Let τ be a type of algebras. There are several commonly used measurements of the complexity of terms of type τ, including the depth or height of a term and the number of variable symbols appearing in a term. In this paper we formalize these various measurements, by defining a complexity or valuation mapping on terms. A valuation of terms is thus a mapping from the absolutely free term algebra of type τ into another algebra of the same type on which an order relation is defined. We develop the interconnections between such term valuations and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; valuations of terms offer a new method to study complete sublattices of this lattice.
Mathematics Subject Classification (2000):08A15, 08A25.
Keywords:Valuation of terms lattice of varieties k-normalization of a variety
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