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As in classical model theory, the construction of ultraproducts will be a very important tool in our development of modal model theory. Given an arbitrary non-empty index set I and a sequence <𝕬 i :i∈I> of modal structures for the language ML, we define the cartesian product X i∈I 𝕬i to be the following modal structure <B l,L,S,P,M> for ML. For each i, let 𝕬i = <A k i , Ki, Ri, Oi, Ni>. Then set L = X i∈I Ki the ordinary set-theoretic cartesian product, and for l,ĺ ∈L, define l Slĺ iff ∀i∈I[l(i)R iĺ(i)].
KeywordsModal Structure Regular Cardinal Canonical Embedding Existential Closure Elementary Embedding
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