Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991
Association of Mizar Users
The Lattice of Natural Numbers and The Sublattice of it.
The Set of Prime Numbers.

Marek Chmur

Warsaw University, Bialystok
Summary.

Basic properties of the least common multiple
and the greatest common divisor. The lattice of natural numbers
(${\rm L}_{\Bbb N}$)
and the lattice of natural numbers greater than zero
(${\rm L}_{\Bbb N^+}$) are constructed.
The notion of
the sublattice of the lattice of natural numbers is given.
Some facts about it are proved.
The last part of the article deals with some properties of prime numbers
and with the notions of the set of prime numbers and the $n$th prime
number. It is proved that the set of prime numbers is infinite.
MML Identifier:
NAT_LAT
The terminology and notation used in this paper have been
introduced in the following articles
[10]
[6]
[12]
[11]
[1]
[9]
[2]
[14]
[4]
[3]
[7]
[13]
[5]
[8]
Contents (PDF format)
Bibliography
 [1]
Grzegorz Bancerek.
Cardinal numbers.
Journal of Formalized Mathematics,
1, 1989.
 [2]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
Journal of Formalized Mathematics,
1, 1989.
 [3]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
Journal of Formalized Mathematics,
1, 1989.
 [4]
Czeslaw Bylinski.
Binary operations.
Journal of Formalized Mathematics,
1, 1989.
 [5]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
 [6]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
 [7]
Agata Darmochwal.
Finite sets.
Journal of Formalized Mathematics,
1, 1989.
 [8]
Rafal Kwiatek.
Factorial and Newton coefficients.
Journal of Formalized Mathematics,
2, 1990.
 [9]
Rafal Kwiatek and Grzegorz Zwara.
The divisibility of integers and integer relatively primes.
Journal of Formalized Mathematics,
2, 1990.
 [10]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
 [11]
Andrzej Trybulec.
Subsets of real numbers.
Journal of Formalized Mathematics,
Addenda, 2003.
 [12]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
 [13]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
 [14]
Stanislaw Zukowski.
Introduction to lattice theory.
Journal of Formalized Mathematics,
1, 1989.
Received April 26, 1991
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