On equalizer-flat and pullback-flat acts

Normak, Peeter

doi:10.1007/bf02575023

Cited by 29 publications

(14 citation statements)

References 6 publications

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“…Properties (P) and (E) for S-acts are introduced implicitly in [1] and were given their present names in [4]. For S-acts, their significance is that if A S is such that the functor A S ⊗ − preserves pullbacks (resp.…”

Section: Properties (P) and (E) For S-posetsmentioning

confidence: 99%

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Lazard's Theorem forS‐posets

Bulman-Fleming

Laan

2005

Mathematische Nachrichten

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In 1971, inspired by the work of Lazard and Govorov for modules over a ring, Stenström proved that the strongly flat right acts AS over a monoid S (that is, the acts that are directed colimits of finitely generated free acts) are those for which the functor AS ⊗ − (from the category of left S-acts into the category of sets) preserves pullbacks and equalizers. He also provided interpolation-type conditions (now referred to in the literature as Property (P) and Property (E)) characterizing strong flatness. Unlike the situation for modules over a ring, strong flatness is strictly stronger than (mono-) flatness (wherein the functor AS ⊗ − is required only to preserve monomorphisms). The study of flatness properties of partially ordered monoids acting on partially ordered sets was initiated by S. M. Fakhruddin in the 1980s, and has been continued recently in the paper "Indecomposable, projective, and flat S-posets" by Shi, Liu, Wang, and Bulman-Fleming, Comm. Algebra 33, 235-251 (2005). In that paper, a criterion for the equality of elements in a tensor product of S-posets is given and a version of Property (P) is presented that, as in the unordered case, implies flatness and is implied by projectivity. The present paper introduces a corresponding Property (E) and establishes an analogue of the Lazard-Govorov-Stenström theorem in the context of S-posets.

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Section: Properties (P) and (E) For S-posetsmentioning

confidence: 99%

“…Such acts are now usually called either strongly flat or pullback flat (see [3]). Interpolation-type conditions, later dubbed Condition (P) and Condition (E) in [4], were also introduced in [1] which together characterize strong flatness. As for modules, strong flatness of S-acts is strictly weaker than projectivity.…”

Section: Introductionmentioning

confidence: 99%

Lazard's Theorem forS‐posets

Bulman-Fleming

Laan

2005

Mathematische Nachrichten

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“…Normak [13] was the first to consider Condition (P ) on its own. In [14], Stenstrom studied right S-acts which are colimits of finitely generated free acts.…”

Section: Introductionmentioning

confidence: 99%

Characterization of monoids by condition (GP)

Abbasi

Golchin

Saany

2016

Asian-European J. Math.

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“…If A is a right S-set and B is a left S-set the tensor product A ® B has been studied extensively (see, for example, [7] or [2]). We will use the following terminology, which is consistent with that appearing in [14], [10] and [8]. Let E(S) denote the set of idempotents of a monoid S. If x e S the principal right ideal xS is projective if and only if there exists ueE(S) such that xu = x, and whenever xs = xt {s,teS) it follows that us = ut (see [12]).…”

Section: Introductionmentioning

confidence: 99%