## On the closure operator and the closure system of many sorted sets (1996)

Venue: | Journal of Formalized Mathematics |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Korniłowicz96onthe,

author = {Artur Korniłowicz},

title = {On the closure operator and the closure system of many sorted sets},

journal = {Journal of Formalized Mathematics},

year = {1996},

volume = {8}

}

### OpenURL

### Abstract

Summary. 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.

### Citations

1278 | Tarski Grothendieck set theory
- Trybulec
- 1990
(Show Context)
Citation Context ...s function between many sorted sets and in [9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles =-=[11]-=-, [5], [15], [10], [16], [2], [4], [3], [6], [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote... |

1257 | Functions and their basic properties
- Byliński
- 1990
(Show Context)
Citation Context ...d sets and in [9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], =-=[2]-=-, [4], [3], [6], [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many s... |

1187 | Properties of subsets
- Trybulec
- 1990
(Show Context)
Citation Context ...between many sorted sets and in [9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], =-=[15]-=-, [10], [16], [2], [4], [3], [6], [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B... |

1012 | Functions from a set to a set
- Byliński
- 1990
(Show Context)
Citation Context ... in [9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], [4], =-=[3]-=-, [6], [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted sets... |

988 | Relations and their basic properties
- Woronowicz
- 1990
(Show Context)
Citation Context ... sorted sets and in [9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], =-=[16]-=-, [2], [4], [3], [6], [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote m... |

725 | Some basic properties of sets
- Byliński
- 1990
(Show Context)
Citation Context ...tion between many sorted sets and in [9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], =-=[5]-=-, [15], [10], [16], [2], [4], [3], [6], [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets... |

432 | Partial functions
- Byliński
- 1990
(Show Context)
Citation Context ...s and in [9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], =-=[4]-=-, [3], [6], [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted... |

311 | Finite sets
- Darmochwał
- 1990
(Show Context)
Citation Context ...9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], [4], [3], =-=[6]-=-, [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted sets inde... |

303 | Families of sets
- Padlewska
- 1990
(Show Context)
Citation Context ...n many sorted sets and in [9] is many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], =-=[10]-=-, [16], [2], [4], [3], [6], [12], [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M de... |

195 | Many-sorted sets
- Trybulec
- 1993
(Show Context)
Citation Context ...sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], [4], [3], [6], [12], =-=[13]-=-, [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted sets indexed by I, an... |

147 | Function domains and Frænkel operator
- Trybulec
- 1990
(Show Context)
Citation Context ... many sorted function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], [4], [3], [6], =-=[12]-=-, [13], [14], [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted sets indexed by... |

126 | Many sorted algebras
- Trybulec
- 1996
(Show Context)
Citation Context ... function from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], [4], [3], [6], [12], [13], =-=[14]-=-, [1], [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted sets indexed by I, and f , ... |

51 | Subalgebras of many sorted algebra. Lattice of subalgebras
- Burakowska
- 1996
(Show Context)
Citation Context ...ion from a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], [4], [3], [6], [12], [13], [14], =-=[1]-=-, [8], and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted sets indexed by I, and f , f1 de... |

6 | Certain facts about families of subsets of many sorted sets
- Korniłowicz
- 1995
(Show Context)
Citation Context ...y sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], [4], [3], [6], [12], [13], [14], [1], [8], and =-=[7]-=- provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted sets indexed by I, and f , f1 denote functions... |

6 | Definitions and basic properties of boolean and union of many sorted sets
- Korniłowicz
- 1995
(Show Context)
Citation Context ...rom a many sorted set into a many sorted set. MML Identifier:CLOSURE2. WWW:http://mizar.org/JFM/Vol8/closure2.html The articles [11], [5], [15], [10], [16], [2], [4], [3], [6], [12], [13], [14], [1], =-=[8]-=-, and [7] provide the notation and terminology for this paper. 1. PRELIMINARIES We adopt the following convention: i, x, I denote sets, A, B, M denote many sorted sets indexed by I, and f , f1 denote ... |

2 | On the many sorted closure operator and the many sorted closure system
- Korniłowicz
- 1996
(Show Context)
Citation Context ...ute of Mathematics Warsaw University Białystok Summary. 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... |