Database Reference
In-Depth Information
Notice that this property is satisfied if the poset R is a complete lattice, where
d
c , for example.
With this final result we demonstrated a concrete difference between the in-
tuitionistic logic of standard topoi and our superintuitionistic, or intermediate,
(Jankov's or De Morgan) logic of our weak monoidal topos for the database map-
ping systems.
=
b
9.3
Review Questions
1. What are topoi and what are the most known examples for them? Why is the DB
category not a topos (provide at least two main reasons)? What is the relation-
ship between the database metric space and the database lattice? Why does the
maximal distance between two databases mean that these two databases have no
view in common? Does it mean that the intersection of the sets of values (from
the universe
) contained in these two databases is empty? Can we choose an-
other metrics, based on the merging instead on the matching operation?
2. The main idea, in order to define the subobject classifier in the DB category,
is to interpret the true and false arrows of the commutative diagrams for the
subobject classifier by the top and bottom values, respectively, in the complete
algebraic database lattice L DB . Is the interpretation of this database lattice, as
the set of the “logic values” of a particular many-valued propositional database
logic, appropriate in order to provide the underlying propositional logic? Can
the meet and join lattice operators be interpreted by the logic operators of con-
junction and disjunction? Is it possible to obtain the pullback diagram for the
logic dual (with “false” arrow) so that it has the same vertex LimP
U
=
A
×
B
B ?
3. Enumerate the topos properties which are still valid in the DB category. Enu-
merate the most important features of the topos which are not satisfied in the
DB category. What is the consequence of the non-well-pointedness property in
the DB category w.r.t. the extensionality principle (which is valid in the Set cat-
egory of sets and functions between them)? Explain why in a standard topos if
its initial object is equal (up to isomorphism) to its terminal object then all ob-
jects are isomorphic, and hence such categories degenerate? Why doesn't this
degeneration happen in the DB category, where from the duality property we
have that the initial and terminal objects are equal to the zero object
instead of Υ
×
0 ?Can
we then consider that DB is a kind of a generalization of a standard topos where
the isomorphism of the initial and terminal objects does not cause the degen-
eration, and why is such a generalization important not only for the database
mappings applications but also for the general theoretical point of view?
4. Demonstrate that the subobject classifier pullback diagram with A
=
Ω
×
Ω ,
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