Transformace diskrétních uzávěrových systémů

Transformations of Discrete Closure Systems

journal article - other

en

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''.

Diskrétní systémy jako množiny, monoidy, grupy jsou známými kategoriemi. Interní struktura posledních dvou je definována algebraickým operátorem. V této práci se zaměřujeme na diskrétní systémy které jsou chrakterizovány unárními operátory, především uzávěrovýn operátorem. Studujema transformace diskrétních systémů dané zobrazeními mezi pšíslučnými potenčními množinami.

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''.

uzávěr, výběr, operátor, spojitost, kategorie, funkce

closure; choice; operator; continuous; category; function

2013

01.06.2013

0236-5294

138

4

386–405

20