On pomonoid of partial transformations of a poset

Bana Al Subaiei

doi:10.1515/math-2023-0161

Article Open Access

On pomonoid of partial transformations of a poset

Bana Al Subaiei

Published/Copyright: December 16, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

The main objective of this article is to study the ordered partial transformations PO ( X ) of a poset X . The findings show that the set of all partial transformations of a poset with a pointwise order is not necessarily a pomonoid. Some conditions are implemented to guarantee that PO ( X ) is a pomonoid and this pomonoid is denoted by PO ↑ ( X ) . Moreover, we determine the necessary conditions in order that the partial order-embedding transformations define the ordered version of the symmetric inverse monoid. The findings show that this set is an inverse pomonoid and we will denote it by ℐPO ↑ ( X ) . In case the order on the poset X is total, we explore some properties of PO ↑ ( X ) and ℐPO ↑ ( X ) , including regressive, unitary, and reversible.

Keywords: posets; pomonoids; partial transformations; inverse pomonoid

MSC 2010: 20M20; 06F05; 20M10

1 Introduction and preliminaries

Semigroups of transformations play a role in semigroup theory similar to the role of permutation groups in group theory. For a set X , we let PT ( X ) denote the monoid (under composition) of all partial transformations of X (i.e., mappings whose domain and image are the subsets of X ). The submonoid of PT ( X ) of all full transformations of X (i.e., mappings from X into X ) is denoted by T ( X ) . The inverse submonoid of all full injective transformations of X is denoted by ℐ ( X ) . These monoids are very important in the theory of semigroups since every semigroup (resp., inverse semigroup) can be embedded in some T ( X ) (resp., ℐ ( X ) ). This fact constitutes an analogy to Cayley’s theorem in group theory. Cayley’s theorem says that every group can be embedded in some symmetric group Sym ( X ) of all permutations on X .

Throughout this work, we will write mappings on the right and compose them from left to right. This means that for f : A → B and g : B → C , we will write x f , instead of f ( x ) , and x ( f g ) , instead of ( g f ) ( x ) . Now, suppose that X is a poset (i.e., a partially ordered set). We say that a transformation f in PT ( X ) is order-preserving (monotone) if x ≤ y implies x f ≤ y f , for all x , y ∈ dom ( f ) . Also, we say that a transformation f in PT ( X ) is order-embedding whenever that x ≤ y if and only if x f ≤ y f , for all x , y ∈ dom ( f ) . Note that the product (composition) of two-order-preserving transformations and two-order-embedding transformations is also order-preserving and order-embedding, respectively. As usual, the submonoid of PT ( X ) of all partial order-preserving transformations of X will be denoted by PO ( X ) , while the monoid PO ( X ) ∩ T ( X ) of all full transformations of X that preserve the order will be denoted by O ( X ) . These monoids have been widely studied when X is totally ordered, namely, in [1–6].

Considerable attention has been paid over the years to full, injective, partial, and partial injective transformations of a set. While there are advancements being made in the study of these concepts (cf. [7]), the ordered versions have not yet been investigated. Some researchers have studied these concepts on totally ordered sets and have defined them as full and partial order-preserving transformations (see, for instance, [1,2,8,9]). Also, some studies have considered the order-decreasing and increasing of full, partial, and partial one-to-one transformations of a totally ordered set (cf. [3,10–12]). However, none of these studies has looked at these concepts as partially ordered semigroups (or monoids) for any poset X . Al Subaiei in [13] has only studied the full transformations of a poset X for some specific partially ordered relations on X . To complete this circle of ideas, our purpose here is to focus on the partial order-preserving (monotone) transformations and partial order-embedding transformations of a poset X .

Next, we recall some background. A posemigroup (resp., pomonoid) is a semigroup (resp., monoid) S partially ordered by ≤ , such that ≤ is compatible with the semigroup (resp., monoid) operation, i.e., for all x , y , z ∈ S , x ≤ y implies z x ≤ z y and x z ≤ y z . For more details about monoids and pomonoids, we refer the reader to [14,15]. In [16], Sohail has considered the full transformations of a poset, when he studied the ordered representations of a pomonoid. He has stated that the sets O ( X ) ⊂ T ( X ) are pomonoids with respect to the usual composition of transformations and pointwise order (i.e., for f , g ∈ O ( X ) , f ≼ g if and only if x f ≤ x g for all x ∈ X ). It is worth mentioning that Nasir’s work is mainly focused on the representations of a pomonoid and not the study of the transformations. As partial transformations of a poset X are not considered yet when X is not a totally ordered set, so a natural question arises: whether PO ( X ) does constitute a pomonoid when equipped with the pointwise order? Recall that the pointwise order on PO ( X ) is defined as follows:

(1) ∀ f , g ∈ PO ( X ) , f ≼ g ⇔ dom ( f ) ⊆ dom ( g ) and ∀ x ∈ dom ( f ) , x f ≤ x g .

It is worth noting that generalizing these concepts to the ordered case requires more conditions to guarantee the “pomonoid structure” of the set of partial transformations of a poset.

Any subset X of a poset Y is called a subposet since the partial order relation is inherited from Y . Let X be a subposet of a poset Y . In general, the set X ↑ Y = { y ∈ Y : x ≤ y for some x ∈ X } is known as the upper/upward closure of X . It is clear that X ⊆ X ↑ Y and this inclusion relation may be strict in general. Moreover, X ↑ Y may be different from the set of upper bounds of X . The following example illustrates these facts.

Example 1.1

Consider the poset Y = { a , b , c , d , e , f } equipped with the following order relation ≤ 1 as in Figure 1.

Also, consider the set X = { a , b , e } . Hence, we have X ↑ Y = { a , b , e , c , d } and the set of upper bounds of X is { b , d } . Then, it is clear that X ↑ Y ⊋ X . However, if the order relation on Y is ≤ 2 , which is defined as in Figure 2. Then, X ↑ Y = { a , b , e } , while the set of upper bounds of X is empty. Hence, X ↑ Y = X .

$Figure 1 Partially ordered relation ≤ 1 {\le }_{1} on Y Y .$

Figure 1

Partially ordered relation ≤ 1 on Y .

$Figure 2 Partially ordered relation ≤ 2 {\le }_{2} on Y Y .$

Figure 2

Partially ordered relation ≤ 2 on Y .

Kemprasit [17] studied some properties of the partial transformation of poset X , such as the idempotent elements, shift of an element and regressive. An element e in any semigroup S is called idempotent, when e 2 = e and usually E ( S ) denote the set of all idempotent elements in S . The set of idempotents of the partial transformation monoid on poset X , denoted by E ( PT ( X ) ) , is defined as: E ( PT ( X ) ) = { f ∈ PT ( X ) ∣ im ( f ) ⊆ dom ( f ) , and x f = x ∀ x ∈ im ( f ) } . The shift of an element f ∈ PT ( X ) is the set S ( f ) = { x ∈ dom ( f ) ∣ x f ≠ x } . The element f ∈ PT ( X ) is said to be regressive if for every x ∈ dom ( f ) , x f ≤ x . Gould and Shaheen [18] studied the concept of unitary in posemigroup, and then, Al Subaiei and Renshaw [19,20] generalized this concept further. Let U be a subpomonoid of a pomonoid S and let v , u ∈ U and s ∈ S . U is said to be an upper strongly right pounitary in S when v ≤ s u implies s ∈ U . Moreover, U is said to be a lower strongly right pounitary in S when s u ≤ v , implies s ∈ U . U is said to be a right unitary in S when s u = v , implies s ∈ U . Left-sided versions are defined dually.

Reversibility was studied in the literature for a pomonoid (see, for example, [21,22]). Let S be a pomonoid, then S is right reversible, if for any s , s ′ ∈ S , we have S s ∩ S s ′ ≠ ∅ . However, a pomonoid S is called weakly right reversible whenever for any t , t ′ ∈ S , we have S t ∩ ( S t ′ ] ≠ ∅ . The set ( S t ] ≔ { s ∈ S : s ≤ k , k ∈ S t } is called the down-set of S . Weakly left reversible is defined dually.

This article is organized as follows: In Section 2, we construct an example showing that PO ( X ) with the pointwise order is not a pomonoid (Example 2.1). Thus, we have added some conditions, and we have considered the set PO ↑ ( X ) ≔ { f ∈ PO ( X ) ∣ dom ( f ) ↑ = dom ( f ) } . The main result of this section is Theorem 2.4, which states that ( PO ↑ ( X ) , ≼ ) is a pomonoid, when it is equipped with the pointwise order (1). In Section 3, we consider the monoid POℰ ( X ) of partial order-embedding transformations on a poset X . It is shown that this monoid is not a pomonoid with respect to the pointwise order. So we restrict ourselves to the set POℰ ↑ ( X ) ≔ { f ∈ POℰ ( X ) ∣ im ( f ) ↑ = im ( f ) } . We establish in Theorem 3.3 that ℐPO ↑ ( X ) is a subpomonoid of PO ↑ ( X ) . Theorem 3.4 states that ℐPO ↑ ( X ) is an inverse pomonoid. Section 4 is devoted to the study of the pomonoids PO ↑ ( X ) and ℐPO ↑ ( X ) in case X is a finite toset (i.e., a totally ordered set). In Proposition 4.1, we prove that if X is a finite toset, then ∣ ℐPO ↑ ( X ) ∣ = ∣ X ∣ + 1 . Moreover, Remark 4.3 ensures that, if X is a finite toset, then ℐPO ↑ ( X ) ⊊ E ( PO ↑ ( X ) ) . It is shown that if X is a finite toset, then ℐPO ↑ ( X ) is right reversible, and for any f ∈ ℐPO ↑ ( X ) , S ( f ) = ∅ and f is regressive (Remark 4.3). Theorem 4.4 states that ℐPO ↑ ( X ) is a right and left unitary in PO ↑ ( X ) , when X is a finite toset. Table 1 contains all the notations used in this article.

Table 1

Descriptions for notations

Notations	Descriptions
X	Poset
PT ( X )	The monoid of all partial transformations of X
T ( X )	The monoid of all full transformations of X
ℐ ( X )	The inverse monoid of all full injective transformations of X
Pℐ ( X )	The inverse monoid of all partial injective transformations of X
PO ( X )	The set of all partial order-preserving transformations of X
O ( X )	The pomonoid of all full order-preserving transformations of X
X ↑	The upper closure of X .
dom ( f ) ↑	( dom ( f ) ) ↑
im ( f ) ↑	( im ( f ) ) ↑
PO ↑ ( X )	≔ { f ∈ PO ( X ) ∣ dom ( f ) ↑ = dom ( f ) }
POℰ ↑ ( X )	≔ { f ∈ POℰ ( X ) ∣ im ( f ) ↑ = im ( f ) }
ℐPO ↑ ( X )	≔ { f ∈ PO ↑ ( X ) ∣ f is an order embedding , im ( f ) ↑ = im ( f ) }

2 Ordered partial transformations of a poset

Let ( X , ≤ ) be a poset and equip PO ( X ) with the following pointwise order relation:

∀ f , g ∈ PO ( X ) , f ≼ g ⇔ dom ( f ) ⊆ dom ( g ) , and ∀ x ∈ dom ( f ) , x f ≤ x g .

The next example shows that ( PO ( X ) , ≼ ) is not, in general, a pomonoid.

Example 2.1

Let X = { a , b , c } be a poset with the partial order ≤ 3 as in Figure 3.

$Figure 3 Partial ordered relation ≤ 3 {\le }_{3} on X X .$

Figure 3

Partial ordered relation ≤ 3 on X .

Our task is to show that the monoid ( PO ( X ) , ≼ ) is not a pomonoid. Let f = a b c a a a , g = a b and h = b b . One can easily check that all these mappings are in PO ( X ) . However, ≼ is not compatible with the composition. Indeed, we have g ≼ f but g h ⋠ f h since dom ( g h ) = { a } ⊈ dom ( f h ) = ∅ . Therefore, PO ( X ) with the pointwise order is not a pomonoid (Figure 3).

It is worth noting that in the previous example, we have dom ( h ) ↑ = { a , b } ⊋ dom ( h ) = { b } . So, Example 2.1 encourages us to add some conditions in order to equip the monoid PO ( X ) with a pomonoid structure. To this end, let us consider the following set:

PO ↑ ( X ) ≔ { f ∈ PO ( X ) ∣ dom ( f ) ↑ = dom ( f ) } .

The set PO ↑ ( X ) will be called the set of ordered partial transformations of X . It is obvious that the identity map 1 X ∈ PO ↑ ( X ) . We start our investigation with the following straightforward lemma. We include a proof for the sake of completeness.

Lemma 2.2

Let f ∈ PO ( X ) . Then, ( Y f − 1 ) ↑ ⊆ ( Y ↑ ) f − 1 for any Y ⊆ im ( f ) .

Proof

Let x ∈ ( Y f − 1 ) ↑ . Then, there exists a ∈ Y f − 1 such that a ≤ x . As f is an order-preserving map, then a f ≤ x f . But a f ∈ Y . This implies that x f ∈ Y ↑ and so x ∈ ( Y ↑ ) f − 1 . This completes the proof.□

The following lemma is straightforward.

Lemma 2.3

Let f , h , g ∈ PO ( X ) . If f ≼ g , then h f ≼ h g .

Theorem 2.4

( PO ↑ ( X ) , ≼ ) is a pomonoid.

Proof

First of all, we need to show that PO ↑ ( X ) is a monoid. To this end, let f , g ∈ PO ↑ ( X ) . Then, dom ( f ) ↑ = dom ( f ) and dom ( g ) ↑ = dom ( g ) . Thus,

dom ( f g ) ↑ = ( ( dom ( g ) ∩ im ( f ) ) f − 1 ) ↑ ⊆ ( ( dom ( g ) ∩ im ( f ) ) ↑ ) f − 1 (Lemma 2.2) ⊆ ( dom ( g ) ↑ ) f − 1 ∩ ( im ( f ) ↑ ) f − 1 = ( dom ( g ) ) f − 1 ∩ ( im ( f ) ) f − 1 = ( dom ( g ) ∩ im ( f ) ) f − 1 = dom ( f g ) .

Thus, dom ( f g ) = dom ( f g ) ↑ . This shows that PO ↑ ( X ) is a submonoid of PO ( X ) and hence a monoid.

Next, we show that ≼ is compatible with the composition in PO ↑ ( X ) . Let f , g , h ∈ PO ↑ ( X ) such that f ≼ g . Our task is to show that f h ≼ g h and h f ≼ h g . Let x ∈ dom ( f h ) = ( dom ( h ) ∩ im ( f ) ) f − 1 . Then, x ∈ dom ( f ) and x f ∈ dom ( h ) . As x f ≤ x g , then x g ∈ dom ( h ) ↑ . But dom ( h ) ↑ = dom ( h ) , which yields that x g ∈ dom ( h ) . Hence, x g ∈ dom ( h ) ∩ im ( g ) . This shows that x ∈ ( dom ( h ) ∩ im ( g ) ) g − 1 = dom ( g h ) . Therefore, dom ( f h ) ⊆ dom ( g h ) . Now, using the fact that h is an order-preserving map and f ≼ g , we obtain readily x ( f h ) ≤ x ( g h ) for any x ∈ dom ( f h ) . Therefore, f h ≼ g h . For h f ≼ h g , it follows directly from Lemma 2.3.□

3 Order-embedding partial transformations of a poset

In this section, we consider the following monoid:

Pℐ ( X ) ≔ { f ∈ PT ( X ) ∣ f is injective } = PT ( X ) ∩ ℐ ( X ) .

We set

POℰ ( X ) ≔ { f ∈ PT ( X ) ∣ f is an order embedding }

POℰ ↑ ( X ) ≔ { f ∈ POℰ ( X ) ∣ im ( f ) ↑ = im ( f ) } ,

and

ℐPO ↑ ( X ) ≔ { f ∈ PO ↑ ( X ) ∣ f is an order embedding , im ( f ) ↑ = im ( f ) } .

The subset POℰ ( X ) of Pℐ ( X ) is also a monoid and this is clear from the definition of POℰ ( X ) . Moreover, ℐPO ↑ ( X ) = PO ↑ ( X ) ∩ POℰ ↑ ( X ) . In this section, we consider the order version of the inverse monoid of all partial injective transformations of X .

Proposition 3.1

The set POℰ ↑ ( X ) is a monoid.

Proof

It is clear that POℰ ↑ ( X ) is a subset of the monoid Pℐ ( X ) . Let f , g ∈ POℰ ↑ ( X ) . We need to show that f g ∈ POℰ ↑ ( X ) . It is clear that f g ∈ POℰ ( X ) . Let x ∈ im ( f g ) ↑ = ( ( im ( f ) ∩ dom ( g ) ) g ) ↑ . Thus, there exists y ∈ ( im ( f ) ∩ dom ( g ) ) g such that y ≤ x . Write y = a g for some a ∈ im ( f ) ∩ dom ( g ) , and let d ∈ dom ( f ) such that a = d f . As a g ≤ x , then x ∈ im ( g ) ↑ . But, by assumption, im ( g ) = im ( g ) ↑ , so x = b g for some b ∈ dom ( g ) . Now, we obtain a g ≤ b g . Since, g is an order-embedding, we infer that a ≤ b . But, a ∈ im ( f ) . So b ∈ im ( f ) ↑ . Hypothetically, im ( f ) ↑ = im ( f ) . Thus, b = c f for some c ∈ dom ( f ) . It follows that x = c ( f g ) ∈ ( im ( f ) ∩ dom ( g ) ) g = im ( f g ) . Therefore, im ( f g ) ↑ ⊆ im ( f g ) . As the reverse inclusion is always true, we conclude that im ( f g ) ↑ = im ( f g ) . This completes the proof.□

From the previous proposition, we can have the following result.

Corollary 3.2

POℰ ↑ ( X ) is a submonoid of POℰ ( X ) .

Theorem 3.3

The following hold true:

ℐPO ↑ ( X ) is a subpomonoid of PO ↑ ( X ) .
ℐPO ↑ ( X ) is a submonoid of ℐ ( X ) .

Proof

Clearly, ℐPO ↑ ( X ) ⊆ POℰ ↑ ( X ) . Using Proposition 3.1, it is easy to show that ℐPO ↑ ( X ) is a submonoid of POℰ ↑ ( X ) . Using similar argument to Theorem 2.4, we can show that f h ≼ g h and h f ≼ h g for every f ≼ g and f , g , h ∈ ℐPO ↑ ( X ) .
It is clear by using similar argument as in case 1.□

Theorem 3.4

ℐPO ↑ ( X ) is an inverse pomonoid.

Proof

Let f ∈ ℐPO ↑ ( X ) . Then, f is an order-embedding map, dom ( f ) = dom ( f ) ↑ and im ( f ) = im ( f ) ↑ . From Theorem 3.3, we know that ℐPO ↑ ( X ) is a pomonoid and ℐPO ↑ ( X ) is a submonoid of ℐ ( X ) . Hence, there exists f − 1 ∈ ℐ ( X ) . The proof will be completed if we show that f − 1 ∈ ℐPO ↑ ( X ) . It is well known that for any φ ∈ ℐ ( X ) , dom ( φ ) = im ( φ − 1 ) and im ( φ ) = dom ( φ − 1 ) . As f − 1 ∈ ℐ ( X ) , we obtain immediately im ( f − 1 ) = dom ( f ) = dom ( f ) ↑ = im ( f − 1 ) ↑ and dom ( f − 1 ) = im ( f ) = im ( f ) ↑ = dom ( f − 1 ) ↑ . It obvious that f − 1 is an order-embedding. Hence, f − 1 ∈ ℐPO ↑ ( X ) . The proof is complete.□

Example 3.5

Consider the poset X = { a , b , c } as in Example 2.1. Then, we have

PO ↑ ( X ) = a 0 = 0 , a 1 = a b c a b c , a 2 = a b c a a a , a 3 = a b c a a c , a 4 = a b c a a b , a 5 = a b c b b b , a 6 = a b c b b a , a 7 = a b c b b c , a 8 = a b c c c c , a 9 = a b c c c a , a 10 = a b c c c b , a 11 = a b a b , a 12 = a b a a , a 13 = a b b b , a 14 = a b c c , a 15 = a c a c , a 16 = a c a b , a 17 = a c a a , a 18 = a c b a , a 19 = a c b b , a 20 = a c b c , a 21 = a c c a , a 22 = a c c b , a 23 = a c c c , a 24 = a a , a 25 = a b , a 26 = a c , a 27 = c a , a 28 = c b , a 29 = c c .
ℐPO ↑ ( X ) = a 0 = 0 , a 1 = a b c a b c , a 11 = a b a b , a 15 = a c a c , a 21 = a c c a , a 24 = a a , a 26 = a c , a 27 = c a , a 29 = c c .

Example 3.6

Let X = { 1 , 2 } with the natural order relation. Then, one can easily check that

PO ↑ ( X ) = a 0 = 0 , a 1 = 1 2 1 2 , a 2 = 1 2 1 1 , a 3 = 1 2 2 2 , a 4 = 2 1 , a 5 = 2 2 .
ℐPO ↑ ( X ) = { a 0 = 0 , a 1 = 1 2 1 2 , a 5 = 2 2 } .

Moreover, the order on PO ↑ ( X ) is as in Figure 4.

$Figure 4 Ordered relation ≼ \preccurlyeq on PO ↑ ( X ) {{\mathcal{PO}}}^{\uparrow }\left(X) .$

Figure 4

Ordered relation ≼ on PO ↑ ( X ) .

Example 3.7

Let X = { 1 , 2 , 3 } with the natural ordered relation. Then, we have

PO ↑ ( X ) = a 0 = 0 , a 1 = 1 2 3 1 2 3 , a 2 = 1 2 3 1 1 1 , a 3 = 1 2 3 2 2 2 , a 4 = 1 2 3 3 3 3 , a 5 = 1 2 3 1 1 2 , a 6 = 1 2 3 1 1 3 , a 7 = 1 2 3 2 2 3 , a 8 = 1 2 3 2 3 3 , a 9 = 1 2 3 1 2 2 , a 10 = 1 2 3 1 3 3 , a 11 = 2 3 1 1 , a 12 = 2 3 2 2 , a 13 = 2 3 3 3 , a 14 = 2 3 1 2 , a 15 = 2 3 1 3 , a 16 = 2 3 2 3 , a 17 = 3 1 , a 18 = 3 2 , a 19 = 3 3 .
ℐPO ↑ ( X ) = a 0 = 0 , a 1 = 1 2 3 1 2 3 , a 16 = 2 3 2 3 , a 19 = 3 3 .

4 Totally ordered poset

In this section, we study PO ↑ ( X ) and ℐPO ↑ ( X ) in case X is a toset (i.e., a totally ordered set). Note that even when X is a toset, the aforementioned monoids differ from the semigroup of order-preserving partial transformation which is known in the literature [1–3]. To see this, consider the poset X in Example 3.6 and note that the mappings 1 2 and 1 1 do not belong to PO ↑ ( X ) . However, these maps are elements of the semigroup of order-preserving partial transformation.

Proposition 4.1

Let X be a finite toset of cardinality n. Then, the following conditions hold true:

If f ∈ ℐPO ↑ ( X ) , then, dom ( f ) = im ( f ) and a f = a for any a ∈ dom ( f ) .
∣ ℐPO ↑ ( X ) ∣ = ∣ X ∣ + 1 .

Proof

Write X = { a 1 < a 2 < ⋯ < a n } as a chain and let f ∈ ℐPO ↑ ( X ) . Since dom ( f ) ↑ = dom ( f ) and im ( f ) ↑ = im ( f ) , then the domain and image of f will be one of the following sequences: X 1 = X , … , X i = { a i < a i + 1 < ⋯ < a n } , … , X n − 1 = { a n − 1 < a n } and X n = { a n } . Moreover, since f is an order-embedding and X is totally ordered, a f = a for any a ∈ dom ( f ) . Therefore, dom ( f ) = im ( f ) = X i .
It follows from the first assertion that
ℐPO ↑ ( X ) = 0 , a 1 a 2 … a n a 1 a 2 … a n , … , a n − 1 a n a n − 1 a n , a n a n .

Therefore, ∣ ℐPO ↑ ( X ) ∣ = n + 1 . This completes the proof.□

We derive the following corollary.

Corollary 4.2

Let X be a finite toset. If f ∈ ℐPO ↑ ( X ) and g ∈ PO ↑ ( X ) \ ℐPO ↑ ( X ) , then f g = g f = k , where k satisfies the following conditions:

k ∈ PO ↑ ( X ) \ ℐPO ↑ ( X ) .
dom ( k ) ⊆ dom ( g ) .
a k = a g for any a ∈ dom ( k ) .

Remark 4.3

lf X is a finite toset of cardinality n , then ℐPO ↑ ( X ) ⊊ E ( PO ↑ ( X ) ) . To see this, we know that
ℐPO ↑ ( X ) = { 0 , a 1 a 2 … a n a 1 a 2 … a n , … , a n − 1 a n a n − 1 a n , a n a n } .
Also, It follows from the definition of E ( PO ↑ ( X ) ) , that ℐPO ↑ ( X ) ⊆ E ( PO ↑ ( X ) ) . Take f = a n − 1 a n a n − 1 a n − 1 ∈ PO ↑ ( X ) . Clearly, f ∈ E ( PO ↑ ( X ) ) , but f ∉ ℐPO ↑ ( X ) . This completes the proof.
If X is a finite toset, then for any f ∈ ℐPO ↑ ( X ) , S ( f ) = ∅ .
If X is a finite toset, then any f in ℐPO ↑ ( X ) is regressive.
It is worth noting that not every element in PO ↑ ( X ) is regressive. To see this, take a 13 = 2 3 3 3 in Example 3.7. Clearly, a 13 is not regressive since 2 a 13 = 3 ≰ 2 .

Theorem 4.4

If X is a finite toset, then ℐPO ↑ ( X ) is a right and left unitary in PO ↑ ( X ) .

Proof

We know from Theorem 3.3 that ℐPO ↑ ( X ) is a subpomonoid of the pomonoid PO ↑ ( X ) . According to Corollary 4.2, we have that if g f ∈ ℐPO ↑ ( X ) for any f ∈ ℐPO ↑ ( X ) and any g ∈ PO ↑ ( X ) , then g ∈ ℐPO ↑ ( X ) . Therefore, ℐPO ↑ ( X ) is a right unitary in PO ( X ) . In similar way, ℐPO ↑ ( X ) is a left unitary in PO ( X ) .□

Remark 4.5

The pomonoid ℐPO ↑ ( X ) is not an upper strongly right (or left) pounitary in PO ↑ ( X ) . To see this, let X = { 1 , 2 , 3 , 4 } . Then,
2 3 2 3 ≤ 1 2 3 4 1 2 3 4 2 3 3 3 = 2 3 3 3 .
The pomonoid ℐPO ↑ ( X ) is not a lower strongly right (or left) pounitary in PO ↑ ( X ) . Take, for instance:
1 2 3 4 1 2 3 4 2 3 1 2 = 2 3 1 2 ≤ 2 3 2 3 .

Proposition 4.6

If X is a finite toset, then ℐPO ↑ ( X ) is right reversible.

Proof

As 0 ∈ ℐPO ↑ ( X ) , then for all a ∈ ℐPO ↑ ( X ) , we have 0 ∈ ℐPO ↑ ( X ) a . Therefore, 0 ∈ ℐPO ↑ ( X ) a ≠ ∅ . This completes the proof.□

Remark 4.7

Consider Example 3.7. It follows from Propostion 4.6 that ℐPO ↑ ( X ) is right reversible.

Proposition 4.8

If X is a finite toset, then ℐPO ↑ ( X ) is weakly right reversible.

Proof

The conclusion follows directly, from [21, Theorem 4.6], since the empty map 0 is both a zero and a minimum element.□

5 Conclusion

In this article, we found the suitable partially ordered relation, which guarantees that the partial transformation monoid of a poset X is a pomoind. Also, we described the inverse pomonoid from the partial transformation monoid of a poset X . Then, our attention goes to study some properties when X is a toset.

Acknowledgement

The author would like to thank the reviewers for their careful reading and valuable suggestions.

Funding information: The author declares that there is no funding available for this article.
Conflict of interest: The author states no conflict of interest.
Data availability statement: Not applicable.

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Received: 2023-07-12

Revised: 2023-11-22

Accepted: 2023-11-22

Published Online: 2023-12-16

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

Articles in the same Issue

https://doi.org/10.1515/math-2023-0161

Keywords for this article

posets; pomonoids; partial transformations; inverse pomonoid

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