Publication detail

Transformations of Discrete Closure Systems

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"
}