Banaschewski’s theorem for S-posets: regular injectivity and completeness

Ebrahimi, Mohsen; Mahmoudi, Mojgan; Rasouli, H.

doi:10.1007/s00233-010-9207-4

Cited by 17 publications

(23 citation statements)

References 11 publications

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“…As the two categories Pos and Pos-S have no non-trivial injective objects (see [8]), so by Corollary 3.5 we immediately obtain the following result: Theorem 3.6. The categories Pos/B and Pos-S/B have no non-trivial injective object.…”

Section: Injectivity and Regular Injectivitymentioning

confidence: 89%

“…It is well known that both categories Pos and Pos-S have enough regular injective objects (see [4,8]). In the following, we give a positive answer to the first question above regarding the two categories Pos/B and Pos-S/B.…”

Section: Regular Injectivity and Completenessmentioning

confidence: 99%

“…In [8], it has been shown that, if S is a pogroup, then regular injective object and complete S-poset in Pos-S are the same. By this fact and Lemma 3.4, we deduce: Proof.…”

Section: Regular Injectivity and Completenessmentioning

confidence: 99%

“…Then, in view of Theorem 3.15, and Theorem 4.2 of [8], it is enough to show that S B (f ) is a complete poset. In fact, since by assumption S is a pogroup, it is a regular injective object in the category Pos-S (see [8]). Let {g i : i ∈ I} be an arbitrary subset of S B (f ) (see Remark 3.16).…”

Section: Injectivity In Pos-s/b In Which B Is Endowed With Trivial Acmentioning

confidence: 99%

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Regular Injectivity and Exponentiability in the Slice Categories of Actions of Pomonoids on Posets

Farsad¹,

Madanshekaf²

2015

Journal of the Korean Mathematical Society

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Abstract. For a pomonoid S, let us denote Pos-S the category of S-posets and S-poset maps. In this paper, we consider the slice category Pos-S/B for an S-poset B, and study some categorical ingredients. We first show that there is no non-trivial injective object in Pos-S/B. Then we investigate injective objects with respect to the class of regular monomorphisms in this category and show that Pos-S/B has enough regular injective objects. We also prove that regular injective objects are retracts of exponentiable objects in this category. One of the main aims of the paper is to draw attention to characterizing injectivity in the category Pos-S/B under a particular case where B has trivial action. Among other things, we also prove that the necessary condition for a map (an object) here to be regular injective is being convex and present an example to show that the converse is not true, in general.

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Section: Injectivity and Regular Injectivitymentioning

confidence: 89%

Section: Regular Injectivity and Completenessmentioning

confidence: 99%

“…In [8], it has been shown that, if S is a pogroup, then regular injective object and complete S-poset in Pos-S are the same. By this fact and Lemma 3.4, we deduce: Proof.…”

Section: Regular Injectivity and Completenessmentioning

confidence: 99%

Section: Injectivity In Pos-s/b In Which B Is Endowed With Trivial Acmentioning

confidence: 99%

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Regular Injectivity and Exponentiability in the Slice Categories of Actions of Pomonoids on Posets

Farsad¹,

Madanshekaf²

2015

Journal of the Korean Mathematical Society

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“…That is why researchers study different types of injectivity with respect to different types of embeddings instead of general monomorphisms. For example, in [6,9,12], and [13] injectivity of S-posets with respect to regular monomorphisms (embeddings) has been studied and [6] shows that there are enough regular injective S-posets.…”

mentioning

confidence: 99%

Injectivity of $$S$$ S -posets with respect to down closed regular monomorphisms

Shahbaz

Mahmoudi

2015

Semigroup Forum

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No need to say that the study of injectivity with respect to different classes of monomorphisms is crucial in any category. In this paper, the notion of injectivity with respect to down closed embeddings in the category of S-posets, posets with a monotone action of a pomonoid S on them, is studied. We give a criterion, like the Baer condition for injectivity of modules, or Skornjakov criterion for injectivity of S-sets, for down closed injectivity. Also, we consider such injectivity for S itself, and its (po)ideals. Further, we investigate if this kind of injectivity is preserved or reflected by products, coproducts, direct sums, and order sums of S-posets. Finally, we define and study some kinds of weak injectivity with respect to down closed embeddings.

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Completion of S-posets

Rasouli

2012

Semigroup Forum

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Banaschewski’s theorem for S-posets: regular injectivity and completeness

Cited by 17 publications

References 11 publications

Regular Injectivity and Exponentiability in the Slice Categories of Actions of Pomonoids on Posets

Regular Injectivity and Exponentiability in the Slice Categories of Actions of Pomonoids on Posets

Injectivity of $$S$$ S -posets with respect to down closed regular monomorphisms

Completion of S-posets

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