Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

## Product of Families of Groups and Vector Spaces

Anna Lango
Warsaw University, Bialystok
Grzegorz Bancerek
Polish Academy of Sciences, Institute of Mathematics, Warsaw

### Summary.

In the first section we present properties of fields and Abelian groups in terms of commutativity, associativity, etc. Next, we are concerned with operations on \$n\$-tuples on some set which are generalization of operations on this set. It is used in third section to introduce the \$n\$-power of a group and the \$n\$-power of a field. Besides, we introduce a concept of indexed family of binary (unary) operations over some indexed family of sets and a product of such families which is binary (unary) operation on a product of family sets. We use that product in the last section to introduce the product of a finite sequence of Abelian groups.

#### MML Identifier: PRVECT_1

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

#### Contents (PDF format)

1. Abelian Groups and Fields
2. The \$n\$-Product of a Binary and a Unary Operation
3. The \$n\$-Power of a Group and of a Field
4. Sequences of Non-empty Sets
5. The Product of Families of Operations
6. The Product of Families of Groups

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