2,682 research outputs found
Partial orderings with the weak Freese-Nation property
A partial ordering P is said to have the weak Freese-Nation property (WFN) if
there is a mapping f:P ---> [P]^{<= aleph_0} such that, for any a, b in P, if a
<= b then there exists c in f(a) cap f(b) such that a <= c <= b. In this note,
we study the WFN and some of its generalizations. Some features of the class of
BAs with the WFN seem to be quite sensitive to additional axioms of set theory:
e.g., under CH, every ccc cBA has this property while, under b >= aleph_2,
there exists no cBA with the WFN
Enriched Stone-type dualities
A common feature of many duality results is that the involved equivalence
functors are liftings of hom-functors into the two-element space resp. lattice.
Due to this fact, we can only expect dualities for categories cogenerated by
the two-element set with an appropriate structure. A prime example of such a
situation is Stone's duality theorem for Boolean algebras and Boolean
spaces,the latter being precisely those compact Hausdorff spaces which are
cogenerated by the two-element discrete space. In this paper we aim for a
systematic way of extending this duality theorem to categories including all
compact Hausdorff spaces. To achieve this goal, we combine duality theory and
quantale-enriched category theory. Our main idea is that, when passing from the
two-element discrete space to a cogenerator of the category of compact
Hausdorff spaces, all other involved structures should be substituted by
corresponding enriched versions. Accordingly, we work with the unit interval
and present duality theory for ordered and metric compact Hausdorff
spaces and (suitably defined) finitely cocomplete categories enriched in
Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction
A (v,0)-semilattice is ultraboolean, if it is a directed union of finite
Boolean (v,0)-semilattices. We prove that every distributive (v,0)-semilattice
is a retract of some ultraboolean (v,0)-semilattices. This is established by
proving that every finite distributive (v,0)-semilattice is a retract of some
finite Boolean (v,0)-semilattice, and this in a functorial way. This result is,
in turn, obtained as a particular case of a category-theoretical result that
gives sufficient conditions, for a functor , to admit a right inverse. The
particular functor used for the abovementioned result about ultraboolean
semilattices has neither a right nor a left adjoint
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