346 research outputs found
Idempotent analysis and continuous semilattices
In this survey article we give a brief overview of various aspects of the recently emerging field of idempotent analysis and suggest potential connections with domain theory. © 2004 Published by Elsevier B.V
Domain theory and mirror properties in inverse semigroups
Inverse semigroups are a class of semigroups whose structure induces a
compatible partial order. This partial order is examined so as to establish
mirror properties between an inverse semigroup and the semilattice of its
idempotent elements, such as continuity in the sense of domain theory.Comment: 15 pages. The final publication is available at www.springerlink.com.
See http://link.springer.com/article/10.1007%2Fs00233-012-9392-4?LI=tru
Approximately multiplicative maps from weighted semilattice algebras
We investigate which weighted convolution algebras , where
is a semilattice, are AMNM in the sense of Johnson (JLMS, 1986). We give an
explicit example where this is not the case. We show that the unweighted
examples are all AMNM, as are all where has either
finite width or finite height. Some of these finite-width examples are
isomorphic to function algebras studied by Feinstein (IJMMS, 1999).
We also investigate when is an AMNM pair in
the sense of Johnson (JLMS, 1988), where denotes the algebra of
2-by-2 complex matrices. In particular, we obtain the following two contrasting
results: (i) for many non-trivial weights on the totally ordered semilattice
, the pair is not
AMNM; (ii) for any semilattice , the pair is AMNM.
The latter result requires a detailed analysis of approximately commuting,
approximately idempotent matrices.Comment: AMS-LaTeX. v3: 31 pages, additional minor corrections to v2. Final
version, to appear in J. Austral. Math. Soc. v4: small correction of
mis-statement at start of Section 4 (this should also be fixed in the journal
version
Galois Connections between Semimodules and Applications in Data Mining
In [1] a generalisation of Formal Concept Analysis was introduced
with data mining applications in mind, K-Formal Concept Analysis,
where incidences take values in certain kinds of semirings, instead
of the standard Boolean carrier set. A fundamental result was missing
there, namely the second half of the equivalent of the main theorem of
Formal Concept Analysis. In this continuation we introduce the structural
lattice of such generalised contexts, providing a limited equivalent
to the main theorem of K-Formal Concept Analysis which allows to interpret
the standard version as a privileged case in yet another direction.
We motivate our results by providing instances of their use to analyse
the confusion matrices of multiple-input multiple-output classifiers
Towards a generalisation of formal concept analysis for data mining purposes
In this paper we justify the need for a generalisation of Formal
Concept Analysis for the purpose of data mining and begin the
synthesis of such theory. For that purpose, we first review semirings and
semimodules over semirings as the appropriate objects to use in abstracting
the Boolean algebra and the notion of extents and intents, respectively.
We later bring to bear powerful theorems developed in the field
of linear algebra over idempotent semimodules to try to build a Fundamental
Theorem for K-Formal Concept Analysis, where K is a type of
idempotent semiring. Finally, we try to put Formal Concept Analysis in
new perspective by considering it as a concrete instance of the theory
developed
A non-commutative generalization of Stone duality
We prove that the category of boolean inverse monoids is dually equivalent to
the category of boolean groupoids. This generalizes the classical Stone duality
between boolean algebras and boolean spaces. As an instance of this duality, we
show that the boolean inverse monoid associated with the Cuntz groupoid is the
strong orthogonal completion of the polycyclic (or Cuntz) monoid and so its
group of units is a Thompson group
Independent resolutions for totally disconnected dynamical systems I: Algebraic case
This is the first out of two papers on independent resolutions for totally
disconnected dynamical systems. In the present paper, we discuss independent
resolutions from an algebraic point of view. We also present applications to
group homology and cohomology. This first paper sets the stage for our second
paper, where we explain how to use independent resolutions in K-theory
computations for crossed products attached to totally disconnected dynamical
systems.Comment: 26 pages; minor changes improving the expositio
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