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

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. Reduction relations. Journal of Formalized Mathematics, 7, 1995.
[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[5] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Journal of Formalized Mathematics, 8, 1996.
[6] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
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[10] Malgorzata Korolkiewicz. Homomorphisms of many sorted algebras. Journal of Formalized Mathematics, 6, 1994.
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[12] Beata Madras. Product of family of universal algebras. Journal of Formalized Mathematics, 5, 1993.
[13] Robert Milewski. Lattice of congruences in many sorted algebra. Journal of Formalized Mathematics, 8, 1996.
[14] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 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] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[18] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[19] Wolfgang Wechler. \em Universal Algebra for Computer Scientists, volume 25 of \em EATCS Monographs on TCS. Springer--Verlag, 1992.
[20] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[21] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.
[22] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Journal of Formalized Mathematics, 1, 1989.

Received February 9, 1996

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