3,197 research outputs found
Quantale Modules and their Operators, with Applications
The central topic of this work is the categories of modules over unital
quantales. The main categorical properties are established and a special class
of operators, called Q-module transforms, is defined. Such operators - that
turn out to be precisely the homomorphisms between free objects in those
categories - find concrete applications in two different branches of image
processing, namely fuzzy image compression and mathematical morphology
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning
Several logical operators are defined as dual pairs, in different types of
logics. Such dual pairs of operators also occur in other algebraic theories,
such as mathematical morphology. Based on this observation, this paper proposes
to define, at the abstract level of institutions, a pair of abstract dual and
logical operators as morphological erosion and dilation. Standard quantifiers
and modalities are then derived from these two abstract logical operators.
These operators are studied both on sets of states and sets of models. To cope
with the lack of explicit set of states in institutions, the proposed abstract
logical dual operators are defined in an extension of institutions, the
stratified institutions, which take into account the notion of open sentences,
the satisfaction of which is parametrized by sets of states. A hint on the
potential interest of the proposed framework for spatial reasoning is also
provided.Comment: 36 page
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