OAL09

Tutorials	There will be four tutorials, composed of three lectures each, given by R. N. Ball (University of Denver (CO), USA) Probable Topics: Let κ be a regular cardinal or the symbol ∞. A κ-set is a set of cardinality strictly less than κ, and a κ-frame is a bounded lattice in which κ-sets have joins which commute with binary meets, and κ-morphisms respect κ-joins, binary meets, and top and bottom. Thus an ω₁-frame is what is usually termed a σ-frame, and an ∞-frame is a what is usually termed a frame. A κ-frame is regular if each of its elements is the join of a κ-set of elements well below it. Regular κ-frames are moncoreflective in regular λ-frames for $κ ≤ λ$ , so that a regular frame may be viewed as the union of these coreflections. Furthermore, these coreflections carry a great deal of information about the frame containing them. For example, a lot of information about a completely regular frame is carried by the smallest of these coreflections, the cozero of the frame. What is interesting and important is that membership in the cozero part can be determined by the intrinsic properties of the element itself, rather than by reference to its containment in some otherwise unspecified regular ω₁-subframe; this is Johnstone's notion of a scale. We will show that this remains true for any κ. An interesting notion here is that of a κ-briar, a particularly prickly poset, and of the κ-briar patch, the free frame generated by the κ-briar. This serves as the free regular κ-frame over a single generator, and is the κ-counterpart of a scale. Aside from the obvious cardinal generalizations of the theory of completely regular frames and their cozero parts, there arises from these ideas a number of useful invariants for the study of frames. One of the many apparent consequences of this generalization is the κ-Lindelöf coreflection of a frame, a construct which sheds light on the structure of the assembly NL of nuclei on a frame L. B. Banaschewski (McMaster University, Canada) Probable Topics: The notion of an essential extension will be considered in the full subcategories CRFrm, C, and K of the category Frm of all frames, given, respectively, by: the complete regular frames, the completely regular frames complete in their real uniformity, and the compact completely regular frames. Specifically, the essentially complete objects will be characterized and the existence of unique (up to isomorphism) essential completions will be established for each of these categories. W. Wm. McGovern (Bowling Green State (OH), USA) Probable topics: With the study of algebraic frames hitting a new high in the literature, we aim to survey the recent results with an eye towards the applications of the theory to other algebraic arenas. In particular, we hope to motivate the discussion through the use of several well-studied examples: the frame of radical ideals of commutative ring with identity; the frame of convex l-subgroups of a lattice-ordered group; the frame of multiplicative filters and noetherian filters of a commutative ring with identity; the frame of z-ideals of C(X), etc. A. Pultr (Charles University, Czech Republic) Probable Topics: (i) Sublocales (As surjective homomorphisms, congruences, nuclei); sublocales vs. subspaces in the spatial case. (ii) Enrichment of the localic structure; uniformity and nearness; uniform sublocales; completion. Advantages of the point-free approach. (iii) Approximation and special filters. Approximate maps. General remarks on filters apt to represent something like a point (prime and completely prime filters, shaving filters, Cauchy filters),
Featured Lectures	The following have agreed to give a talk during the workshop: F. Dashiell (UCLA, USA) A. W. Hager (Wesleyan University (CT), USA) P. Jipsen (Chapman University (CA), USA) P. T. Johnstone (Cambridge University, UK) J. J. Madden (Louisiana State (LA), USA) C. J. Mulvey (University of Sussex, UK) J. Picado (University of Coimbra, Portugal)

Tutorials

There will be four tutorials, composed of three lectures each, given by

R. N. Ball (University of Denver (CO), USA)

Probable Topics:
Let κ be a regular cardinal or the symbol ∞. A κ-set is a set of cardinality strictly less than κ, and a κ-frame is a bounded lattice in which κ-sets have joins which commute with binary meets, and κ-morphisms respect κ-joins, binary meets, and top and bottom. Thus an ω₁-frame is what is usually termed a σ-frame, and an ∞-frame is a what is usually termed a frame. A κ-frame is regular if each of its elements is the join of a κ-set of elements well below it.

Regular κ-frames are moncoreflective in regular λ-frames for $κ ≤ λ$ , so that a regular frame may be viewed as the union of these coreflections. Furthermore, these coreflections carry a great deal of information about the frame containing them. For example, a lot of information about a completely regular frame is carried by the smallest of these coreflections, the cozero of the frame. What is interesting and important is that membership in the cozero part can be determined by the intrinsic properties of the element itself, rather than by reference to its containment in some otherwise unspecified regular ω₁-subframe; this is Johnstone's notion of a scale. We will show that this remains true for any κ. An interesting notion here is that of a κ-briar, a particularly prickly poset, and of the κ-briar patch, the free frame generated by the κ-briar. This serves as the free regular κ-frame over a single generator, and is the κ-counterpart of a scale. Aside from the obvious cardinal generalizations of the theory of completely regular frames and their cozero parts, there arises from these ideas a number of useful invariants for the study of frames. One of the many apparent consequences of this generalization is the κ-Lindelöf coreflection of a frame, a construct which sheds light on the structure of the assembly NL of nuclei on a frame L.
B. Banaschewski (McMaster University, Canada)

Probable Topics:
The notion of an essential extension will be considered in the full subcategories CRFrm, C, and K of the category Frm of all frames, given, respectively, by: the complete regular frames, the completely regular frames complete in their real uniformity, and the compact completely regular frames. Specifically, the essentially complete objects will be characterized and the existence of unique (up to isomorphism) essential completions will be established for each of these categories.
W. Wm. McGovern (Bowling Green State (OH), USA)

Probable topics:
With the study of algebraic frames hitting a new high in the literature, we aim to survey the recent results with an eye towards the applications of the theory to other algebraic arenas. In particular, we hope to motivate the discussion through the use of several well-studied examples: the frame of radical ideals of commutative ring with identity; the frame of convex l-subgroups of a lattice-ordered group; the frame of multiplicative filters and noetherian filters of a commutative ring with identity; the frame of z-ideals of C(X), etc.
A. Pultr (Charles University, Czech Republic)

Probable Topics:
(i) Sublocales (As surjective homomorphisms, congruences, nuclei); sublocales vs. subspaces in the spatial case.
(ii) Enrichment of the localic structure; uniformity and nearness; uniform sublocales; completion. Advantages of the point-free approach.
(iii) Approximation and special filters. Approximate maps. General remarks on filters apt to represent something like a point (prime and completely prime filters, shaving filters, Cauchy filters),

Featured
Lectures

The following have agreed to give a talk during the workshop:

F. Dashiell (UCLA, USA)
A. W. Hager (Wesleyan University (CT), USA)
P. Jipsen (Chapman University (CA), USA)
P. T. Johnstone (Cambridge University, UK)
J. J. Madden (Louisiana State (LA), USA)
C. J. Mulvey (University of Sussex, UK)
J. Picado (University of Coimbra, Portugal)

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Last modified: Tue Nov 10 14:08:49 EST 2009