Compactness in first order Łukasiewicz logic

For a subset K ⊆ [0, 1], the notion of K-satisfiability is a generalization of the usual satisfiability in first order fuzzy logics. A set Γ of closed formulas in a first order language τ is K-satisfiable, if there exists a τ-structure $\mathcal{M}$ such that ∥ σ ∥ ${}_\mathcal{M}$  ∈ K, for any σ ∈ Γ. As a consequence, the usual compactness property can be replaced by the K-compactness property. In this paper, the K-compactness property for Łukasiewicz first order logic is investigated. Using the ultraproduct construction, it is proved that for any closed subset K and set Γ of closed formulas, Γ is K-satisfiable if and only if it is finitely K-satisfiable.