16 research outputs found
Semigroup Closures of Finite Rank Symmetric Inverse Semigroups
We introduce the notion of semigroup with a tight ideal series and
investigate their closures in semitopological semigroups, particularly inverse
semigroups with continuous inversion. As a corollary we show that the symmetric
inverse semigroup of finite transformations of the rank
is algebraically closed in the class of (semi)topological inverse
semigroups with continuous inversion. We also derive related results about the
nonexistence of (partial) compactifications of classes of semigroups that we
consider.Comment: With the participation of the new coauthor - Jimmie Lawson - the
manuscript has been substantially revised and expanded. Accordingly, we have
also changed the manuscript titl
Quantitative Concept Analysis
Formal Concept Analysis (FCA) begins from a context, given as a binary
relation between some objects and some attributes, and derives a lattice of
concepts, where each concept is given as a set of objects and a set of
attributes, such that the first set consists of all objects that satisfy all
attributes in the second, and vice versa. Many applications, though, provide
contexts with quantitative information, telling not just whether an object
satisfies an attribute, but also quantifying this satisfaction. Contexts in
this form arise as rating matrices in recommender systems, as occurrence
matrices in text analysis, as pixel intensity matrices in digital image
processing, etc. Such applications have attracted a lot of attention, and
several numeric extensions of FCA have been proposed. We propose the framework
of proximity sets (proxets), which subsume partially ordered sets (posets) as
well as metric spaces. One feature of this approach is that it extracts from
quantified contexts quantified concepts, and thus allows full use of the
available information. Another feature is that the categorical approach allows
analyzing any universal properties that the classical FCA and the new versions
may have, and thus provides structural guidance for aligning and combining the
approaches.Comment: 16 pages, 3 figures, ICFCA 201