opencv morphological dilation filter as maximum filter

Like the definition of median filter, I can define "maximum filter" as for a local window

e.g. dst(x,y) = max(3x3 local window pixels)

But I cannot find such a filter in opencv, the closest one is "dilate" function

Then I use the default configuration of "dilate" function, but the result is incorrect compare to my brute force implementation of the maximum filter.

I found that for 3x3 case, the equivalent dilate configuration is to use a 1x1 rectangular structure element, that is

dilate(src, dst, getStructuringElement(MORPH_RECT, Size(1,1)))

My questions are:

1. It seemed counter-intuitive for me that I thought a structure element is equivalent to a local window. But now the local window's size is only 1x1 pixel?

2. What if my maximum filter's local window expands(e.g. 5x5, 7x7)? Is there any relationship between the maximum filter and dilation filter?

An implementation example in Mathematica:

But Dilation[ ] also accepts a structuring kernel: