Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996 Association of Mizar Users

On the Many Sorted Closure Operator and the Many Sorted Closure System


Artur Kornilowicz
Institute of Mathematics, Warsaw University, Bialystok

MML Identifier: CLOSURE1

The terminology and notation used in this paper have been introduced in the following articles [13] [4] [16] [12] [17] [2] [3] [6] [5] [14] [15] [1] [9] [10] [11] [8] [7]

Contents (PDF format)

  1. Preliminaries
  2. Properties of Many Sorted Closure Operators
  3. On the Many Sorted Closure Operator and the Many Sorted Closure System

Bibliography

[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. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.
[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. Extensions of mappings on generator set. Journal of Formalized Mathematics, 7, 1995.
[9] Malgorzata Korolkiewicz. Homomorphisms of many sorted algebras. Journal of Formalized Mathematics, 6, 1994.
[10] Beata Madras. Product of family of universal algebras. Journal of Formalized Mathematics, 5, 1993.
[11] Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto. Preliminaries to circuits, I. Journal of Formalized Mathematics, 6, 1994.
[12] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[13] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[14] Andrzej Trybulec. Many-sorted sets. Journal of Formalized Mathematics, 5, 1993.
[15] Andrzej Trybulec. Many sorted algebras. Journal of Formalized Mathematics, 6, 1994.
[16] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[17] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received February 7, 1996

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