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Discussiones Mathematicae General Algebra and Applications 22(1) (2002) 87-100
DOI: https://doi.org/10.7151/dmgaa.1049

EQUATIONAL BASES FOR WEAK MONOUNARY VARIETIES

 Grzegorz Bińczak

Institute of Mathematics, Warsaw University of Technology
pl. Politechniki 1, 00-661 Warszawa, Poland
e-mail: binczak@mini.pw.edu.pl

Abstract

It is well-known that every monounary variety of total algebras has one-element equational basis (see [5]). In my paper I prove that every monounary weak variety has at most 3-element equational basis. I give an example of monounary weak variety having 3-element equational basis, which has no 2-element equational basis.

Keywords: partial algebra, weak equation, weak variety, regular equation, regular weak equational theory, monounary algebras.

2000 AMS Mathematics Subject Classifications: 08A55, 08B05.

Bibliography

[1] G. Bińczak, A characterization theorem for weak varieties, Algebra Universalis 45 (2001), 53-62.

[2] P. Burmeister, A Model - Theoretic Oriented Approach to Partial Algebras, Akademie-Verlag, Berlin 1986.

[3] G. Grätzer, Universal Algebra, (the second edition), Springer-Verlag, New York 1979.

[4] H. Höft, Weak and strong equations in partial algebras, Algebra Universalis 3 (1973), 203-215.

[5] E. Jacobs and R. Schwabauer, The lattice of equational classes of algebras with one unary operation, Amer. Math. Monthly 71 (1964), 151-155.

[6] L. Rudak, A completness theorem for weak equational logic, Algebra Universalis 16 (1983), 331-337.

[7] L. Rudak, Algebraic characterization of conflict-free varieties of partial algebras, Algebra Universalis 30 (1993), 89-100.

Received 19 April 2002
Revised 2 July 2002