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 of closed formulas in a first order language τ is K -satisfiable, if there exists a τ-structureM such that ‖σ ‖M∈K , for any σ∈ . As a consequence, the usual compactness property can be replaced by the K -compactness property. In this paper, the K -compactness property for Lukasiewicz first order logic is investigated. Using the ultraproduct construction, it is proved that for any closed subset K and set of closed formulas, is K -satisfiable if and only if it is finitely K -satisfiable.