Detail publikace

# Transformations of Discrete Closure Systems

ŠLAPAL, J. PFALTZ, J.

Originální název

Transformations of Discrete Closure Systems

Anglický název

Transformations of Discrete Closure Systems

Jazyk

en

Originální abstrakt

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''.

Anglický abstrakt

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''.

Dokumenty

BibTex

``````
@article{BUT91785,
author="Josef {Šlapal} and John {Pfaltz}",
title="Transformations of Discrete Closure Systems",
annote="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''.",
chapter="91785",
number="4",
volume="138",
year="2013",
month="june",
pages="386--405",
type="journal article - other"
}``````