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I have two overlapping SHPs. One is with two noncontinous polygons - values 0 is blue and 1 is green. Now I have 9 meters grid for the same area, but I need to "fill" the grid with values from previsous shapefile - not by averaged, but by maximum value (it means when one small polygon with value 1 is underlaying the grid, then the whole grid cell has new value 1). Below is the picture. I'm afraid all this spatial join tools does it only by averaged values.

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Look at the answer here: How to get the "Join attributes by location" features on a QGIS Mac version . Use the Join by Location (Summary) tool . you can find it in the Processing Toolbox.

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First create a virtual field (using Field Calculator), use the tools in there to populate the virtual field with "Max(##)" values. The do a spatial join attributes.

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