### LANS Publications

# "Tarski Theorems on Self-Dual Equational Bases for Groups"

R. Padmanabhan, W. McCune

Preprint ANL/MCS-P1092-0903

Preprint Version: [pdf]

We present independent self-dual equational bases of arbitrarily large finite sizes for the equational theory of groups treated as varieties of various well-known types. Here the dual of a term f is the mirror reflection of f. For each type of group theory, we provide an independent self-dual basis with n

identities for n = 2, 3, 4. Then we develop a simple algorithmic procedure to construct independent self-dual equational bases of arbitrary finite sizes in such a way that the new larger equational bases depend explicitly on the initial bases of small sizes. Applying this ``expansion'' procedure, we show that every finitely based variety of groups can be defined by an independent

self-dual set of n identities for all n >= 2. Apart from generalizing the various theorems of Alfred Tarski who initiated this topic in the late 1960s, these proofs also provide explicitly the bases and hence may be construed as the first constructive proof of Tarski's theorems as well.