Volume 6, 1994

University of Bialystok

Copyright (c) 1994 Association of Mizar Users

**Andrzej Trybulec**- Warsaw University, Bialystok

- The aim of this work is to provide a bridge between the theory of context-free grammars developed in [10], [6] and universally free manysorted algebras([14]. The third scheme proved in the article allows to prove that two homomorphisms equal on the set of free generators are equal. The first scheme is a slight modification of the scheme in [6] and the second is rather technical, but since it was useful for me, perhaps it might be useful for somebody else. The concept of flattening of a many sorted function $F$ between two manysorted sets $A$ and $B$ (with common set of indices $I$) is introduced for $A$ with mutually disjoint components (pairwise disjoint function - the concept introduced in [13]). This is a function on the union of $A$, that is equal to $F$ on every component of $A$. A trivial many sorted algebra over a signature $S$ is defined with sorts being singletons of corresponding sort symbols. It has mutually disjoint sorts.

Contents (PDF format)

- [1]
Grzegorz Bancerek.
K\"onig's theorem.
*Journal of Formalized Mathematics*, 2, 1990. - [2]
Grzegorz Bancerek.
K\"onig's Lemma.
*Journal of Formalized Mathematics*, 3, 1991. - [3]
Grzegorz Bancerek.
Sets and functions of trees and joining operations of trees.
*Journal of Formalized Mathematics*, 4, 1992. - [4]
Grzegorz Bancerek.
Joining of decorated trees.
*Journal of Formalized Mathematics*, 5, 1993. - [5]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Grzegorz Bancerek and Piotr Rudnicki.
On defining functions on trees.
*Journal of Formalized Mathematics*, 5, 1993. - [7]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [8]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [9]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [10]
Patricia L. Carlson and Grzegorz Bancerek.
Context-free grammar --- part I.
*Journal of Formalized Mathematics*, 4, 1992. - [11]
Malgorzata Korolkiewicz.
Homomorphisms of many sorted algebras.
*Journal of Formalized Mathematics*, 6, 1994. - [12]
Andrzej Nedzusiak.
$\sigma$-fields and probability.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Andrzej Nedzusiak.
Probability.
*Journal of Formalized Mathematics*, 2, 1990. - [14]
Beata Perkowska.
Free many sorted universal algebra.
*Journal of Formalized Mathematics*, 6, 1994. - [15]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [16]
Andrzej Trybulec.
Many-sorted sets.
*Journal of Formalized Mathematics*, 5, 1993. - [17]
Andrzej Trybulec.
Many sorted algebras.
*Journal of Formalized Mathematics*, 6, 1994. - [18]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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