Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996
Association of Mizar Users
The Correspondence Between Monotonic Many Sorted Signatures
and Well-Founded Graphs. Part I
-
Czeslaw Bylinski
-
Warsaw University, Bialystok
-
Piotr Rudnicki
-
University of Alberta, Edmonton
Summary.
-
We prove a number of auxiliary facts about graphs, mainly about
vertex sequences of chains and oriented chains. Then we define
a graph to be {\em well-founded} if for each vertex in the graph
the length of oriented chains ending at the vertex is bounded.
A {\em well-founded} graph does not have directed cycles or infinite
descending chains.
In the second part of the article we prove some auxiliary facts
about free algebras and locally-finite algebras.
This work was partially supported by NSERC Grant OGP9207.
The terminology and notation used in this paper have been
introduced in the following articles
[27]
[17]
[30]
[2]
[1]
[25]
[4]
[31]
[14]
[16]
[15]
[19]
[11]
[21]
[23]
[20]
[3]
[6]
[7]
[5]
[8]
[18]
[12]
[28]
[29]
[13]
[22]
[26]
[24]
[9]
[10]
-
Some properties of graphs
-
Some properties of many sorted algebras
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.
Introduction to trees.
Journal of Formalized Mathematics,
1, 1989.
- [4]
Grzegorz Bancerek.
K\"onig's theorem.
Journal of Formalized Mathematics,
2, 1990.
- [5]
Grzegorz Bancerek.
Cartesian product of functions.
Journal of Formalized Mathematics,
3, 1991.
- [6]
Grzegorz Bancerek.
K\"onig's Lemma.
Journal of Formalized Mathematics,
3, 1991.
- [7]
Grzegorz Bancerek.
Sets and functions of trees and joining operations of trees.
Journal of Formalized Mathematics,
4, 1992.
- [8]
Grzegorz Bancerek.
Joining of decorated trees.
Journal of Formalized Mathematics,
5, 1993.
- [9]
Grzegorz Bancerek.
Subtrees.
Journal of Formalized Mathematics,
6, 1994.
- [10]
Grzegorz Bancerek.
Terms over many sorted universal algebra.
Journal of Formalized Mathematics,
6, 1994.
- [11]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
Journal of Formalized Mathematics,
1, 1989.
- [12]
Grzegorz Bancerek and Piotr Rudnicki.
On defining functions on trees.
Journal of Formalized Mathematics,
5, 1993.
- [13]
Ewa Burakowska.
Subalgebras of many sorted algebra. Lattice of subalgebras.
Journal of Formalized Mathematics,
6, 1994.
- [14]
Czeslaw Bylinski.
Functions and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
- [15]
Czeslaw Bylinski.
Functions from a set to a set.
Journal of Formalized Mathematics,
1, 1989.
- [16]
Czeslaw Bylinski.
Partial functions.
Journal of Formalized Mathematics,
1, 1989.
- [17]
Czeslaw Bylinski.
Some basic properties of sets.
Journal of Formalized Mathematics,
1, 1989.
- [18]
Patricia L. Carlson and Grzegorz Bancerek.
Context-free grammar --- part I.
Journal of Formalized Mathematics,
4, 1992.
- [19]
Agata Darmochwal.
Finite sets.
Journal of Formalized Mathematics,
1, 1989.
- [20]
Agata Darmochwal and Yatsuka Nakamura.
The topological space $\calE^2_\rmT$. Arcs, line segments and special polygonal arcs.
Journal of Formalized Mathematics,
3, 1991.
- [21]
Krzysztof Hryniewiecki.
Graphs.
Journal of Formalized Mathematics,
2, 1990.
- [22]
Malgorzata Korolkiewicz.
Homomorphisms of many sorted algebras.
Journal of Formalized Mathematics,
6, 1994.
- [23]
Yatsuka Nakamura and Piotr Rudnicki.
Vertex sequences induced by chains.
Journal of Formalized Mathematics,
7, 1995.
- [24]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Preliminaries to circuits, II.
Journal of Formalized Mathematics,
6, 1994.
- [25]
Andrzej Nedzusiak.
$\sigma$-fields and probability.
Journal of Formalized Mathematics,
1, 1989.
- [26]
Beata Perkowska.
Free many sorted universal algebra.
Journal of Formalized Mathematics,
6, 1994.
- [27]
Andrzej Trybulec.
Tarski Grothendieck set theory.
Journal of Formalized Mathematics,
Axiomatics, 1989.
- [28]
Andrzej Trybulec.
Many-sorted sets.
Journal of Formalized Mathematics,
5, 1993.
- [29]
Andrzej Trybulec.
Many sorted algebras.
Journal of Formalized Mathematics,
6, 1994.
- [30]
Zinaida Trybulec.
Properties of subsets.
Journal of Formalized Mathematics,
1, 1989.
- [31]
Edmund Woronowicz.
Relations and their basic properties.
Journal of Formalized Mathematics,
1, 1989.
Received February 14, 1996
[
Download a postscript version,
MML identifier index,
Mizar home page]