Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

**Czeslaw Bylinski**- Warsaw University, Bialystok
**Piotr Rudnicki**- University of Alberta, Edmonton

- The graph induced by a many sorted signature is defined as follows: the vertices are the symbols of sorts, and if a sort $s$ is an argument of an operation with result sort $t$, then a directed edge $[s, t]$ is in the graph. The key lemma states relationship between the depth of elements of a free many sorted algebra over a signature and the length of directed chains in the graph induced by the signature. Then we prove that a monotonic many sorted signature (every finitely-generated algebra over it is locally-finite) induces a {\em well-founded} graph. The converse holds with an additional assumption that the signature is finitely operated, i.e. there is only a finite number of operations with the given result sort.

This work was partially supported by NSERC Grant OGP9207.

Contents (PDF format)

- [1]
Grzegorz Bancerek.
The fundamental properties of natural numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek.
Introduction to trees.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Grzegorz Bancerek.
K\"onig's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [4]
Grzegorz Bancerek.
K\"onig's Lemma.
*Journal of Formalized Mathematics*, 3, 1991. - [5]
Grzegorz Bancerek.
Sets and functions of trees and joining operations of trees.
*Journal of Formalized Mathematics*, 4, 1992. - [6]
Grzegorz Bancerek.
Joining of decorated trees.
*Journal of Formalized Mathematics*, 5, 1993. - [7]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Ewa Burakowska.
Subalgebras of many sorted algebra. Lattice of subalgebras.
*Journal of Formalized Mathematics*, 6, 1994. - [9]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Czeslaw Bylinski and Piotr Rudnicki.
The correspondence between monotonic many sorted signatures and well-founded graphs. Part I.
*Journal of Formalized Mathematics*, 8, 1996. - [14]
Agata Darmochwal.
Finite sets.
*Journal of Formalized Mathematics*, 1, 1989. - [15]
Krzysztof Hryniewiecki.
Graphs.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Yatsuka Nakamura and Piotr Rudnicki.
Vertex sequences induced by chains.
*Journal of Formalized Mathematics*, 7, 1995. - [17]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Preliminaries to circuits, I.
*Journal of Formalized Mathematics*, 6, 1994. - [18]
Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec, and Pauline N. Kawamoto.
Preliminaries to circuits, II.
*Journal of Formalized Mathematics*, 6, 1994. - [19]
Andrzej Nedzusiak.
$\sigma$-fields and probability.
*Journal of Formalized Mathematics*, 1, 1989. - [20]
Beata Perkowska.
Free many sorted universal algebra.
*Journal of Formalized Mathematics*, 6, 1994. - [21]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [22]
Andrzej Trybulec.
Tuples, projections and Cartesian products.
*Journal of Formalized Mathematics*, 1, 1989. - [23]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [24]
Andrzej Trybulec.
Many sorted algebras.
*Journal of Formalized Mathematics*, 6, 1994. - [25]
Andrzej Trybulec.
Subsets of real numbers.
*Journal of Formalized Mathematics*, Addenda, 2003. - [26]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [27]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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