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Top and Bottom Varieties of Pseudo-MV Algebras
by
W. Charles Holland
University of Colorado
Pseudo-MV-algebras are the non-commutative generalization of MV-algebras, which are the algebras associated with Lukasiewicz multi-valued logic. They are closely associated with lattice-ordered groups. These subjects arose in the early twentieth century inspired, partly, by problems arising from quantum mechanics.
Varieties of these algebras are equationally defined classes. The lattice of varieties of pseudo-MV-algebras is essentially identical to the lattice of varieties of lattice-ordered groups with a specified strong unit. The lattice of varieties of (commutative) MV-algebras was completely described by Komori (1981). Much, but not all, is known about the lattice of varieties of lattice-ordered groups. Varieties of the non-commutative pseudo-MV-algebras have only recently been investigated. I will describe those varieties which are"near the top" and those "near the bottom" of the lattice of all varieties of pseudo-MV-algebras.
This is on-going work with A. Dvurecenskij, A. Glass, and M. Darnel.
References
1. Holland, W. C., Small varieties of lattice-ordered groups
and MV-algebras, in Contributions to General Algebra 16(2005),
107-114, Heyn, Klagenfurt.
2. Holland, W. C., Covers of the Boolean variety in the lattice of varieties of unital lattice-ordered groups and GMV-algebras, Izvrannie Voprosi Algebra (2007), Memorial Volume for N. Ya. Medvedev, 208-217.
3. Holland, W. C., and Dvurecenskij, A., Top varieties of generalized MV-algebras and unital lattice-ordered groups, Comm. Algebra 35(2007), no. 11, 3370-3390..
4. Holland, W. C., and Dvurecenskij, A., Komori's characterization and top varieties of GMV-algebras, (to appear in Algebra Universalis).
5. Holland, W. C., and Dvurecenskij, A., Covers of the Abelian variety of generalized MV-algebras , (in progress).
6. Holland, W. C., and Glass, A., Top periodic varieties of unital lattice-ordered groups, (in progress).
7. Holland, W. C., and Darnel, M., Bottom varieties of unital lattice-ordered gro
Date received: May 23, 2008
Copyright © 2008 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 # caxi-08.