Equivalence for Varieties in General and for BOOL in Particular

The varieties W equivalent to a given variety V are characterized in a purely categorical way. In fact they are described as the models of those Lawvere theories which are Morita equivalent to the Lawvere theory of V which therefore are characterized first. Along this way the conceptual meanings of the n–th matrix power construction of a variety and McKenzie’s σ– modification of classes of algebras [22] become transparent. Besides other applications not only the well known equivalences between the varieties Pm of Post algebras of fixed orders m and the variety BOOL of Boolean algebras are obtained; moreover it can be shown that the varieties Pm are the only varieties equivalent to BOOL. The results then are generalized to quasivarieties and more general classes of algebras. Mathematics Subject Classification (1991): Primary 08B99, 08C05, Secondary 03C05, 03E05, 06D25