I feel like I'm missing something obvious. What is the best way to split (cut) a set of polygons with some lines?
If I use the Intersector followed by the AreaBuilder I end up with a lot of additional areas created by the lines, and I have a few hoops to jump through to get the attributes back.
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Unless I misunderstand the problem, you should get what you need by first clipping the lines by the areas, then running the clipped lines and the areas into an AreaBuilder. The clipping will prevent the dangling ends of your lines from forming extra areas. You can recover your attributes by running an AreaOnAreaOverlayer.