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Minimal subvarieties of involutive residuated lattices
by
Daisuke Souma
Japan Advanced Institute of Science and Technology
It is known that classical logic CL is the single maximal consistent logic over intuitionistic logic Int, which is moreover the single one even over the substructural logic FLew. On the other hand, if we consider maximal consistent logic over a weaker logic the number of them can be uncountably many. Since the subvariety lattice of a given variety V of residuated lattices is dually isomorphic to the lattice of logics over the corresponding substructural logic L(V), the number of maximal consistent logics is equal to the number of minimal subvarieties (atoms) of the subvariety lattice of V.
Tsinakis and Wille have shown that there exist uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. In the present talk, we will show that while there exist uncountable many atoms in the subvariety lattice of the variety Vm of bounded representable involutive residuated lattices with mingle axiom x2 ≤ x, only two atoms exists in the subvariety lattice of the variety Vi of bounded representable involutive residuated lattices with the idempotency x = x2.
To prove the latter, it suffices to construct two algebras in Vi and to show that every subvarieties of Vi contains the variety generated by either of these two. To prove the former, we employ a method, introduced by Galatos and Raftery, of constructing an involutive residuated lattice from a given residuated lattice, which preserves always mingle axiom. By taking suitable uncountably many upper bounded residuated lattices with mingle axiom and applying the above method, we can get residuated lattices in Vm. So, what remains to us is to show that every distinct algebra among them generates a distinct variety which is an atom of Vm.
We don't know whether the similar technique can work well for the subvariety lattice of the variety of bounded involutive residuated lattices with contraction axiom x ≤ x2.
Date received: April 27, 2007
Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caug-13.