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Unit and kernel systems in algebraic frames
 Alg. Univ
"... Abstract. One considers the poset of dense, coherent frame quotients of an algebraic frame with the finite intersection property, which are compact. It is shown that there is a smallest such, the frame of delements. However, unless the frame is already compact there is no largest such quotient. Wit ..."
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Cited by 5 (5 self)
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Abstract. One considers the poset of dense, coherent frame quotients of an algebraic frame with the finite intersection property, which are compact. It is shown that there is a smallest such, the frame of delements. However, unless the frame is already compact there is no largest such quotient. With the additional assumption of disjointification on the frame, one then studies the maximal ideal spaces of these quotients and the relationship to covers of compact spaces. Several applications are considered, with considerable attention to the frame quotients defined by extension of ideals of a commutative ring A to a ring extension; this type of frame quotient is considered both with and without an underlying lattice structure on the rings. Our interest in this subject springs from a fascination with the theory of covers in compact spaces, and, in particular, those covers which can be constructed as a structure space. Algebraically, the ideas which will be presented here borrow heavily from the context and language of rings of fractions of a commutative ring with identity. To be more precise, given a commutative ring A with identity, it is the relationship between a multiplicative set S of elements of A which are not zerodivisors and the ideals of A which are disjoint to S which we propose to view more generally. Some of the ideas of this paper have been germinating for a decade. Recently, however, it has become apparent that the appropriate context for this discussion is that of frame theory. In this we follow the modus operandi of the recent articles [MZ03] and [M05], by presenting a general theory in algebraic frames, followed by applications to a number of latticealgebraic situations. We devote some attention to the role of maximal ideals in the interplay of units and kernels. In Sections 5 and 6 the socalled structure space covers are considered. The discussion given here is not the full story of these structure space covers, however. Further atttention will be paid to the subject elsewhere.
Epicompletion in frames with skeletal maps, I: Compact regular frames
"... Abstract. A frame homomorphism h: A − → B is skeletal if x ⊥ ⊥ = 1 in A implies that h(x) ⊥ ⊥ = 1 in B. It is shown that, in KRegS, the category of compact regular frames with skeletal maps, the subcategory SPRegS, consisting of the frames in which every polar is complemented, coincides with the e ..."
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Cited by 4 (4 self)
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Abstract. A frame homomorphism h: A − → B is skeletal if x ⊥ ⊥ = 1 in A implies that h(x) ⊥ ⊥ = 1 in B. It is shown that, in KRegS, the category of compact regular frames with skeletal maps, the subcategory SPRegS, consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in KRegS. Further, SPRegS is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame εA of PA, the boolean algebra of polars of A. This research grew out of an interest in generalizing the related notions of (i) essential extensions of latticeordered groups and (ii) irreducible surjections between compact Hausdorff spaces. As the title suggests, this is the first installment of two articles, dealing with the process of epicompletion in frames; the second, [MZ06b], will be concerned with coherent archimedean frames. The tools, which will be employed here and in [MZ06b], originate in the work on nuclear typings in [MZ06a], and the first steps in the direction of projectable hulls are already taken in [HM06]. The authors are grateful to Professor Bernhard Banaschewski for pointing out the work in [BaP96], which
Dimension in algebraic frames, II: applications to frames of ideals
 in C(X). Submitted. Jorge Martínez & Eric R. Zenk
"... Abstract. This paper continues the investigation into Krullstyle dimensions in algebraic frames. Let L be an algebraic frame. dim(L) is the supremum of the lengths k of sequences p0 < p1 < · · · < pk of (proper) prime elements of L. Recently, Th. Coquand, H. Lombardi and M.F. Roy have formulated ..."
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Cited by 3 (3 self)
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Abstract. This paper continues the investigation into Krullstyle dimensions in algebraic frames. Let L be an algebraic frame. dim(L) is the supremum of the lengths k of sequences p0 < p1 < · · · < pk of (proper) prime elements of L. Recently, Th. Coquand, H. Lombardi and M.F. Roy have formulated a characterization which describes the dimension of L in terms of the dimensions of certain boundary quotients of L. This paper gives a purely frametheoretic proof of this result, at once generalizing it to frames which are not necessarily compact. This result applies to the frame Cz(X) of all zideals of C(X), provided the underlying Tychonoff space X is Lindelöf. If the space X is compact, then it is shown that the dimension of Cz(X) is at most n if and only if X is scattered of CantorBendixson index at most n + 1. If X is the topological union of spaces Xi, then the dimension of Cz(X) is the supremum of the dimensions of the Cz(Xi). This and other results apply to the frame of all dideals Cd(X) of C(X), however, not the characterization in terms
Nuclear typings of frames vs spatial selectors
 Appl. Categ. Struct
"... Abstract. Nuclei which are defined over a class of frames are called nuclear typings. There is the dual notion of a spatial selector, and the relationship between nuclear typings and spatial selectors emanates from the duality between spatial frames and sober spaces. Especially interesting is the in ..."
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Cited by 2 (2 self)
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Abstract. Nuclei which are defined over a class of frames are called nuclear typings. There is the dual notion of a spatial selector, and the relationship between nuclear typings and spatial selectors emanates from the duality between spatial frames and sober spaces. Especially interesting is the interplay between typings that are wellbehaved with respect to certain frame quotients and selectors which similarly behave well in passage to closed sets.
Sublattices generated by polars
 Comm. Algebra
"... Abstract. In an algebraic frame L the complete sublattice CP(L) generated by the polars of L is studied, in comparison with FP(L), the subframe generated by the polars. It is shown, by an example from the theory of ℓgroups, that these are distinct, in general. The relationship between FP(L), CP(L), ..."
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Cited by 1 (0 self)
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Abstract. In an algebraic frame L the complete sublattice CP(L) generated by the polars of L is studied, in comparison with FP(L), the subframe generated by the polars. It is shown, by an example from the theory of ℓgroups, that these are distinct, in general. The relationship between FP(L), CP(L), and other established constructs, closely related to the boolean algebra of polars, is also studied. This investigation into sublattices generated by the polars of a frame began as an effort to generalize to frames the ideas contained in [XZ97] for latticeordered groups. It has evolved into a study of the subframe of regular elements of an algebraic frame, and, more specifically, of the question of when this subframe coincides with the regular coreflection of an algebraic frame. This paper deals with the polars of an algebraic frame, per sé, and with various sublattices that are closely related to the polars. Issues having to do with regularity are discussed in [MZ06b]. Let G be a latticeordered group. In [XZ97] is introduced the complete sublattice Cc(G) of the frame of all convex ℓsubgroups of G, denoted C(G), generated by the polars of G. This clearly includes all the joins of polars, but, as we will show, not every member of Cc(G) is a join of polars. Let P(G) denote the boolean algebra of polars of G, and Cf (G) denote the subframe of C(G) generated by P(G). In general, Cc(G) � = Cf (G), and we give an example (in §2) showing that Cf (G) is, in general, not closed under settheoretic intersection. It is an open question whether Cf (G) is an algebraic lattice. We generalize the constructs described in the preceding paragraph to algebraic frames, although the intuition guiding these ideas comes squarely from latticeordered groups. The first section reviews the necessary background in frame theory. It also compares the new constructs to some closely related sublattices recently studied in the literature (e.g., [MZ03]). The distinction between the various sublattices introduced in §1 is left for §4 and §5; a number of examples is given in the latter, for rings of continuous functions. In §3 the branching of primes is related to suprema of polars. The concluding section lists the main open questions. 2 Jorge Martínez
Epicompletion in Frames with Skeletal Maps, IV: Strongly Joinfit Frames
"... Abstract. Earlier work has shown that there is a monoreflection ψ of the category of compact normal, joinfit frames with skeletal frame maps in the subcategory consisting of strongly projectable frames. This article extends the domain of ψ to the strongly joinfit frames. The saturation nucleus s is ..."
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Abstract. Earlier work has shown that there is a monoreflection ψ of the category of compact normal, joinfit frames with skeletal frame maps in the subcategory consisting of strongly projectable frames. This article extends the domain of ψ to the strongly joinfit frames. The saturation nucleus s is a reflection with respect to weakly closed frame maps, in the subcategory of subfit frames. Moreover, s · ψ = ψ · s, on compact normal, joinfit frames with skeletal, weakly closed frame maps, and s · ψ is an epireflection, but not a monoreflection, in the subcategory of strongly projectable, regular frames, all of which are epicomplete. This article builds on work in [MZ08a, MZ08b, M08b]. It is part of an ongoing investigation into epicompletion in various categories of compact frames with skeletal frame homomorphisms. In [MZ08a] it was shown that the absolute of a compact regular frame represents the functorial epicompletion. The epicomplete objects here are the strongly projectable compact regular frames. The objective, in the long run, being to abstract the work of Conrad in [C71] on essential closures of archimedean ℓgroups, and Carrera’s contribution in [Cr04] on the
Feebly Projectable Algebraic Frames and Multiplicative Filters of Ideals
"... was shown that Reg(3) is equivalent to the more familiar condition known as projectability. In this article we show that there is a nice property, which we call feebly projectable, that is between Reg(3) and Reg(4). In the main section of the article we apply our notions to the frame of multiplicati ..."
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was shown that Reg(3) is equivalent to the more familiar condition known as projectability. In this article we show that there is a nice property, which we call feebly projectable, that is between Reg(3) and Reg(4). In the main section of the article we apply our notions to the frame of multiplicative filters of ideals in a commutative ring with unit and give characterizations of several wellknown classes of commutative rings. Key words algebraic frame · feebly projectable · flatly projectable · multiplicative filter of ideals · clean ring Mathematics Subject Classifications (2000) 06D22 · 13F99 1
© Birkhäuser Verlag Basel/Switzerland 2010 Feebly projectable ℓgroups Algebra Universalis
, 2010
"... Abstract. In the article [17], we introduced and investigated feebly and flatly projectable frames. In this article, we apply these two properties to latticeordered groups. An example is constructed to illustrate that the two properties are distinct, which solves a question from [17]. We also inves ..."
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Abstract. In the article [17], we introduced and investigated feebly and flatly projectable frames. In this article, we apply these two properties to latticeordered groups. An example is constructed to illustrate that the two properties are distinct, which solves a question from [17]. We also investigate these properties with respect to archimedean ℓgroups with weak order unit, as well as commutative semiprime frings. 1.