Quantifier Free Definability on Infinite Algebras

An operation f : An→A on the domain A of an algebra A is definable if there exists a first order logic formula Φ(x̅, y) with parameters from A such that for all α̅ ∈ A
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and b ∈ A we have f(α̅) = b iff A |= Φ(α̅, b). The goal of this paper is to study definability of operations by quantifier-free formulas on countable infinite algebras from computability and model-theoretic definability points of view.