The Laterally σ-Complete Reflection of an Archimedean Lattice-Ordered Group

Hager, Anthony W.; Martínez, Jorge

doi:10.1007/978-94-011-5640-0_9

Cited by 19 publications

(29 citation statements)

References 11 publications

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“…As is pointed out in the similar development in [HM99a], eA may be replaced by any laterally α-complete -group B in which A is essentially embedded. For the proof of Corollary 2.1.3, B also has to be a subring of eA, an assumption which is omitted in [HM99a, Corollary 2.4].…”

Section: Remarks 214ºmentioning

confidence: 93%

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On α-regular archimedean f-rings

Hager

Martínez

2011

Mathematica Slovaca

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ABSTRACT. For a commutative ring A with identity, and for infinite cardinals α as well as the symbol ∞, which indicates the situation in which there are no cardinal restrictions, one defines A to be α-regular if for each subset D of A, with |D| < α and de = 0, forThis paper studies α-regular archimedean f -rings, relative to lateral α-completeness. The main result is that the operator l(α) that gives the lateral α-completion commutes with b, the reflection that closes an f -ring with respect to bounded inversion. An f -ring is α-regular if and only if it has bounded inversion and is laterally α-complete, and the operator that creates the α-regular hull isIt is shown that the space m r(α)A of all maximal -ideals of r(α)A is the same as that of the α-projectable hull. Finally, r(α)A contains the ring of α-quotients, and necessary conditions are given for them to coincide.

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Section: Remarks 214ºmentioning

confidence: 93%

“…Corollary 2.4 of [HM99a] then guarantees that σA is again an f -ring. The reader should note, however, that in this context neither βA, nor σA, nor the composite of Theorem 2.2.3 below are generally essential extensions of A; what makes the theorem work anyway is the fact that βA is an epimorphic extension of A.…”

Section: ò ø óò ² ê ñ ö × 222ºmentioning

confidence: 95%

On α-regular archimedean f-rings

Hager

Martínez

2011

Mathematica Slovaca

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“…(See [18].) Indeed, as is shown in [13], if Y is zero-dimensional, then G is local if and only if for each clopen set U of Y, and each g G, the element of D(Y), which agrees with g on U, and is 0 on the complement, belongs to G (Remark 2.3(c), [13]). It is shown also in Theorem 2.2 of that article that if G is projectable, then it is local.…”

Section: Projectable Singular L-groups and Modulesmentioning

confidence: 99%

Singular Archimedean lattice-ordered groups

Hager¹,

Martínez²

1998

Algebra Universalis

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W stands for the category of all archimedean l-groups with designated weak unit. The subcategory W s of all groups with singular weak unit is analyzed as a full subcategory of W which is both epireflective and monocoreflective. A general technique for ''contracting'' monoreflections of a category A to a monocoreflective subcategory B is developed and then applied to W s to show that: (i) the projectable hull in W s is a monoreflection; (ii) essential hulls in W s are formed by simply taking the lateral completion, and G is essentially closed in this category if and only if G= D(X, Z), where X is compact, Hausdorff and extremally disconnected; (iii) the maximum monoreflection on W s , denoted i s , is obtained by contracting the maximum monoreflection i on W, and G is epicomplete in W s precisely when G is laterally |-complete; (iv) the maximum essential reflection on W s , denoted m s , is the contraction of the maximum essential reflection m on W.

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“…The relatively uniform completion (ru-completion, for short) of an archimedean lattice ordered group has been investigated by Hager and Martinez [8], Ball and Hager [1], Martinez [12],Černák and Lihová [5], and Jakubík andČernák [11]. The ru-completion of an archimedean lattice ordered group G is denoted by G ω 1 .…”

Section: Introductionmentioning

confidence: 99%