Department of Mathematics
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For a given equational theory Σ, Alfred Tarski defined the set ∇(Σ) as

where |Σ₀| is the cardinality of the set Σ₀. Tarski's Interpolation Theorem states that ∇(Σ) is always an interval and that it is unbounded if Σ has an identity of the form f=x where the term f contains the variable x at least twice. Thus, in particular, for any equational theory Σ of groups of type (2) with the binary operation of, say the right division, , we have ∇(Σ) = [1, ∞). Tarski's 1975 proof is existential—he uses topological tools like closure operations to show the existence of an independent basis of cardinality n for all n. In this talk, we give constructive proofs for Tarksi's "unbounded theorem" for several examples of group-like varieties by constructing independent bases with n for all n. Then we compare this with lattice-like varieties.

1. Alfred Tarski, Equational logic and equational theories of algebras. Contributions to Math. Logic, North-Holland, Amsterdam, 1968.
2. George McNulty, Minimum bases for equational theories of groups and rings, Ann. Pure Appl. Logic 127 (2004), 131--153.
3. D. Kelly and R. Padmanabhan, Irredundant self-dual bases for self-dual lattice varieties. Algebra Universalis 52 (2004), 501Ð517
4. William McCune and R. Padmanabhan, Tarski Theorems, Argonne National Lab Lecture Notes, 2006. (produced under Contract No. W-31-109-ENG-38 with the U.S. Department of Energy).