(2) Let x β γ ∈ X α γ . Then x β γ = y α γ for some y ∈ X . Hence, ( x β γ , y α γ ) ∈ E , which implies that ( x β , y α ) ∈ E . We obtain x β , y α ∈ A ∩ Y for some class A ∈ X / E . If γ ∣ A ∩ Y is injective, then x β = y α ∈ X α , which implies that x β = x α . Thus, x β γ = x α γ . Now, suppose that ∣ ( A ∩ Y ) γ ∣ = 1 . By Theorem 2.1 (3), we obtain y α = z β for some z ∈ Y . Moreover, we have z α = z β since z β = y α ∈ X α and α ≤ β . Thus, y α = z α , which implies that ( y , z ) ∈ E . In addition, we obtain z β = y α ∈ A ∩ Y from which it follows that ( x β , z β ) ∈ E since x β ∈ A ∩ Y . Hence, ( x , z ) ∈ E and so ( x , y ) ∈ E . We have ( x α , y α ) ∈ E and x α ∈ A ∩ Y since y α ∈ A ∩ Y . Therefore, x α γ = x α γ ∣ A ∩ Y = x β γ ∣ A ∩ Y = x β γ since ∣ ( A ∩ Y ) γ ∣ = 1 .

(3) Let A ∈ X / E . By Theorem 2.1 (3), A α ⊆ ( A ∩ Y ) β and so A α γ ⊆ ( A ∩ Y ) β γ .□

By the above two theorems, we obtain the following two corollaries which appear in [3,4,9].

Corollary 3.3

Let γ ∈ T E ∗ ( X ) . Then the following statements hold.

  1. γ is left compatible if and only if for each A ∈ X / E , A γ ∈ X / E .

  2. γ is right compatible if and only if for each A ∈ X / E , γ ∣ A is injective or ∣ A γ ∣ = 1 .

Corollary 3.4

Let γ ∈ T ( X , Y ) . Then the following statements hold.

  1. γ is left compatible if and only if Y γ = Y .

  2. γ is right compatible if and only if γ ∣ Y is injective or ∣ Y γ ∣ = 1 .

4 Maximal and minimal elements

In this section, we characterize maximal and minimal elements and show that every element of T E ∗ ( X , Y ) must lie between maximal and minimal elements. Furthermore, we obtain the existence of an upper cover and a lower cover.

Sanwong and Sommanee [2] defined a subset F of T ( X , Y ) by

F = { α ∈ T ( X , Y ) : X α ⊆ Y α }

and proved that F is the largest regular subsemigroup of T ( X , Y ) . Moreover, in [10], the authors defined the set F E ∗ = F ∩ T E ∗ ( X , Y ) , equivalently,

F E ∗ = { α ∈ T E ∗ ( X , Y ) : X α ⊆ Y α } = { α ∈ T E ∗ ( X , Y ) : X α = Y α } .

It is clear that F E ∗ is an appropriate extension of F in the sense that if E = X × X , then F E ∗ = F . In addition, if X = Y , then F E ∗ = T E ∗ ( X ) . It is straightforward to verify that

F E ∗ = { α ∈ T E ∗ ( X , Y ) : A α = ( A ∩ Y ) α for all A ∈ X / E } .

In [10], they showed that F E ∗ is a right ideal of T E ∗ ( X , Y ) and so it is a subsemigroup of T E ∗ ( X , Y ) . In this section, F E ∗ plays an essential role in the characterization of maximal and minimal elements, as follows.

Lemma 4.1

Let α ∈ T E ∗ ( X , Y ) . If α ∉ F E ∗ , then α is maximal.

Proof

Let α ∉ F E ∗ and β ∈ T E ∗ ( X , Y ) be such that α ≤ β . Assume that α ≠ β . Let x α ∈ X α . Then x ∈ A for some A ∈ X / E from which it follows that x α ∈ A α ⊆ ( A ∩ Y ) β by Theorem 2.1 (3). Hence, x α = a β for some a ∈ A ∩ Y implies a β = x α ∈ X α . By Theorem 2.1 (2), x α = a β = a α ∈ ( A ∩ Y ) α ⊆ Y α . We obtain X α ⊆ Y α , which contradicts to α ∉ F E ∗ . Therefore, α = β .□

Theorem 4.2

Let α ∈ T E ∗ ( X , Y ) . Then α is maximal if and only if one of the following statements hold.

  1. α ∉ F E ∗ .

  2. For each E -class A , A α ∈ Y / E Y or α ∣ A is injective.

Proof

( ⇒ ) We prove by contrapositive. Let α ∈ F E ∗ . Assume that there is a class A ∈ X / E such that A α ⊊ B for some B ∈ Y / E Y and α ∣ A is not injective. We note that A α = ( A ∩ Y ) α since α ∈ F E ∗ . Then we can find distinct two elements p ∈ A and q ∈ A ∩ Y such that p α = q α . Now, we choose b ∈ B ⧹ A α and define β : X → Y by

x β = b if x = p , x α otherwise .

It is easy to verify that β ∈ T E ∗ ( X , Y ) and α ≠ β . Now, we prove that α ≤ β .

  1. Let ( x β ) β − 1 ∈ π ( β ) . We assert that ( x β ) β − 1 ⊆ ( x α ) α − 1 ∈ π ( α ) . Indeed, let a ∈ ( x β ) β − 1 . Then a β = x β . If x = p , then a β = p β = b , which implies that a = p . Hence, a α = p α = x α and so a ∈ ( x α ) α − 1 . If x ≠ p , then a β = x β = x α , which implies that a ≠ p . Thus, a α = a β = x α and so a ∈ ( x α ) α − 1 . Therefore, π ( β ) refines π ( α ) .

  2. Let x β ∈ X α . Obviously, x ≠ p and so x α = x β .

  3. It is clear that C α = ( C ∩ Y ) α = ( C ∩ Y ) β if C ∈ X / E with C ≠ A . To show that A α ⊆ ( A ∩ Y ) β , let a α ∈ A α . Since a α ∈ A α = ( A ∩ Y ) α , there is y ∈ A ∩ Y such that a α = y α . If y = p , then a α = p α = q α = q β ∈ ( A ∩ Y ) β . If y ≠ p , then a α = y α = y β ∈ ( A ∩ Y ) β .

Thus, α is not maximal.

( ⇐ ) If α ∉ F E ∗ , then α is maximal by Lemma 4.1. Let α ∈ F E ∗ . Suppose that for each E -class A , A α ∈ Y / E Y or α ∣ A is injective. Let β ∈ T E ∗ ( X , Y ) be such that α ≤ β . Assume that α ≠ β . Then there is x ∈ X such that x α ≠ x β . Let x ∈ A for some A ∈ X / E . If A α = B ∈ Y / E Y , then B = A α ⊆ ( A ∩ Y ) β ⊆ A β , which implies that x β ∈ A β = B = A α ⊆ X α . By Theorem 2.1 (2), we obtain x α = x β , which is a contradiction. Hence, α ∣ A is injective. Moreover, since A α ⊆ ( A ∩ Y ) β , we have x α = y β for some y ∈ A ∩ Y . Again by Theorem 2.1 (2), we obtain x α = y β = y α , which implies x = y since α ∣ A is injective. Thus, x α = x β , which is also a contradiction. Therefore, α = β .□

Now, we show that every element in T E ∗ ( X , Y ) lies below a maximal element.

Theorem 4.3

Let α ∈ T E ∗ ( X , Y ) . Then there exists a maximal element β ∈ T E ∗ ( X , Y ) such that α ≤ β .

Proof

It is clear that if α ∉ F E ∗ , then α is maximal and α ≤ α . Now, we suppose that α ∈ F E ∗ and α is not maximal. Let { A i : i ∈ I } be the class of all E -classes such that A i α ⊊ B i for some B i ∈ Y / E Y and α ∣ A i is not injective. We note by Theorem 4.2 that such class is nonempty since α is not maximal. For each i ∈ I , we define

C i ( α ) = { x α − 1 : x ∈ B i and ∣ x α − 1 ∣ > 1 } .

It is clear that C i ( α ) is nonempty since α ∣ A i is not injective and C i ( α ) ⊆ π ( α ) for each i ∈ I . We claim that for each C ∈ C i ( α ) , C ∩ Y ≠ ∅ . Indeed, let C ∈ C i ( α ) . Then C = x α − 1 for some x ∈ B i . Hence, x ∈ X α = Y α , which implies that x = y α for some y ∈ Y and so y ∈ x α − 1 ∩ Y = C ∩ Y ≠ ∅ .

For each i ∈ I , we define a function β i : A i → B i as follows. For each C ∈ C i ( α ) , choose d C ∈ C ∩ Y . We can see that C ⧹ { d C } is nonempty since ∣ C ∣ > 1 . Consider the following two cases.

Case 1:

∣ ⋃ C ∈ C i ( α ) ( C ⧹ { d C } ) ∣ ≥ ∣ B i ⧹ A i α ∣ .

Then there is an injection

γ i : B i ⧹ A i α → ⋃ C ∈ C i ( α ) ( C ⧹ { d C } ) .

Let x γ i − 1 = { g x } for each x ∈ im γ i . Define β i : A i → B i by

x β i = g x if x ∈ im γ i , x α otherwise .

Clearly, β i is surjective.

Case 2:

∣ ⋃ C ∈ C i ( α ) ( C ⧹ { d C } ) ∣ < ∣ B i ⧹ A i α ∣ .

Then there is an injection

μ i : ⋃ C ∈ C i ( α ) ( C ⧹ { d C } ) → B i ⧹ A i α .

Define β i : A i → B i by

x β i = x μ i if x ∈ dom μ i , x α otherwise .

Clearly, β i is injective.

Finally, we define a function β : X → Y by

x β = x β i if x ∈ A i , x α otherwise .

It is easy to see that β ∈ T E ∗ ( X , Y ) and α ≠ β . Moreover, β satisfies Theorem 4.2, which implies that β is maximal. Now, we show that α ≤ β by using Theorem 2.1.

(1) Let ( x β ) β − 1 ∈ π ( β ) . If x ∉ A i for all i ∈ I , then x β = x α . We claim that ( x β ) β − 1 ⊆ ( x α ) α − 1 . Indeed, let z ∈ ( x β ) β − 1 . Then z β = x β , which implies that ( x β , z β ) ∈ E . Thus, ( x , z ) ∈ E and so z ∉ A i for all i ∈ I . Hence, z β = z α from which it follows that z α = x α and then z ∈ ( x α ) α − 1 . On the other hand, if x ∈ A i for some i ∈ I , then x β = x β i . We consider the following two cases.

If A i satisfies case 1 as above, then we first suppose that x ∈ im γ i . Then x β = g x . We claim that ( x β ) β − 1 = { x } . Clearly, x ∈ ( x β ) β − 1 . Let y ∈ ( x β ) β − 1 . Then y β = x β = g x and y ∈ A i , which implies that y ∈ im γ i . Thus,

y γ i − 1 = { g y } = { y β } = { x β } = { g x } = x γ i − 1

and so x = y . Hence, ( x β ) β − 1 = { x } ⊆ ( x α ) α − 1 ∈ π ( α ) . Now, we assume that x ∉ im γ i . Let z ∈ ( x β ) β − 1 . Then z β = x β = x α from which it follows that z ∈ A i and z ∉ im γ i . Hence, z α = z β = x α and so z ∈ ( x α ) α − 1 ∈ π ( α ) . Therefore, π ( β ) refines π ( α ) .

If A i satisfies case 2 as above, then we suppose that x ∈ dom μ i . Then x β = x μ i . We claim that ( x β ) β − 1 = { x } . Clearly, x ∈ ( x β ) β − 1 . Let y ∈ ( x β ) β − 1 . Then y β = x β = x μ i and y ∈ A i , which implies that y ∈ dom μ i . Thus,

y μ i = y β = x β = x μ i

and so x = y since μ i is injective. Hence, ( x β ) β − 1 = { x } ⊆ ( x α ) α − 1 ∈ π ( α ) . Now, we assume that x ∉ dom μ i . Let z ∈ ( x β ) β − 1 . Then z β = x β = x α from which it follows that z ∈ A i and z ∉ dom μ i . Hence, z β = z α and so z α = x α implies z ∈ ( x α ) α − 1 ∈ π ( α ) . Therefore, π ( β ) refines π ( α ) .

(2) By the definition of β , it is easy to see that for each x ∈ X , x α = x β if x β ∈ X α .

(3) Let A ∈ X / E . We assert that ( A ∩ Y ) α ⊆ ( A ∩ Y ) β . Indeed, let x α ∈ ( A ∩ Y ) α , where x ∈ A ∩ Y . If A ≠ A i for all i ∈ I , then x α = x β ∈ ( A ∩ Y ) β . Assume that A = A i for some i ∈ I . We consider the following two cases.

For A i satisfies case 1 as above, it is clear that x α = x β ∈ ( A i ∩ Y ) β if x ∉ im γ i . Now, we assume that x ∈ im γ i . Then x ∈ C ⧹ { d C } for some C ∈ C i ( α ) . We can see that x α = d C α since x , d C ∈ C ∈ π ( α ) . Moreover, since d C ∉ im γ i , we obtain x α = d C α = d C β ∈ ( C ∩ Y ) β ⊆ ( A i ∩ Y ) β .

For A i satisfies case 2 as above, it is clear that x α = x β ∈ ( A i ∩ Y ) β if x ∉ dom μ i . Now, we assume that x ∈ dom μ i . Then x ∈ C ⧹ { d C } for some C ∈ C i ( α ) . We can see that x α = d C α since x , d C ∈ C ∈ π ( α ) . Moreover, since d C ∉ dom μ i , we obtain x α = d C α = d C β ∈ ( C ∩ Y ) β ⊆ ( A i ∩ Y ) β .

Therefore, ( A ∩ Y ) α ⊆ ( A ∩ Y ) β and so A α = ( A ∩ Y ) α ⊆ ( A ∩ Y ) β since α ∈ F E ∗ .□

The proof of the above theorem can be illustrated by the following example.

Example 4.4

Let X = { 1 , 2 , 3 , … } , Y = { 2 , 4 , 6 , … } and

X / E = { { 1 , 2 , 3 , 4 } , { 5 , 6 } , { 7 , 8 , 9 , 10 , 11 , 12 } , { 13 , 14 , 15 , 16 , … } } .

Let α ∈ T E ∗ ( X , Y ) be defined by

α = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 … 2 2 2 2 8 8 6 6 6 6 6 6 14 14 16 16 18 18 20 20 … .

According to the proof of Theorem 4.3, let A 1 = { 1 , 2 , 3 , 4 } and A 2 = { 5 , 6 } . It is easy to see that A 1 α = { 2 } ⊊ { 2 , 4 } = B 1 and A 2 α = { 8 } ⊊ { 8 , 10 , 12 } = B 2 . Moreover, α ∣ A 1 and α ∣ A 2 are not injective. By the notation defined in the proof of Theorem 4.3, we obtain C 1 ( α ) = { { 1 , 2 , 3 , 4 } } and C 2 ( α ) = { { 5 , 6 } } . We choose d C 1 = 4 ∈ C 1 ∩ Y and d C 2 = 6 ∈ C 2 ∩ Y and then ∣ { 1 , 2 , 3 , 4 } ⧹ { d C 1 } ∣ = ∣ { 1 , 2 , 3 } ∣ ≥ ∣ { 4 } ∣ = ∣ B 1 ⧹ A 1 α ∣ and ∣ { 5 , 6 } ⧹ { d C 2 } ∣ = ∣ { 5 } ∣ < ∣ { 10 , 12 } ∣ = ∣ B 2 ⧹ A 2 α ∣ . By Case 1 given in the proof of the above theorem, there is an injection γ 1 : { 4 } → { 1 , 2 , 3 } , say γ : 4 ↦ 3 . Define β 1 : A 1 → B 1 by

β 1 = 1 2 3 4 2 2 4 2 .

Similarly, by using case 2, there is an injection μ 2 : { 5 } → { 10 , 12 } , say γ : 5 ↦ 10 . We define β 2 : A 2 → B 2 by

β 2 = 5 6 10 8 .

Finally, we have

β = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 … 2 2 4 2 10 8 6 6 6 6 6 6 14 14 16 16 18 18 20 20 … .

Then α ≤ β and β is maximal by Theorem 4.2.

Now, we deal with minimal elements.

Theorem 4.5

Let α ∈ T E ∗ ( X , Y ) . Then α is minimal if and only if ∣ A α ∣ = 1 for each A ∈ X / E .

Proof

( ⇒ ) We prove by contrapositive. Assume that ∣ A α ∣ > 1 for some A ∈ X / E . We can find p ∈ ( A ∩ Y ) α since A ∩ Y is nonempty. Write X / E ⧹ { A } = { A i : i ∈ I } and choose a i ∈ A i ∩ Y for all i ∈ I . Define a function β : X → Y by

β = A A i p a i α .

It is easy to verify that β ∈ T E ∗ ( X , Y ) and β ≠ α . We show that β ≤ α .

  1. Let ( x α ) α − 1 ∈ π ( α ) . We claim that ( x α ) α − 1 ⊆ ( x β ) β − 1 ∈ π ( β ) . Let a ∈ ( x α ) α − 1 . Then a α = x α . We note that ( a , x ) ∈ E . If x ∈ A , then a ∈ A and a β = p = x β , which implies that a ∈ ( x β ) β − 1 . If x ∈ A i for some i ∈ I , then a ∈ A i and a β = a i α = x β . Hence, a ∈ ( x β ) β − 1 . We conclude that π ( α ) refines π ( β ) .

  2. Let x α ∈ X β . Then x α = y β for some y ∈ X . If y ∈ A , then x α = y β = p ∈ ( A ∩ Y ) α . We obtain x ∈ A since α ∈ T E ∗ ( X , Y ) . Hence, x β = p = x α . If y ∈ A i , then x α = y β = a i α ∈ ( A i ∩ Y ) α . Similarly, we see that x ∈ A i and thus x β = a i α = x α .

  3. Clearly, A β = { p } ⊆ ( A ∩ Y ) α and A i β = { a i α } ⊆ ( A i ∩ Y ) α for all i ∈ I .

Thus, α is not minimal.

( ⇐ ) Suppose that ∣ A α ∣ = 1 for each A ∈ X / E . Let β ≤ α be such that β ≠ α . There exists x ∈ X be such that x α ≠ x β . We have x ∈ A for some A ∈ X / E and ∣ A α ∣ = 1 , which implies A α = { x α } . By Theorem 2.1 (3), x β ∈ A β ⊆ ( A ∩ Y ) α ⊆ A α = { x α } . Hence, x α = x β , which is a contradiction. Therefore, α = β .□

Theorem 4.6

Let α ∈ T E ∗ ( X , Y ) . Then there exists a minimal element β ∈ T E ∗ ( X , Y ) such that β ≤ α .

Proof

Assume that α is not minimal. Then ∣ A α ∣ > 1 for some A ∈ X / E . Let β be defined as in the proof of Theorem 4.5. Then β < α . Moreover, it is clear that ∣ C β ∣ = 1 for each C ∈ X / E . Hence, β is minimal.□

By the above results, we may conclude the following corollaries which extend the results in [3,4,9].

Corollary 4.7

Let α ∈ T E ∗ ( X ) . Then the following statements hold.

  1. α is maximal if and only if for each E -class A , A α ∈ X / E or α ∣ A is injective.

  2. α is minimal if and only if ∣ A α ∣ = 1 for each A ∈ X / E .

  3. There exists a maximal element β ∈ T E ∗ ( X ) such that α ≤ β .

  4. There exists a minimal element β ∈ T E ∗ ( X ) such that β ≤ α .

Corollary 4.8

Let α ∈ T ( X , Y ) . Then the following statements hold.

  1. α is maximal if and only if α ∉ F or X α = Y or α is injective.

  2. α is minimal if and only if ∣ X α ∣ = 1 .

  3. There exists a maximal element β ∈ T ( X , Y ) such that α ≤ β .

  4. There exists a minimal element β ∈ T ( X , Y ) such that β ≤ α .

Finally, the following results are concerned with the existence of an upper cover and a lower cover for elements of T E ∗ ( X , Y ) . Recall that an element b in a semigroup S is called an upper cover for a ∈ S if a < b and there exists no c ∈ S such that a < c < b . A lower cover is defined dually.

Theorem 4.9

Let α ∈ T E ∗ ( X , Y ) . If α is not maximal, then α has an upper cover.

Proof

Suppose that α is not maximal. Then there is a class A ∈ X / E such that A α ⊊ B for some B ∈ Y / E Y and α ∣ A is not injective. Moreover, we have α ∈ F E ∗ . Let β be defined as in the proof of Theorem 4.2, and we will show that β is an upper cover of α . Suppose that α < γ ≤ β for some γ ∈ T E ∗ ( X , Y ) . Then by Corollary 1.1, we conclude that X α ≠ X γ . We obtain X α ⊂ X γ ⊆ X β = X α ∪ { b } and thus X γ = X β . It follows from Corollary 1.1 that γ = β . Hence, β is an upper cover of α .□

Theorem 4.10

Let α ∈ T E ∗ ( X , Y ) . If α is not minimal, then α has a lower cover.

Proof

Suppose that α is not minimal. We have two cases to consider.

Case 1: α ∈ F E ∗ . There exists an E -class A such that ∣ A α ∣ > 1 . We can find two distinct elements p , q ∈ A α . Define β : X → Y by

x β = p if x ∈ q α − 1 , x α otherwise .

It is straightforward to show that β ∈ T E ∗ ( X , Y ) and β ≠ α . Now, we prove that β ≤ α by using Theorem 2.1.

  1. Let ( x α ) α − 1 ∈ π ( α ) . To show that ( x α ) α − 1 ⊆ ( x β ) β − 1 , let y ∈ ( x α ) α − 1 . Then y α = x α . First, we assume that x ∉ q α − 1 . Then x α = x β . If y ∈ q α − 1 , then x α = y α = q , which contradicts to x ∉ q α − 1 . Hence, y ∉ q α − 1 and so y β = y α = x β . We obtain y ∈ ( x β ) β − 1 ∈ π ( β ) . Now, suppose that x ∈ q α − 1 . Then x α = q and x β = p . We obtain y α = x α = q , which implies that y ∈ q α − 1 . Thus, y β = p = x β and so y ∈ ( x β ) β − 1 ∈ π ( β ) . Therefore, π ( α ) refines π ( β ) .

  2. Let x α ∈ X β . Then x α = y β for some y ∈ X . If y ∈ q α − 1 , then x α = y β = p and hence x ∈ p α − 1 ≠ q α − 1 . We have x ∉ q α − 1 and so x α = x β . If y ∉ q α − 1 , then x α = y β = y α ≠ q , which implies that x ∉ q α − 1 . Thus, x α = x β .

  3. Let C ∈ X / E . To show that C β ⊆ ( C ∩ Y ) α , let c β ∈ C β . If c ∈ q α − 1 , then c ∈ A , which implies that C = A . Hence, c β = p ∈ A α ⊆ ( A ∩ Y ) α = ( C ∩ Y ) α since α ∈ F E ∗ . On the other hand, if c ∉ q α − 1 , then c β = c α ∈ C α ⊆ ( C ∩ Y ) α .

To show that β is a lower cover for α , let γ ∈ T E ∗ ( X , Y ) be such that β ≤ γ < α . Then by Theorem 2.1, X α ⧹ { q } = X β ⊆ X γ ⊂ X α , which implies X β = X γ . By Corollary 1.1 (1), we conclude that β = γ . Consequently, β is a lower cover.

Case 2: α ∉ F E ∗ . Let { A i : i ∈ I } be the set of all E -classes such that ∣ A i α ∣ > 1 and π A i ( α ) ⧹ π ˜ A i ( α ) ≠ ∅ . We can see that { A i : i ∈ I } is nonempty since α ∉ F E ∗ and α is not minimal. For each i ∈ I , we choose and fix a i ∈ ( A i ∩ Y ) α and define a function β i : A i → A i α by

x β i = a i if x ∈ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) , x α otherwise .

Let β : X → Y be defined by

x β = x β i if x ∈ A i , x α otherwise .

It is clear that β ∈ T E ∗ ( X , Y ) and β ≠ α . We aim to prove that β ≤ α by Theorem 2.1.

  1. Let ( x α ) α − 1 ∈ π ( α ) . To show that ( x α ) α − 1 ⊆ ( x β ) β − 1 , let y ∈ ( x α ) α − 1 . Then y α = x α . We note that ( x , y ) ∈ E .

First, we assume that x ∉ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for all i ∈ I . Then x α = x β . If y ∈ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for some i ∈ I , then y ∈ A i and ( x α ) α − 1 ∩ Y = ( y α ) α − 1 ∩ Y = ∅ . Hence, x ∈ A i and x ∈ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) , which is a contradiction. Thus, y ∉ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for all i ∈ I . Then y β = y α = x α = x β and hence y ∈ ( x β ) β − 1 ∈ π ( β ) .

Now, suppose that x ∈ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for some i ∈ I . Then x ∈ A i and x β = a i . We note that y ∈ A i since ( x , y ) ∈ E . If y ∉ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) , then ( x α ) α − 1 ∩ Y = ( y α ) α − 1 ∩ Y ≠ ∅ . Hence, x ∈ π ˜ A i ( α ) , which is a contradiction. Hence, y ∈ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) . We have y β = a i = x β and so y ∈ ( x β ) β − 1 ∈ π ( β ) . We conclude that π ( α ) refines π ( β ) .

  1. Let x α ∈ X β . Then x α = y β for some y ∈ X . If y ∈ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for some i ∈ I , then x α = y β = a i ∈ ( A i ∩ Y ) α . There is b ∈ A i ∩ Y such that x α = b α from which it follows that x ∈ ( b α ) α − 1 ∈ π ˜ A i ( α ) . We have x ∉ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) and so x α = x β . If y ∉ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for all i ∈ I , then x α = y β = y α and hence ( x α ) α − 1 ∩ Y = ( y α ) α − 1 ∩ Y ≠ ∅ . We conclude that x ∉ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for all i ∈ I . Consequently, x α = x β .

  2. Let C ∈ X / E . To show that C β ⊆ ( C ∩ Y ) α , let c β ∈ C β . If c ∈ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for some i ∈ I , then A i = C and c β = a i ∈ ( A i ∩ Y ) α = ( C ∩ Y ) α . If c ∉ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for all i ∈ I , then c β = c α . Moreover, we have c ∈ π ˜ C ( α ) , which implies that c ∈ ( d α ) α − 1 for some ( d α ) α − 1 ∈ π ˜ C ( α ) . Hence, there is y ∈ ( d α ) α − 1 ∩ Y and so c α = d α = y α . It is clear that y ∈ C . We obtain c β = c α = y α ∈ ( C ∩ Y ) α .

To show that β is a lower cover for α , let γ ∈ T E ∗ ( X , Y ) be such that β < γ < α . Then X β ⊂ X γ ⊂ X α and so there is x γ ∈ X γ ⧹ X β . Moreover, since γ ≠ α , we obtain x γ ∈ X γ ⊆ Y α from which it follows that there exists y ∈ Y such that x γ = y α . It is clear that y ∉ ⋃ ( π A i ( α ) ⧹ π ˜ A i ( α ) ) for all i ∈ I . Hence, x γ = y α = y β ∈ X β , which is a contradiction. Consequently, β is a lower cover.□

Example 4.11

Let X = { 1 , 2 , 3 , … } , Y = { 2 , 4 , 6 , … } and X / E = { { 1 , 2 , 3 , 4 } , { 5 , 6 , 7 , 8 } , { 9 , 10 , 11 , … } } . We define α ∈ T E ∗ ( X , Y ) by

α = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 … 2 2 4 2 6 8 8 8 10 10 12 12 14 14 16 16 18 18 20 20 … .

It is clear that α ∉ F E ∗ . Let A 1 = { 1 , 2 , 3 , 4 } and A 2 = { 5 , 6 , 7 , 8 } . We can see that π A 1 ( α ) = { { 1 , 2 , 4 } , { 3 } } , π ˜ A 1 ( α ) = { { 1 , 2 , 4 } } , π A 2 ( α ) = { { 5 } , { 6 , 7 , 8 } } and π ˜ A 2 ( α ) = { { 6 , 7 , 8 } } , which imply that π A 1 ( α ) ⧹ π ˜ A 1 ( α ) = { { 3 } } ≠ ∅ and π A 2 ( α ) ⧹ π ˜ A 2 ( α ) = { { 5 } } ≠ ∅ . Now, we choose 2 = 2 α ∈ ( A 1 ∩ Y ) α and 8 = 6 α ∈ ( A 2 ∩ Y ) α . By Case 2 in the proof of Theorem 4.10, we define

β 1 = 1 2 3 4 2 2 2 2 and β 2 = 5 6 7 8 8 8 8 8 .

Finally, we have

β = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 … 2 2 2 2 8 8 8 8 10 10 12 12 14 14 16 16 18 18 20 20 …

is a lower cover of α .

We end this section by briefly summarizing our paper and point out future work. In [3,4,9], the natural partial order on T ( X , Y ) and T E ∗ ( X ) was determined. This paper is concerned with the same problems for the semigroup T E ∗ ( X , Y ) , that is, we study the compatibility, maximality and minimality of its elements in this semigroup under the natural partial order. Moreover, in the last section of our paper, we show that every element of T E ∗ ( X , Y ) lies between maximal and minimal elements. These are results that we have not found in the studies of other transformation semigroups. We also prove that for every α ∈ T E ∗ ( X , Y ) , if α is not maximal, then it has an upper cover; and if α is not minimal, then it has a lower cover. However, in [4], the authors investigated the greatest lower bound and the least upper bound of two elements of T E ∗ ( X ) , it is then natural to ask for this property on T E ∗ ( X , Y ) , which is an open question.

Acknowledgements

The author is grateful to Prof. Jintana Sanwong for her collaboration. The author thanks the referee for his/her careful review and for a number of helpful suggestions.

Part of this work was presented in the Arbeitstagung Allgemeine Algebra conference (AAA99), Siena, Italy, February 21–23, 2020.

  1. Funding information: This research was supported by the Research Fund for DPST Graduate with First Placement, Year 2016, under grant No. 027/2559 and Chiang Mai University.

  2. Conflict of interest: The author states no conflict of interest.

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Received: 2021-06-09
Revised: 2021-09-15
Accepted: 2021-09-20
Published Online: 2021-12-31

© 2021 Kritsada Sangkhanan, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0109
Keywords for this article
transformation semigroup; equivalence; partial order; compatibility; maximal and minimal
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BY 4.0

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  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
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  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
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  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
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  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
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  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
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  38. Inviscid, zero Froude number limit of the viscous shallow water system
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  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
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  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
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  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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