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Image-finite first-order structures

One of the well known difficulties (see [4, p. 236] or [1]) when dealing with first-order many-valued logics is the requirement to consider safe structures, i.e., those ones which have the necessary infima and suprema for computing the values of all first-order formulas. Besides trivial cases like witnessed structures (which include finite ones) [2], in general it is quite difficult to show that a particular structure is safe. Another quite simple case, and apparently not previously considered in the literature, of safe structures is the one provided by what we call here image-finite. By definition, an structure is image-finite when, for each one of the predicate symbols in the vocabulary, its interpretation only takes a finite number of values in the many-valued chain considered. As previously pointed, it is not difficult to prove that all image-finite structures are safe (indeed witnessed), and another straightforward result is the following one. Lemma. Let us assume that K1 and K2 are two classes of MTL-chains generating the same variety. Then, image-finite structures over chains in K1 and image-finite structures over chains in K2 share the same 1-valid sentences. The previous result can be used to prove the statements below. Theorem (Łukasiewicz Standard Semantics). The following families of first-order structures have the same set of 1-valid sentences.