# zbMATH — the first resource for mathematics

Category equivalence preserves unification type. (English) Zbl 0902.08014
Let $$\Lambda =\{f_{i}(x)=g_{i}(x): 1\leq i\leq k\}$$ be a system of equations in the variables $${x}=(x_1, \dots , x_r)$$. A solution of $$\Lambda$$ in a variety $${\mathcal V}$$ is a homomorphism $$\alpha : F_{r}(x_1, \dots , x_r) \to F_{r}(Z)$$ (where $$Z$$ is countably infinite) such that $$(f_{i}, g_{i}) \in \text{ker }\alpha$$, $$i=1, \dots , k$$. Write $$\text{Sol}_{\mathcal V}(\Lambda)$$ for the set of all solutions. Let $$\text{Sol}_{\mathcal V}(\Lambda)/ \sim$$ be the partially ordered set of equivalence classes of solutions. A system $$\text{Sol}_{\mathcal V}(\Lambda)/ \sim$$ is called unitary (finitary, infinitary) if the set of minimal elements in $$\text{Sol}_{\mathcal V}(\Lambda)/ \sim$$ consists of one element (finitely, infinitely many elements), and if every element in $$\text{Sol}_{\mathcal V}(\Lambda)/ \sim$$ lies above some minimal element. A system $$\Sigma$$ is called nullary if none of the preceding cases occurs. The unification type of a variety $${\mathcal V}$$ is the worst of the above possibilities which occurs for some system $$\Lambda$$ of equations over $${\mathcal V}$$.
The aim of this paper is to show that equivalent varieties $${\mathcal V}$$ and $${\mathcal W}$$ have the same unification type, in fact for every system of equations $$\Lambda$$ over $${\mathcal V}$$ there is a system of equations $$\Lambda '$$ over $${\mathcal W}$$ such that $$\text{Sol}_{\mathcal V}(\Lambda)/ \sim _{\mathcal V}$$ and $$\text{Sol}_{\mathcal W}(\Lambda ')/ \sim _{\mathcal W}$$ are isomorphic partially ordered sets. The proof of the main theorem is based on Ralph McKenzie’s algebraic criterion for the equivalence as categories of varieties. The last section of the paper deals with some applications of the main result for varieties generated by finite ordered sets.
Reviewer: J.Paseka (Brno)

##### MSC:
 08B20 Free algebras 08B05 Equational logic, Mal’tsev conditions 03C05 Equational classes, universal algebra in model theory 18C05 Equational categories
Full Text:
##### References:
 [1] Albert, M. H. andLawrence, J.,Unification in varieties of groups I: nilpotent groups. Canadian Journal of Mathematics,46 (1994), 1135-1149. · Zbl 0827.20047 · doi:10.4153/CJM-1994-064-8 [2] Albert, M. H. andWillard, R.,Unification in locally finite varieties, preprint 1992. [3] Burris, S. andSankappanavar, H. P.,A Course in Universal Algebra, Springer-Verlag, New York, 1981. · Zbl 0478.08001 [4] Nipkow, T.,Unification in primal algebras, their powers and their varieties, J.A.C.M.,37 (1990), 742-776. · Zbl 0711.68092 [5] MacLane, S.,Categories for the Working Mathematician, Springer-Verlag, New York, 1971. · Zbl 0705.18001 [6] McKenzie, R.,Algebraic Morita theorem for varieties, preprint, August 1992. [7] McKenzie, R., McNulty, G. andTaylor, W.,Algebras, Lattices, Varieties, Wadsworth/Brooks-Cole, Monterrey, 1987. [8] Siekmann, J.,Unification theory, J. Symbolic Computation,7 (1989), 207-274. · Zbl 0678.68098 · doi:10.1016/S0747-7171(89)80012-4 [9] Taylor, W.,The fine spectrum of a variety, Algebra Universalis,5 (1975), 263-303. · Zbl 0336.08004 · doi:10.1007/BF02485261
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.