Valuations of terms 
 
Authors:  Email author" target="_blank">K?DeneckeEmail author S?L?Wismath 
 
Institution:  (1) Institute of Mathematics, University of Potsdam, PF 601553, 14415 Potsdam, Germany;(2) Dept. of Mathematics and Computer Science, University of Lethbridge, T1K 3M4 Alberta, Canada 
 
Abstract:  Let be a type of algebras. There are several commonly used measurements of
the complexity of terms of type , 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 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. 
 
Keywords:  08A15 08A25 
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