Let T be a simple L-theory and let \(T^-\) be a reduct of T to a sublanguage \(L^-\) of L. For variables x, we call an \(\emptyset \)-invariant set \(\Gamma (x)\) in \({\mathcal {C}}\) a universal transducer if for every formula \(\phi ^-(x,y)\in L^-\) and every a,$$\begin{aligned} \phi ^-(x,a)\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma (x)\wedge \phi ^-(x,a)\ L\text{-forks } \text{ over }\ \emptyset . \end{aligned}$$We show that there is a greatest universal transducer \(\tilde{\Gamma }_x\) (for any x) and it is type-definable. In particular, the forking topology on \(S_y(T)\) refines the forking topology on \(S_y(T^-)\) for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that \(\tilde{\Gamma }_x\) is the unique universal transducer that is \(L^-\)-type-definable with parameters. If \(T^-\) is a theory with the wnfcp (the weak nfcp) and T is the theory of its lovely pairs of models we show that \(\tilde{\Gamma }_x=(x=x)\) and give a more precise description of the set of universal transducers for the special case where \(T^-\) has the nfcp.

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关于简化理论的分叉拓扑

让牛逼是一个简单的大号-理论，让\（T ^ - \）是一个简牛逼的子语言- \（\ L ^）的大号。对于变量x，如果对于每个公式\（\ phi ^-，则将\（{ emptyset \）-不变量集\（\ Gamma（x）\）在\（{\ mathcal {C}} \）中称为通用换能器（x，y）\ in L ^-\）和每个a，$$ \ begin {aligned} \ phi ^-（x，a）\ L ^-\ text {-forks} \ text {over} \ \ emptyset \ \ text {iff} \ Gamma（x）\ wedge \ phi ^-（x，a）\ L \ text {-forks} \ text {over} \ \ emptyset。\ end {aligned} $$我们展示了最棒的通用换能器\（\ tilde {\ Gamma} _x \）（对于任何x），并且它是类型定义的。特别是，\（S_y（T）\）上的分支拓扑针对所有y改进了\（S_y（T ^-）\）上的分支拓扑。此外，我们根据Stone空间上的某些拓扑描述了通用换能器的集合，并表明\（\ tilde {\ Gamma} _x \）是唯一的通用换能器，它可以（L ^-\） -类型定义与参数。如果\（T ^-\）是具有wnfcp（弱nfcp）的理论，而T是其可爱模型对的理论，则表明\（\ tilde {\ Gamma} _x =（x = x）\）并针对\（T ^-\）具有nfcp的特殊情况给出通用换能器集的更精确描述。