Abstract
Let \({\mathcal {K}}\) be a quasivariety. We say that \({\mathcal {K}}\) is a term quasivariety if there exist an operation of arity zero e and a family of binary terms \(\{t_i\}_{i\in I}\) such that for every \(A \in {\mathcal {K}}\), \(\theta \) a \({\mathcal {K}}\)-congruence of A and \(a,b\in A\) the following condition is satisfied: \((a,b)\in \theta \) if and only if \((t_{i}(a,b),e) \in \theta \) for every \(i\in I\). In this paper we study term quasivarieties. For every \(A\in {\mathcal {K}}\) and \(a,b\in A\) we present a description for the smallest \({\mathcal {K}}\)-congruence containing the pair (a, b). We apply this result in order to characterize \({\mathcal {K}}\)-compatible functions on A (i.e., functions which preserve all the \({\mathcal {K}}\)-congruences of A) and we give two applications of this property: (1) we give necessary conditions on \({\mathcal {K}}\) for which for every \(A \in {\mathcal {K}}\) the \({\mathcal {K}}\)-compatible functions on A coincides with a polynomial over finite subsets of A; (2) we give a method to build up \({\mathcal {K}}\)-compatible functions.
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Acknowledgements
This work was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (PIP 11220200100912CO, CONICET-Argentina) and Agencia Nacional de Promoción Científicas y Tecnológica (PICT2019- 2019- 00882, ANPCyT-Argentina).
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San Martín, H.J. On Relative Principal Congruences in Term Quasivarieties. Stud Logica 110, 1465–1491 (2022). https://doi.org/10.1007/s11225-022-10011-8
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DOI: https://doi.org/10.1007/s11225-022-10011-8