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DOMINIONS AND PRIMITIVE POSITIVE FUNCTIONS

Published online by Cambridge University Press:  05 February 2018

MIGUEL CAMPERCHOLI
MIGUEL CAMPERCHOLI*
Affiliation:
CIEM — FACULTAD DE MATEMÁTICA, ASTRONOMÍA Y FÍSICA (FA.M.A.F.) UNIVERSIDAD NACIONAL DE CÓRDOBA — CIUDAD UNIVERSITARIA CÓRDOBA 5000, ARGENTINAE-mail:camper@famaf.unc.edu.ar
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Abstract

Let A ≤ B be structures, and ${\cal K}$ a class of structures. An element b ∈ B is dominated by A relative to ${\cal K}$ if for all ${\bf{C}} \in {\cal K}$ and all homomorphisms g, g' : B → C such that g and g' agree on A, we have gb = g'b. Our main theorem states that if ${\cal K}$ is closed under ultraproducts, then A dominates b relative to ${\cal K}$ if and only if there is a partial function F definable by a primitive positive formula in ${\cal K}$ such that FB(a1,…,an) = b for some a1,…,an ∈ A. Applying this result we show that a quasivariety of algebras ${\cal Q}$ with an n-ary near-unanimity term has surjective epimorphisms if and only if $\mathbb{S}\mathbb{P}_n \mathbb{P}_u \left( {\mathcal{Q}_{{\text{RSI}}} } \right)$ has surjective epimorphisms. It follows that if ${\cal F}$ is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by ${\cal F}$ has surjective epimorphisms.

Keywords

18A2003C0503C6003C4008C15epimorphismdominionepic substructureBeth definabilitydefinable functionIsbell’s Zigzag Theorem
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 40 - 54
Copyright
Copyright © The Association for Symbolic Logic 2018 

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