Morita Equivalence for Unary Varieties

Abstract

The problem of determining all varieties W (categorically) equivalent to a given variety V, which Isbell first posed in the early 1970´s, has been addressed in numerous ways. McKenzie (1996) and Porst (2000) describe a general solution consisting of a two step construction. For a given variety V, first its n-th matrix powers are constructed for natural numbers n, and then McKenzie´s u-modification for idempotent and invertible terms u is applied to the matrix powers. Thus, one obtains all varieties equivalent to V. This thesis contains new descriptions for the matrix power of a (finitary) variety and for its u-modification. The syntax of these new constructions is considerably simpler than the syntax of the original constructions. The matrix power of an arbitrary variety V is characterized by adding just one binary operation to the original operations of the given variety V as well as adding equations to the original ones. The u-modification is more elusive and we have to restrict ourselves to the unary case. A solution is given for a large class of unary varieties. Again a characterization is obtained by adding operations and equations to the originally given ones. Furthermore some special cases like the variety of Boolean algebras and the variety of sets are treated as illustrating examples.