Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996 Association of Mizar Users

## Translations, Endomorphisms, and Stable Equational Theories

Grzegorz Bancerek
Institute of Mathematics, Polish Academy of Sciences

### Summary.

Equational theories of an algebra, i.e. the equivalence relation closed under translations and endomorphisms, are formalized. The correspondence between equational theories and term rewriting systems is discussed in the paper. We get as the main result that any pair of elements of an algebra belongs to the equational theory generated by a set $A$ of axioms iff the elements are convertible w.r.t. term rewriting reduction determined by $A$.\par The theory is developed according to [19].

#### MML Identifier: MSUALG_6

The terminology and notation used in this paper have been introduced in the following articles [14] [9] [18] [1] [20] [22] [21] [3] [6] [8] [7] [4] [2] [17] [15] [12] [16] [10] [11] [13] [5]

#### Contents (PDF format)

1. Endomorphisms and translations
2. Compatibility, invariantness, and stability
3. Invariant, stable, and invariant stable closure
4. Equational theory

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