Journal of Formalized Mathematics
Volume 7, 1995
University of Bialystok
Copyright (c) 1995 Association of Mizar Users

## Minimal Signature for Partial Algebra

Grzegorz Bancerek
Institute of Mathematics, Polish Academy of Sciences

### Summary.

The concept of characterizing of partial algebras by many sorted signature is introduced, i.e. we say that a signature $S$ characterizes a partial algebra $A$ if there is an $S$-algebra whose sorts form a partition of the carrier of algebra $A$ and operations are formed from operations of $A$ by the partition. The main result is that for any partial algebra there is the minimal many sorted signature which characterizes the algebra. The minimality means that there are signature endomorphisms from any signature which characterizes the algebra $A$ onto the minimal one.

#### MML Identifier: PUA2MSS1

The terminology and notation used in this paper have been introduced in the following articles [13] [7] [18] [17] [1] [11] [19] [20] [4] [14] [3] [8] [6] [5] [21] [12] [10] [2] [15] [16] [9]

#### Contents (PDF format)

1. Preliminary
2. Partitions
3. Relations Generated by Operations of Partial Algebra
4. Partitability
5. Signature Morphisms
6. Many Sorted Signature of Partial Algebra

#### Acknowledgments

I would like to thank Professor Andrzej Trybulec for suggesting me the problem solved here.

#### Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. K\"onig's theorem. Journal of Formalized Mathematics, 2, 1990.
[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[9] Jaroslaw Kotowicz, Beata Madras, and Malgorzata Korolkiewicz. Basic notation of universal algebra. Journal of Formalized Mathematics, 4, 1992.
[10] Andrzej Nedzusiak. $\sigma$-fields and probability. Journal of Formalized Mathematics, 1, 1989.
[11] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[12] Konrad Raczkowski and Pawel Sadowski. Equivalence relations and classes of abstraction. Journal of Formalized Mathematics, 1, 1989.
[13] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[14] Andrzej Trybulec. Tuples, projections and Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[15] Andrzej Trybulec. Many-sorted sets. Journal of Formalized Mathematics, 5, 1993.
[16] Andrzej Trybulec. Many sorted algebras. Journal of Formalized Mathematics, 6, 1994.
[17] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[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.
[20] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.
[21] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Journal of Formalized Mathematics, 1, 1989.

Received October 1, 1995