Department of Mathematics
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To prove negative results, we need counterexamples, very often a large class of models from which we could choose or custom-design our counterexamples. Isotopes of rings offer such a rich variety of models to choose from. In this talk, I will present a few examples of say, "non-theorems" from (the Berkeley School of) equational logic which make systematic use of ring models. Here is one such result from Alfred Tarski and Thomas Green: A finitely based equational theory of groups of type ? is one-based iff |?| < 3. In particular, the usual treatment of groups with three fundamental operations of binary multiplication, unary inverse and the nullary identity is two-based but not one-based. Ring isotopes come very handy to prove such a class of theorems - not only in group theory but also in lattice theory.

References

1. A. Tarski, Equational logic and equational theories of algebras. Contributions to Math. Logic, North-Holland, Amsterdam, 1968.
2. R. Padmanabhan, On single equational-axiom systems for abelian groups. J. Austral. Math. Soc. 9 1969 143--152.
3. R. Padmanabhan; Barry Wolk, Equational theories with a minority polynomial. Proc. Amer. Math. Soc. 83 (1981), 238--242.
4. G. McNulty, Minimum bases for equational theories of groups and rings: the work of Tarski and Green, Ann. Pure Appl. Logic 127 (2004).