Abstract
The following classes of equational theories, which are important in unification theory, are presented: permutative, finite, Noetherian, simple, almost collapse free, collapse free, regular, and Ω-free theories. The relationships between the particular theories are shown and the connection between these classes and the unification hierarchy is pointed out. We give an equational theory that always has a minimal set of unifiers for single equations, but there exists a system of two equations which has no minimal set of unifiers. This example suggests that the definition of the unification type of an equational theory has to be changed. Furthermore we study the conditions, under which minimal sets of unifiers always exist.Decidability results about the membership of equational theories to the classes above are presented. It is proved that Noetherianness, simplicity, almost collapse freeness and Ω-freeness are undecidable. We show that it is not possible to decide where a given equational theory resides in the unification hierarchy and where in the matching hierarchy.
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