Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

**Artur Kornilowicz**- Institute of Mathematics, Warsaw University, Bialystok

- In this paper definitions of many sorted closure system and many sorted closure operator are introduced. These notations are also introduced in [9], but in another meaning. In this article closure system is absolutely multiplicative subset family of many sorted sets and in [9] is many sorted absolutely multiplicative subset family of many sorted sets. Analogously, closure operator is function between many sorted sets and in [9] is many sorted function from a many sorted set into a many sorted set.

- Preliminaries
- Set of Many Sorted Subsets of a Many Sorted Set
- Many Sorted Operator corresponding to the Operator on Many Sorted Subsets
- Properties of Closure Operators
- On the Closure Operator and the Closure System

- [1]
Ewa Burakowska.
Subalgebras of many sorted algebra. Lattice of subalgebras.
*Journal of Formalized Mathematics*, 6, 1994. - [2]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Artur Kornilowicz.
Certain facts about families of subsets of many sorted sets.
*Journal of Formalized Mathematics*, 7, 1995. - [8]
Artur Kornilowicz.
Definitions and basic properties of boolean and union of many sorted sets.
*Journal of Formalized Mathematics*, 7, 1995. - [9]
Artur Kornilowicz.
On the many sorted closure operator and the many sorted closure system.
*Journal of Formalized Mathematics*, 8, 1996. - [10]
Beata Padlewska.
Families of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [12]
Andrzej Trybulec.
Function domains and Fr\aenkel operator.
*Journal of Formalized Mathematics*, 2, 1990. - [13]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [14]
Andrzej Trybulec.
Many sorted algebras.
*Journal of Formalized Mathematics*, 6, 1994. - [15]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [16]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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