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The persistence of universal formulae in free algebras

Published online by Cambridge University Press:  17 April 2009

Anthony M. Gaglione
Affiliation:
Departments of Mathematics, United States Naval Academy, Annapolis, Maryland 21402–5000, United States of America.
Dennis Spellman
Affiliation:
Department of Mathematics, Sacred Heart University, Bridgeport, Connecticut 06606, United States of America.
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Abstract

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Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements:

(1) Fr(V) discriminates V. (1*) The Fs(V) satisfy the same universal sentences for all sr. Moreover, we have introduced the concept of strong discrimination in such a way that for a fixed finite cardinal r the following two statements are equivalent:

(2) Fr(V) strongly discriminates V. (2*) The Fs(V) satisfy the same universal formulas for all sr whenever elements of Fr(V) are substituted for the unquantified variables. On the surface (2) and (2*) appear to be stronger conditions than (1) and (1*). However, we have shown that for particular varieties (of groups) (2) and (2*) are no stronger than (1) and (1*).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Baumslag, Gilbert, Neumann, B.H., Neumann, Hanna and Neumann, Peter M., “On varieties generated by a finitely generated group”, Math. Z. 86 (1964), 93122.CrossRefGoogle Scholar
[2]Bell, J.L. and Slomson, A.B., Models and Ultraproducts, (North-Holland, Amsterdam, Second Revised Pringing, 1971).Google Scholar
[3]Gaglione, A. and Spellman, D., “Are some groups more discriminating than others?” (to appear).Google Scholar
[4]Grätzer, G., Universal Algebra, (Van Nostrand, Princeton, 1968).Google Scholar
[5]Gupta, N. and Levin, F., “Generating groups of certain product varieties”, Arch. Math. 30 (1978), 113117.CrossRefGoogle Scholar
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