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Publication detail
ŠLAPAL, J. PFALTZ, J.
Original Title
Transformations of Discrete Closure Systems
Type
journal article - other
Language
English
Original Abstract
Discrete systems such as sets, monoids, groups are familiar categories. The internal structure of the latter two is defined by an algebraic operator. In this paper we concentrate on discrete systems that are characterized by unary operators; these include choice operators $\CHOICE$, encountered in economics and social theory, and closure operators $\CL$, encountered in discrete geometry and data mining. Because, for many arbitrary operators $\OPER$, it is easy to induce a closure structure on the base set, closure operators play a central role in discrete systems. Our primary interest is in functions $f$ that map power sets $2^{\UNIV}$ into power sets $2^{\UNIV'}$, which are called transformations. Functions over continuous domains are usually characterized in terms of open sets. When the domains are discrete, closed sets seem more appropriate. In particular, we consider monotone transformations which are ``continuous'', or ``closed''. These can be used to establish criteria for asserting that ``the closure of a transformed image under $f$ is equal to the transformed image of the closure''.
Keywords
closure; choice; operator; continuous; category; function
Authors
ŠLAPAL, J.; PFALTZ, J.
RIV year
2013
Released
1. 6. 2013
ISBN
0236-5294
Periodical
Acta Mathematica Hungarica
Year of study
138
Number
4
State
Hungary
Pages from
386
Pages to
405
Pages count
20
BibTex
@article{BUT91785, author="Josef {Šlapal} and John {Pfaltz}", title="Transformations of Discrete Closure Systems", journal="Acta Mathematica Hungarica", year="2013", volume="138", number="4", pages="386--405", issn="0236-5294" }