Morphisms in Logic, Topology, and Formal Concept Analysis
Morphisms in Logic, Topology, and Formal Concept Analysis
Master's thesis by Markus Krötzsch
- Supervisor Steffen Hölldobler
- Wissensverarbeitung
- 1 Februar 2005 – 1 Februar 2005
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between logical deductive systems, motivated by the intuition that additional (categorical)
structure is needed to model the interrelations of formal specifications.
This general task necessarily involves considerations in various mathematical disciplines,
some of which might be interesting in their own right and which can be
read independently.
To find suitable morphisms, we review the relationships of formal logic, algebra,
topology, domain theory, and formal concept analysis (FCA). This leads
to a rather complete exposition of the representation theory of algebraic lattices,
including some novel interpretations in terms of FCA and an explicit proof of the
cartestian closedness of the emerging category. It also introduces the main concepts
of “domain theory in logical form” for a particularly simple example.
In order to incorporate morphisms from FCA, we embark on the study of
various context morphisms and their relationships. The discovered connections
are summarized in a hierarchy of context morphisms, which includes dual bonds,
scale measures, and infomorphisms.
Finally, we employ the well-known means of Stone duality to unify the topological
and the FCA-based approach. A notion of logical consequence relation
with a suggestive proof theoretical reading is introduced as a morphism between
deductive systems, and special instances of these relations are identified with morphismsfrom
topology, FCA, and lattice theory. Especially, scale measures are recognized
as topologically continuous mappings, and infomorphisms are identified
both with coherent maps and with Lindenbaum algebra homomorphisms.
