Category equivalence preserves unification type.

*(English)*Zbl 0902.08014Let \(\Lambda =\{f_{i}(x)=g_{i}(x): 1\leq i\leq k\}\) 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.

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 |

##### Keywords:

unification problem; finitary systems of equations; equivalence classes of solutions; unitary systems of equations; infinitary systems of equations; nullary systems of equations
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\textit{M. H. Albert}, Algebra Univers. 36, No. 4, 457--466 (1996; Zbl 0902.08014)

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##### References:

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