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Pointfree pointwise suprema in archimedean l-groups
 
 
Description:  We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of RL, the family of continuous real valued functions on a locale, or pointfree space.
Thus our setting is the category W of archimedean lattice-ordered groups (l-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. This is an appropriate context for this investigation because every W-object can be canonically represented as a subobject of RL.
We prove the appropriate analog of the Nakano-Stone Theorem: a (completely regular) locale L has the feature that RL is conditionally pointwise complete (sigma-complete), i.e., every bounded (countable) family from RL has a pointwise supremum in RL, iff L is boolean (a P-locale).
It is perhaps surprising that pointwise suprema can be characterized purely algebraically, without reference to a representation. They are the context-free suprema, in the sense that the pointwise suprema are precisely those which are preserved by all morphisms out of G. We adopt the latter attribute as the final, representation-free definition of pointwise suprema.
Thus emboldened, we adopt a maximally broad definition of unconditional pointwise completeness (sigma-completeness): a divisible W-object G is pointwise complete (sigma-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete (sigma-complete) W-objects are those of the form RL for L a boolean locale (P-locale). Finally, we show that a W-object G is pointwise sigma-complete iff it is epicomplete.

Joint work with Anthony W. Hager
Date:  2014-09-10
Start Time:   14:30
Speaker:  Richard N. Ball (Univ. Denver, USA)
Institution:  Univ. Denver, USA
Place:  Room 5.5
Research Groups: -Algebra, Logic and Topology
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