Let E be an equational theory and =E be the equality of terms induced by E. A substitution 8 is called an E-unifier of the pair of terms s, t iff sf3 =E tt9. The set of all E-unifiers of s, ? is denoted by UE(s, t). We are mostly interested in complete sets of E-unifiers, i.e., sets of E-unifiers from which UE(s, t) may be generated by instantation. More formally, we extend =E to UE(s, t) and define a quasi-ordering sE on UE(s, t) by l 13 =E r iff xt9 =E xr for all variables x occurring in s or t, l 7 <E B iff there exists a substitution X such that 8 =E T 0 X. A complete set cU,(s, t) of E-unifiers of s, t is defined as (I) cG(s9 t) c UE(& t), (2) for all 0 E U&s, t) there exists a u E cU,(s, t) such that u =& 8. A set of most general E-unifiers &(s, t) is a complete set of E-unifiers of s, t satisfying the minimality condition: (3) for all 6, T in pU,(s, t), u dE r implies u = 7. This notion was introduced by Plotkin [a]. In the same paper he conjectured that there exists an equational theory E and terms s, t such that UE(s, t) does not exist. We say that such a theory is of unification type zero. The first example of a type zero theory is due to Fages and Huet [4,5]. They constructed a theory E and terms s, t such that UE(s, t) contains a strictly decreasing chain with respect to Q~ which is a complete set of E-unifiers of s, t. Under these circumstances, no complete set of E-unifiers of s, t satisfies the minimality condition (see [4,5] or Section 3 where we use Fages-Huet’s method to show that the respective theory has type zero). Schmidt-Schauss [7] and the present author [l] showed that the theory of idempotent semigroups is of unification type zero and in (21 we have proved that almost all varieties of idempotent semigroups are defined by type zero theories. In this case, complete sets of unifiers are not simply chains but we also used a decreasing chain of unifiers without lower bound -which was not complete but satisfied another additional condition-to establish our result. In the present paper we show that these additional conditions are not superfluous, i.e., that it is not sufficient to find a strictly decreasing chain without lower bound in UE(s, t) for some terms s,? to prove that E has unification type zero. But note that-by Zom’s lemma-the existence of such a chain is a necessary condition for E to be type zero. An example which establishes the same result can be found in [3]. The authors of [3] use a theory which has only unary function symbols.
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