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We investigate Grothendieck rings appearing in real geometry, notably forarc-symmetric sets, and focus on the relative case in analogy with theproperties of the ring of algebraically constructible functions defined byMcCrory and Parusinski. We study in particular the duality and link operators,including its behaviour with respect to motivic Milnor fibres with signs.

On relative grothendieck rings and algebraically constructible functions