Department of Mathematics
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The concept of a one-based equational class of algebras, i.e. a class consisting of all models of a single identity, is a model theoretic notion and it was Alfred Tarski who in 1968 first proposed the problem of providing a "purely model-theoretic" characterization for this concept. The inherent difficulty of this problem was clearly brought out by Tarski himself: a mathematical characterization of one- based equational classes cannot be expressed purely in terms of subalgebras, direct products, homomorphisms, isomorphisms, categorical equivalence etc. The problem is still open.

As in a mathematical investigation of any open problem, we need a large class of non-trivial positive examples illustrating the phenomenon in question and an equally interesting class of negative examples. For instance, while the equational theory of rings with unity is one-based (see [3]), the theory of rings (without specifically assuming 1 in the type) has no single axiom!

In this talk, we give a survey of general techniques developed to construct equational theories (of groups, rings, lattices etc.) with single axioms and a large class of negative examples - all discovered during the 70-year period 1938-2008.

References.

1. Alfred Tarski, Ein Beitrag zur Axiomatic der Abelschen Gruppen, Fund. Math. 30 (1938).
2. Alfred Tarski, Equational logic and equational theories of algebras. Contributions to Math. Logic, North-Holland, Amsterdam, 1968.
3. G. Gratzer and R. Padmanabhan, Symmetric difference in abelian groups, Pacific J Math 74 (1978) 339-347.
4. George McNulty, Minimum bases for equational theories of groups and rings, Ann. Pure Appl. Logic 127 (2004), 131--153.
5. William McCune and R. Padmanabhan, Tarski Theorems, Argonne National Lab Lecture Notes, 2005. (produced under Contract No. W-31-109-ENG-38 with the U.S. Department of Energy).