Plotting 2D slices of CylindricalDecomposition objects?

Questions
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1

I want to plot 2D slices of eight objects, K1,...,K8, in (x,y,z) \in \mathbb{R}^3_+. I am happy for the slices to be parallel to the planes.

The objects are returned from CylindricalDecomposition. Currently, I am plotting

RegionPlot3D[{K1, K2, K3, K4, K5, K6, K7, K8},{x,0,4},{y,0,4},{z,0,4}]

I would like something like

RegionPlot[{K1, K2, K3, K4, K5, K6, K7, K8},{x,0,4},{y,0,4},{z=i}]

for different fixed values of i.

One way of doing this would be to add z=i as a constraint to K1,...,K8, but this seems labour intensive.

2

In my original formulation, each K[i] contains the result of a CylindricalDecomposition call with variables \{x,y,z\}.

Thus, the problem with my RegionPlot above is that fixing z=i passes two variables (x and y) and a constant (z=i) to a CylindricalDecomposition expecting three variables. This yields error messages like “1 is not a valid variable”.

Here is a MWE for a solution:

arg1[x_, y_, z_] := {x >= y && y >= z && z >= 0}
RegionPlot3D[Evaluate[CylindricalDecomposition[arg1[x, y, z], {x, y, z}]], {x, 0, 2}, {y, 0, 2}, {z, 0, 2}]
RegionPlot[Evaluate[CylindricalDecomposition[arg1[x, y, 0], {x, y}]], {x, 0, 2}, {y, 0, 2}]