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Let 𝕬 and 𝔅 be modal structures for ML and let m: 𝕬 → 𝔅. If m is a Γ-morphism we say that 𝔅 is a Γ-extension of 𝕬 via m. A set Γ of formulas of ML is said to be regular if it includes all formulas of the forms x = y and Ax1…xn[y1…yn] whenever it comtains A. A formula A of ML is said to be a * *-formula if it is obtained from a formula of the form θ h * (cf. Definition 6.8), where h was a basic formula bundle over 𝕬, by replacing all names of elements of |𝒜 0| by free variables (so that ∧h 2(O) becomes a conjunction of basic formulas of ML).
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