Subpullbacks and Po-flatness Properties of S-posets

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Journal of Sciences, Islamic Republic of Iran 25(4): 369 - 377 (2014) http://jsciences.ut.ac.ir

University of Tehran, ISSN 1016-1104

Subpullbacks and Po-flatness Properties of S-posets

A. Golchin

*

_{and L. Nouri}

Department of Mathematics, Faculty of Sciences,University of Sistan and Baluchestan, Zahedan, Islamic Republic of Iran

Received: 16 April 2014 / Revised: 1 November 2014 / Accepted: 19 November 2014

Abstract

In (Golchin A. and Rezaei P., Subpullbacks and flatness properties of S-posets.

Comm. Algebra.

37 : 1995-2007

(

2009)) study was initiated of flatness properties of

right -posets

over a pomonoid that can be described by surjectivity of

corresponding to certain (sub)pullback diagrams and new properties such as

(

)

and

(

)

were discovered. In this article first of all we describe

po-flatness properties of -posets over pomonoids by po-surjectivity of corresponding

to certain subpullback diagrams. Then we introduce three new Conditions

(

),

(

)

and

(

)

and investigate the relation between them and Conditions

( ), ( ), (

), (

)

, (

) and

(

)

. Also we describe these properties

by po-surjectivity of corresponding to certain subpullback diagrams and describe

Conditions ( ),

(

)

and

(

)

by weak po-surjectivity of corresponding

to certain subpullback diagrams.

Keywords: Ordered monoid; -poset; Subpullback diagram.

*_{Corresponding author: Tel: +989153498782; Fax: +985412447166, Email: agdm@math.usb.ac.ir} Introduction

In [5] the notation ( , , , , ) was introduced to denote the pullback diagram of homomorphisms :SM

→ SQ and : SN → SQ in the category of left -acts,

where is a monoid. Tensoring such a diagram by produces a diagram (in the category of sets) that may or may not be a pullback diagram, depending on whether or not the mapping , obtained via the universal property of pullbacks in the category of sets, is bijective. It was shown that, if we require either bijectivity or surjectivity of for pullback diagrams of certain types, we not only recover most of the well-known forms of flatness, but also obtain Conditions

( ) and ( ) as well. In [4] Golchin and Rezaei extended the results from [5] to -posets and two new Conditions ( ) and ( ) were introduced.

Let be a pomonoid. A poset is called a right -poset if acts on in such a way that ( ) the action is monotonic in each of the variables, ( ) for all , ∈

and ∈ , ( ) = ( ) and 1 = . Left -posets are

defined similarly. The notations and SB will often be

used to denote a right or left -poset and Θ = { } is the one-element right -poset. The -poset morphisms are the orderpreserving maps that also preserve the -action. We denote the category of all right -posets, with -poset maps between them, by Pos- . We recall from [3] that an -poset map : → is called an

embedding if ( ) ≤ ( ) implies ≤ , for

, ∈ .

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categories, a

is subcom for every l

: ⟶

there exists that ℎ =

called subpu

of -posets o realized as = where projections. subpullback ( , , , , ( , , , , subcommuta

in the ca mappings we may take

′ = {( ⨂

with a follows from exists a uni such that, in

diagram

mmutative if eft -poset

and : ⟶

a unique hom and ℎ = llback diagra

or the categor

{( , ) ∈

and are th (Notice that

diagram ( , ). Tensorin

, ) by any r ative diagram

ategory of po

⨂ and

⨂ , ′⨂ ) ∈ ( ⨂ ≤ ′⨂

and the res m the definitio

que monoton the diagram

≤ . If

and all

⟶ such th momorphism

, then diag

m for and ry of posets,

× | ( )

he first and se is possib

( ) will b ng the subp right -poset

osets. For the

⨂ in the ca

) × ( ⨂ ( )}

strictions of th on of subpull ne mapping

f ≤

homomorphis

hat ≤

ℎ: ⟶ s

gram ( ) will . In the categ may in fact

≤ ( )}

econd coordin bly empty.) T be denoted

ullback diagr one gets

e subpullback ategory of po

) | ⨂ (

he projections lbacks that th

: ⨂ ⟶ and sms , such l be gory t be nate The by ram the

k of sets ) s. It here ⟶ ′ thi dia ma ( ⨂ (∃ ⨂ ⨂ (∃ ⨂ if suc ( : ′ { ′′ sat ∈ som am

we have s mapping th

agram ( ,

apping in di

( ⨂

for all ∈

we define po correspondi , , , , ) (∀ , ∈ ) ⨂ ( ) ⇒ ∈ )(∃ ⨂ = ⨂ and (∀ , ∈ ) ⨂ ( ) ⇒ ∈ )(∃ ⨂ ≤ ⨂ respectively.

Obviously, s

⇒ weak po-s We say that for all , ′

= ′′ , ′′

ch that ≤ ) if for a

S( ∪ ) ⟶

( ), then th

, }, such that

⨂ ′ ≤ ′⨂

tisfies Condit

∈ , ≤

me ′′ ∈ a

For a pomo mong flatness p

= ⨂

he correspo

, , , ). I

iagram ( ) i

⨂ ( , )) = (

and ( , ) ∈

-surjectivity a ing to the

as

)(∀ ∈SM)(∀

′ ∈SM)(∃ ′ ∈

, ⨂ ≤

)(∀ ∈SM)(∀

′ ∈SM)(∃ ′ ∈

, ⨂ ≤

surjectivity of surjectivity of

an -poset

∈ and ,

≤ ′, for so

′. An -po ll elements , ⟶ SS and a

here exist

( ′) ≤ ( ⨂ in ⨂ S(

ion ( )

′ implies and , ∈ , onoid the properties of

for = 1,2.

onding to the

It can be se is given by

( ⨂ , ⨂

∈ SP.

and weak po-s e subpullba

∀ ∈SN) ⨂

∈SN)( ( ) ≤

⨂ )]

∀ ∈SN) ⨂

∈SN) ( ( ) ≤

⨂ )] ,

f ⇒ po-s f .

satisfies Co

′ ∈ , ≤

ome ′′ ∈

oset satisf

∈ , all hom

all , ′ ∈ ,

′′ ∈ , ,

′) and ⨂

( ∪ ). A

if for all

= ′′ , ′

such that ≤

following re an -poset.

We shall call

e subpullback

een that the

)

surjectivity of ack diagram

( ) ≤

≤ ( ),

( ) ≤

≤ ( ),

urjectivity of

ondition ( )

≤ ′ ′ implies

and , ∈ , fies Condition momorphisms , if ( ) ≤

∈ , ′, ′ ∈

= ⨂ ,

n -poset

, ′ ∈ and

′′ ≤ ′, for

≤ .

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Subpullbacks and Po-flatness Properties of S-posets

free ⟹projective

⟹ ( ) ⟹ ( ) ⟹ ( )

⇓ ⇓ ⇓

( ) ⟹ ( ) ⟹ ( )

⇓ ⇓ ⇓

( ) ⟹ ( ) ⟹ ( )

⇓ ⇓ ⇓

− ⟹ w. po − f. ⟹ . . − . ⟹ −

. .

⇓ ⇓ ⇓ ( ) ⟹ w. f. ⟹ . . . ⟹ . .

( )

In this article for a pomonoid , using the information from [5, 4] we describe po-flatness and Conditions ( ), ( ) and ( ) of -posets according to po-surjectivity of corresponding to certain subpullback diagrams. We also describe Conditions ( ), ( ) and ( ) introduced in [7, 4] by weak po-surjectivity of corresponding to certain subpullback diagrams.

Results

1. Po-flatness of -posets

In this section for a pomonoid we give equivalences of po-flatness of −posets by po-surjectivity of the mapping corresponding to certain subpullback diagrams.

Recall from [7] that an -poset is called po-flat

(in Pos- ) if and only if, for all embeddingsSB ⟶SC in

the category of left -posets, the induced order-preserving map ⨂ SB⟶ ⨂ SC is embedding, that

is, ⨂ ≤ ′⨂ ′ in ⨂SC implies ⨂ ≤ ′⨂ ′

in ⨂ SB. The -poset is called (principally) weakly

po-flat if the functor ⨂ − preserves embeddings of (principal) left ideals of monoid into , that is, for all (principal) left ideals of monoid , , ′ ∈ , , ′ ∈

, ⨂ ≤ ′⨂ ′ in ⨂ SS implies ⨂ ≤ ′⨂ ′ in

⨂ SI. For more po-flatness properties we refer the

reader to [8].

Lemma 1.1. [7, Lemma 3.5] An -poset is po-flat if and only if for every left -poset SB, any , ′ ∈

and any , ′ ∈ SB, ⨂ ≤ ′⨂ ′ in ⨂SB implies

⨂ ≤ ′⨂ ′ in ⨂ S(Sb ∪ ′).

Theorem 1.2. An -poset is po-flat if and only if the corresponding is po-surjective for every subpullback diagram ( , , , , ), where :SM ⟶SQ

is an embedding of left -posets.

Proof. Suppose is po-flat, : SM ⟶ SQ is an

embedding of left -posets and let ⨂ ( ) ≤ ⨂

( ′) in ⨂ SQ, for , ′ ∈ and , ′ ∈SM. Since

is po-flat, ⨂ is embedding, and so ⨂ ≤

′⨂ ′ in ⨂SM. Since ( ) ≤ ( ), ⨂ = ⨂

and ⨂ ≤ ′⨂ ', the corresponding is

po-surjective for the subpullback diagram ( , , , , ).

Conversely, suppose the corresponding is po-surjective for every subpullback diagram

( , , , , ), where :SM ⟶SQ is an embedding of

left -posets. LetSM be an -poset, , ′ ∈ , , ′ ∈ SM and ⨂ ≤ ′⨂ ′ in ⨂ SM. If : S(Sm ∪ ′)

⟶ SM is the inclusion map, then is an embedding

(since S(Sm ∪ ′) is a -subposet of SM). Also

⨂ ( ) = ⨂ ≤ ′⨂ ′= ⨂ ( ′). By

po-surjectivity of for the subpullback diagram ( ∪

, ∪ ′, , , ), there exist " ∈ and , ′ ∈

S(Sm ∪ ′) such that ( ) ≤ ( ′), ⨂ = "⨂ and

" ⨂ ′ ≤ ′⨂ ′ in ⨂S(Sm ∪ ′). Thus ≤ ′, and

so ⨂ = " ⨂ ≤ "⨂ ′ ≤ ⨂ ′ in ⨂ S(Sm

∪ ′). That is, is po-flat as required.

Similarly, we can prove the following theorems. Theorem 1.3. An -poset is weakly po-flat if and only if the corresponding is po-surjective for every subpullback diagram ( , , , , ), where is a left ideal of and :SI⟶SS is an embedding of left -posets.

Theorem 1.4. An -poset is principally weakly flat if and only if the corresponding is po-surjective for every subpullback diagram ( , , , , ), where ∈ and :S(Ss)⟶SS is an embedding of left

-posets.

We recall from [1] that an element of a pomonoid is called right po-cancellable if ≤ ′ implies

≤ ′, for all , ′ ∈ . An -poset is called

po-torsion free if ≤ ′ implies ≤ ′, whenever

, ′ ∈ and is a right po-cancellable element of .

Theorem 1.5. An -poset is po-torsion free if and only if the corresponding is po-surjective for every subpullback diagram ( , , , , ), where : SS⟶SS is

an embedding of left -posets.

Proof. Suppose is po-torsion free and let : SS⟶ SS be an embedding of left -posets. Then (1) is a right

po-cancellable element of , for if (1)≤ ′ (1), for

, ′ ∈ , then ( ) ≤ ( ′), and so ≤ ′. Suppose

⊗ ( ) ≤ ′ ⊗ ( ) in ⊗ SS, for , ′ ∈ and

, ∈ . Then ( ) ≤ ( ), and so (1) ≤

′ ′ (1). Since (1) is a right po-cancellable element of

and is po-torsion free, we have ≤ ′ ′,and so

⊗ ≤ ⊗ in ⊗ SS. Since ( ) ≤ ( ),

⊗ = ⊗ and ⊗ ≤ ⊗ , is po-surjective

for the subpullback diagram ( , , , , ).

Conversely, suppose the corresponding is po-surjective for every subpullback diagram ( , , , , ) where :SS⟶SS is an embedding of left -posets. Let

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, for , ′ ∈ . If :SS⟶SS is defined by ( ) =

, for every ∈ , then is an embedding, also

⊗ (1) = ⊗ ≤ ′ ⊗ = ′ ⊗ (1) in ⊗SS.

By po-surjectivity of for the subpullback diagram

( , , , , ), there exist " ∈ and , ∈ , such

that ( ) ≤ ( ), ⊗ 1 = " ⊗ and " ⊗ ≤ ′ ⊗ 1 in ⊗SS. Thus ≤ , and so ≤ . Also, =

" and " ≤ ′. Hence = " ≤ " ≤ ′, and so

is po-torsion free.

2. -Posets satisfying Condition (( )) ( )

In this section we give some equivalent conditions on an -poset satisfying Condition (( )) ( ) according to (weak) po-surjectivity of corresponding to certain subpullback diagrams. Note that for -posets

and SB, ⊗ ≤ ⊗ in ⨂SB, for , ′ ∈

and , ′ ∈ SB if and only if there exist , , … , ∈

, , … , ∈SB, , , … , , ∈ , such that

≤

≤ ≤ ... ...

≤ ≤ .

Lemma 2.1. An -poset satisfies Condition ( )

if and only if for all SB and all , ∈ , , ∈ SB,

⊗ ≤ ′ ⊗ ′ in ⨂ SB implies the existence of

" ∈ and , , such that = " , " ≤ ′ and

≤ .

Proof. Suppose satisfies Condition ( ) and let

⊗ ≤ ′ ⊗ ′ in ⨂ SB, for , ∈ and , ∈

SB. Then there exist , , … , ∈ , ,… , ∈SB,

, , … , , ∈ , such that

≤

≤ ≤ ... ...

≤ ≤ .

Since ≤ and satisfies Condition ( ), there exist ∈ and , ′ ∈ , such that = ,

′ ≤ and ≤ . Then

( ) = ( ) ≤ ( ) = ( ) ≤

( ) = ( ) ≤ ( ),

and so we have the following -tossing of length

– 1

≤ ( ′ )

( ) ≤ ( ) ≤ ( ) ... ...

≤ ≤ .

Continuing this procedure we have:

≤

≤ ≤ ′.

Again ≤ implies the existence of ∈ and

, ∈ , such that = , ≤ and ≤

. Since ≤ and ≤ , we have

( ) ≤ ′, and so there exist " ∈ and , ∈

, such that = " , ′′ ≤ ′ and ( ) ≤ .

If = then = = "( ) = " ,

" ≤ ′ and

= ( ) ≤ ( ) = ( ) ≤

( ) ≤ .

The converse is obvious.

Theorem 2.2. For an -poset the following assertions are equivalent:

(1) the corresponding is po-surjective for every subpullback diagram ( , , , , );

(3) the corresponding is po-surjective for every subpullback diagram ( , , , , ), where is a left ideal of ;

(4) the corresponding is po-surjective for every subpullback diagram ( , , , , ), ∈ ;

(7) satisfies Condition ( ).

Proof. Implications (1) ⟹(2) ⟹(3) ⟹ (4)⟹ (5) and (2)⟹(6) are obvious.

(5) ⟹(7). Suppose ≤ ′ ′, for , ′ ∈ and

, ′ ∈ . Then ⊗ ≤ ′ ⊗ ′. If ( ) = and

( ) = ′, for ∈ , then ⊗ (1) = ⊗

≤ ′ ⊗ ′ = ′ ⊗ (1). By assumption there exist

" ∈ and , ∈ , such that ( ) ≤ ( ), ⊗ 1 =

" ⊗ and " ⊗ ≤ ′ ⊗ 1. Thus ≤ ′, = "

and " ≤ ′, and so satisfies Condition ( ) as

required.

(6) ⟹ (7). Suppose ≤ , for , ∈ ,

, ∈ and let SF = × { , }, where ( , ) =

( , ) and ( , ) ≤ ( , ) if and only if s ≤ and =

, for , ∈ and , ∈ { , }. Define : SF ⟶ , as

(1, ) = and (1, ) = . Since ≤ , we have

⊗ ≤ ⊗ , and so ⊗ (1, ) = ⊗ ≤ ⊗

= ⊗ (1, ). By assumption there exist " ∈

and ( , ), ( , ) ∈ SF, such that ( , ) ≤ ( , ),

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Then ⊗ (1, ) = " ⊗ ( , ) implies that there exist

, … , , , … , ∈ , , , … , , , , , … , ,

∈ and ( , ), . . . , ( , ), ( , ), …,

( , ) ∈ SF, such that

≤

≤ (1, ) ≤ ( , )

≤ ( , ) ≤ ( , ) ... ...

≤ " ( , ) ≤ ( , ),

" ≤

≤ ( , ) ≤ ( , )

... ...

≤ ( , ) ≤ (1, ).

Thus = = . . . = = , and so = . Also

= = . . . = = , and so ⊗ 1 = " ⊗ , that is,

= " . Similarly, " ⊗ ( , ) ≤ ′ ⊗ (1, ) implies

that = and " ≤ ′. Since ( , ) ≤ ( , ), we have ≤ ′, and so satisfies Condition ( ) as required.

(7) ⟹ (1). Suppose ⊗ ( ) ≤ ⊗ ( ), for

, ∈ , ∈ SM and ∈ SN. By Lemma 2.1 there

exist " ∈ and , ∈ such that = " , " ≤ ′ and ( ) ≤ ( ). Thus ( ) ≤ ( ). If

′ = and = , then ( ) ≤ ( ), ⊗ =

⊗ = ⊗ = ⊗ ′ and ⊗ ′ =

⊗ = ⊗ ≤ ′ ⊗ . Thus the

corresponding is po-surjective for every subpullback diagram ( , , , , ) as required.

Recall from [7] that an -poset satisfies Condition

( ), if ≤ , for , ∈ and , ′ ∈ implies that there exist " ∈ and , ∈ , such that ≤

, ′′ ≤ ′ and ≤ . Using an argument similar

to that of the proof of Lemma 2.1 we have the following lemma.

Lemma 2.3. An -poset satisfies Condition ( )

if and only if for all SB and all , ∈ , , ′ ∈ SB,

⊗ ≤ ⊗ in ⊗SB implies the existence of

∈ and , ∈ , such that ≤ , ≤

and ≤ .

Using Lemma 2.3 and an argument similar to that of the proof of Theorem 2.2 we have the following theorem.

(1) the corresponding is weakly po-surjective for every subpullback diagram ( , , , , );

(3) the corresponding is weakly po-surjective for

every subpullback diagram ( , , , , ), where is a left ideal of ;

(4) the corresponding is weakly po-surjective for every subpullback diagram ( , , , , ), ∈ ;

(7) satisfies Condition ( ).

3. -posets satisfying Condition (( ) ) ( )

In this section first of all we give some equivalent conditions on an -poset satisfying Condition (( ) ) ( ) according to (weak) po-surjectivity of corresponding to certain subpullback diagram. Then we give a necessary and a sufficient condition for an -poset to satisfy Condition ( ) .

Theorem 3.1. An -poset satisfies Condition

( ) if and only if the corresponding is po-surjective for every subpullback diagram ( , , , , ),

where is a left ideal of .

Proof. Suppose satisfies Condition ( ) , ∶

SI ⟶SS be an -poset morphism and let ⊗ ( ) ≤

⊗ ( ) in ⊗SS, for , ∈ and , ∈ . Then,

( ) ≤ ( ). If = ∪ ⊆ and ℎ = | then

ℎ( ) ≤ ′ℎ( ), and so by assumption there exist

" ∈ , , ∈ , and , ′ ∈ { , }, such that

ℎ( ) ≤ ℎ( ), ⊗ = ⊗ and ⊗ ≤

⊗ in ⊗ SJ. Clearly, , ∈ , ( ′) =

ℎ( ′) ≤ ℎ( ′) = ( ′) and ⊆ implies that

⊗ = ⊗ and ⊗ ≤ ⊗ in ⊗

SI. Thus the corresponding is po-surjective for every

subpullback diagram ( , , , , ).

Conversely, suppose the corresponding is po-surjective for every subpullback diagram ( , , , , ),

:S(Ss∪ ) ⟶SS is an -poset morphism, for , ∈

and let ( ) ≤ ′ ( ), for , ′ ∈ . Then ⊗

( ) ≤ ′ ⊗ ( ) in ⊗ SS. By po-surjectivity of

for the subpullback diagram ( ∪ , ∪

, , , ), there exist " ∈ , , ∈ and ′, ′ ∈

{ , }, such that ( ) ≤ ( ), ⊗ = " ⊗ ′

and " ⊗ ′ ≤ ′ ⊗ in ⊗ S(Ss∪ ). Thus

satisfies Condition ( ) as required.

An -poset satisfies Condition ( ) if for all

, ∈ , all homomorphisms :S(Ss∪ ) ⟶ SS and all

, ∈ , if ( ) ≤ ′ ( ), then there exist

" ∈ , , ∈ , ′, ′ ∈ { , } such that ( ′) ≤

( ′) and ⊗ ≤ " ⊗ ′, " ⊗ ′ ≤ ′ ⊗ in

⊗S(Ss∪ ) (see [4]). Using an argument similar to

that of the proof of Theorem 3.l we can show the following theorem.

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po-surjective for every subpullback diagram ( , , , , ), where is a left ideal of .

( ) if and only if for all , ∈ , all homomorphisms : S(Ss∪ )⟶SS and all , ′ ∈ ,

if ( ) ≤ ′ ( ), then there exist ", , ,

, , ∈ , , , , , , , ′ , ′ , ′ , ′ ,

′ , ′ ∈ , such that either ( ) ≤ ( ) and

" ≤ ′,

=

≤ ≤

= " ≤ ,

" = ′ ′

′ ′ ≤ ′ ′ ′ ≤ ′

′ ′ = ′ ′ ′ ≤ ′

′ ′ ≤ ′ ≤ ′ ,

or ( ) ≤ ( ) and

= " ,

" ≤

= ≤

≤ ′ ≤ ,

or ( ) ≤ ( ) and

=

≤ " ≤ ,

" = ′ ′

′ ′ ≤ ′ ≤ ′ ,

" =

≤ ′ ≤ .

Proof. Necessity. Let ∶S(Ss∪ ) ⟶ be a

homomorphism, for , ∈ and ( ) ≤ ( ), for

, ∈ . By assumption there exist " ∈ , , ∈

and , ∈ { , }, such that ( ) ≤ ( ), ⊗ =

" ⊗ and ′′⊗ ≤ ′⊗ . Then ⊗ = " ⊗

and a" ⊗ ′ ≤ ′ ⊗ , and so there

exist , … , , , … , , , … , ∈ ,

, , … , , , , , … , , , , , …,

, ∈ and , … , , , … , , ′ , …, ′

∈ { , }, such that

≤

≤ ≤

... ...

≤ ≤ ,

≤

≤ ≤

... ...

≤ ≤ ,

≤

≤ ≤

... ...

≤ ≤ .

If = , = ′, = ′, = , = and

= , then there are three cases as follows: Case 1. ′ = . In this case there exist ∈ {1,..., }

and ∈ {1, … , ′}, such that = , =

= … = = = , = , = =

⋯ = = = . Then ( ) ≤ ( ) =

( ) ≤ ( ) ≤ ⋯ ≤

( ) = ( ) ≤ ( ) =

( ) ≤ ( ) = ( ) ≤ ( ) =

( ) ≤ ( ) ≤ ⋯ ≤ ( ) =

( ) ≤ ( ).

Since satisfies Condition ( ) , there exist

∈ and , ∈ , such that = ,

≤ and ( ) ≤ ( ). Since also

≤ ≤ ≤ ≤ ⋯ ≤

=

there exist ∈ and , ∈ , such that

= , ≤ and ≤ . Thus ( ) ≤

( ) and

≤ ′,

=

≤ ≤

= ≤ .

Also,

= ≤ ≤ ≤ ⋯

≤ ≤ ≤

≤ ≤ ≤ ⋯ ≤

=

and since satisfies Condition ( ) , there exist ′ ∈ and ′ , ′ ∈ , such that = ′ ′ ,

′ ′ ≤ and ′ ≤ ′ . On the other hand,

≤ =

≤

≤ ≤ ⋯ ≤

≤ ≤

and so there exist ′ ∈ and ′ , ′ ∈ , such that

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Thus

= ′ ′

′ ′ ≤ ′ ≤ ′

= ′ ′ ≤

′ ′ ≤ ′ ≤ ′ ,

as required.

Case 2. = . In this case there exists ′ ∈ {1,..., ′}, such that ′ = and ′ = ′ =

⋯ = ′ = . Thus

( ) ≤ ( ) = ( ) ≤ ( ) ≤ ⋯

≤ ( ) = ( )

≤ ( ) = ( ) ≤ ( )

= ( ) ≤ ( )

= ( ) ≤ ⋯

≤ ( )

Since satisfies Condition ( ) , there exist

∈ and , ′ ∈ , such that = , ′ ≤ ′ ′

and ( ) ≤ ′ ( ). Since also

′ ′ ≤ ′ ′ = ′ ′ ′ ≤

′ ′ ′ ≤ ⋯ ≤ ′ ′ ′ ≤ ′ ,

there exist ′ ∈ and , ′ ∈ , such that

′ ′ = ′ , ′ ′ ≤ ′ and ≤ ′ . Thus

( ) ≤ ( ) and

= ,

≤

= ≤

′ ′ ≤ ≤ ′ ,

as required.

Case 3. ′ = and ′ = . Then ( ) ≤ ( ),

⊗ = " ⊗ and " ⨂ ≤ ′ ⊗ . Since

= " and satisfies Condition ( ) , there

exist , ∈ and , , , ∈ , such that

=

≤ " ≤ ,

" =

≤ ≤ .

Since " ≤ ′ , there exist ∈ and , ∈ , such that

" =

≤ ≤ ,

Sufficiency. Suppose : S(Ss∪ ) ⟶ is a

homomorphism, for , ∈ and let ( ) ≤ ( ), for

, ′ ∈ . By assumption there are three cases as

follows:

Case 1. ( ) ≤ ( ) and

" ≤ ′, =

≤ ≤

= " ≤ ,

" = ′ ′

′ ′ ≤ ′ ′ ′ ≤ ′

= ≤

≤ ≤ .

Thus

⊗ = ⊗ = ⊗ ≤ ⊗

= ⊗ ≤ ⨂ = ⨂

≤ ⨂ = ⨂ = " ⨂

= "⨂

and

⨂ = ⨂ = ⨂ = ⨂ ≤

⨂ = ⨂ ≤ ⨂ = ⨂ ≤ ⊗

= ⊗ = ⊗ = ′ ⊗ ′ ≤

′ ⨂ ′ = ′ ′ ⨂ ≤ ⨂ ,

and hence ⨂ = "⨂ . Since " ≤ ′, we have

" ⨂ ≤ ′⨂ , and so "⨂ ≤ ′⨂ . Thus

satisfies Condition ( ) as required.

Case 2. ( ) ≤ ( ) and = " ,

" ≤

= ≤

≤ ≤ .

Thus ⨂ = " ⊗ = " ⊗ and

⨂ = ⨂ ≤ ⊗ = ⨂ ≤

⨂ = ⨂ = ⨂ = ⨂ ≤ ⨂ =

⨂ ≤ ⨂ ,

as required.

Case 3. ( ) ≤ ( ) and =

≤ " ≤ ,

" = ′ ′

′ ′ ≤ ′ ≤ ′ ,

" =

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Thus

⊗ = ⊗ = ⊗ ≤ ⊗

= ⊗ ≤ ⊗ = ′′ ⊗

and

⨂ = ⨂ = ⨂ = ⨂ ≤

⨂ = ⨂ ≤ ⨂ ,

and so ⨂ = ⨂ . Also,

⨂ = ⨂ = ⨂ = ⨂ ≤

⨂ = ⨂ ≤ ′⨂ ,

as required.

4. -Posets satisfying Condition (( ) ) ( )

In this section we give some equivalent conditions on an -poset satisfying Condition (( ) )

( ) according to (weak) po-surjectivity of corresponding to certain subpullback diagrams.

(1) the corresponding is po-surjective for every subpullback diagram ( , , , , ), where ∈ ;

(3) satisfies Condition ( ) . Proof. Implication (1) ⟹ (2) is clear.

(2) ⟹ (3). Suppose ≤ ′ , for , ′ ∈ and

∈ . Then, ⊗ ≤ ′ ⊗ in ⊗SS. If : SS⟶SS

is defined as ( )= , for ∈ , then ⊗ ( 1) =

⊗ ≤ ′ ⊗ = ′ ⊗ (1). By po-surjectivity of

for the subpullback diagram ( , , , , ), there exist ′′ ∈ and , ′ ∈ SS, such that ( ) ≤ ( ′),

⊗ 1 = " ⊗ and " ⊗ ′ ≤ ′ ⊗ 1 in ⊗ SS.

Thus ≤ ′ , = " and " ′ ≤ ′, and so satisfies Condition ( ) as required.

(3) ⟹ (1). Suppose ∶ SS ⟶ SS is an -poset

morphism and let ⊗ ( ) ≤ ′ ⊗ ( ′ ) in ⊗SS,

for , ′ ∈ and , , ′ ∈ . Then ( ) ≤ ′ ( ′ ),

and so ( ) ≤ ′ ′ ( ). Since satisfies Condition ( ) there exist " ∈ and , ∈ , such that = " , " ≤ ′ ′ and ( ) ≤ ( ). Thus ( ) ≤ ( ). If = and = , then

⊗ = ⊗ = ′′ ⊗ = " ⊗ = " ⊗

,

and

" ⊗ ′ = " ⊗ = " ⊗ ≤ ′ ′ ⊗ = ′

⊗ ,

and so the corresponding is po-surjective for the subpullback diagram ( , , , , ).

An -poset satisfies Condition ( ) , if

≤ ′, for , ′ ∈ and ∈ , implies that there exist " ∈ and , ∈ , such that ≤ " , " ≤ ′

and ≤ (see [4]). Using an argument similar to that of the proof of Theorem 4.1 we can prove the following theorem.

(1) the corresponding is weakly po-surjective for every subpullback diagram ( , , , , ), where

∈ ;

(3) satisfies Condition ( ) .

5. Relations of the properties

In this section we demonstrate that with one possible exception (( ) ⟹ po-flat), all of the implications in Figure ( ) are strict.

The proof of the following theorem is clear. Theorem 5.1. Let be an ordered group. Then all -posets satisfy Condition ( ).

Theorem 5.2. [4, Theorem 4.6.] For any pomonoid

, there exists an -poset that does not satisfy Condition

( ).

Recall from [7], [8], [1], [4], that an -poset satisfies Condition ( ), if ≤ ′ , for , ′ ∈ and

, ∈ , implies that there exist " ∈ and , ∈ , such that = " , " = ′ and ≤ . An -poset is called flat (in SPOS) if and only if, for all embeddings SB ⟶SC in the category of left -posets,

the induced order-preserving map ⊗ SB ⟶ ⊗SC

is injective. An -poset is called weaklyflat if the induced morphism ⊗ ⟶ ⊗ is injective for all embeddings of left ideals into SS. A pomonoid is

called weaklyright reversible in case ∩ ( ] ≠ ∅, for all , ∈ (if is a subset of a poset , ( ]: = { ∈

∶ ≤ for some ∈ } is the down-set of , or the

order ideal generated by ). An -poset satisfies

Condition ( ) if the corresponding is surjective for every subpullback diagram ( , , , , ), where is a left ideal of .

Using [4, Theorem 6.2.] the following theorem is clear.

Theorem 5.3. For a pomonoid the following assertions are equivalent:

(1) Θ satisfies Condition ( ); (2) Θ satisfies Condition ( ); (3) Θ satisfies Condition ( ); (4) Θ is po-flat;

(5) Θ is flat;

(9)

(9) Θ is w (10) Θ is (11) is w

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