Journal of Formalized Mathematics
Volume 6, 1994
University of Bialystok
Copyright (c) 1994 Association of Mizar Users

## A Scheme for Extensions of Homomorphisms of Many Sorted Algebras

Andrzej Trybulec
Warsaw University, Bialystok

### Summary.

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.

#### MML Identifier: MSAFREE1

The terminology and notation used in this paper have been introduced in the following articles [15] [9] [18] [19] [7] [8] [5] [13] [1] [2] [3] [4] [10] [6] [12] [16] [17] [14] [11]

Contents (PDF format)

#### Bibliography

[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.

Received December 13, 1994