Colors do not really exist. Light with different frequencies exists and the interpretation of them as colors is something our brain does. It is not just that we inteprtie diffrent wavelengths but the combination of wavelength as colors.
Our eyes have 3 types of cones with different sensitivities at different frequencies and one type of rod that is primary for low-light vision. So our color interpretation is from 4 signals https://midimagic.sgc-hosting.com/huvision.htm
We can fool the eye but emitting just red green and blue light that tries to stimulate one cone each and fool them. That is how a display work, yellow is created by sending out red and green light that stimulates the two relevant cones just like yellow light do
We are not sure how other animals interpret light as colors, you can really be sure another human’s integrity colors the same way as you do. Human interpretation is likely similar to other humans but it might not be exactly the same.
So how a Mantis shrimp interpret the information from the eye is unknown to use. The do have more receptors for the same visible range that we do so you will have an advantage in telling colors apart. But they also see into the ultraviolet range and can detect light polarization. So how they perceive the environment from that data is not something we really know. You can test what light is can see and what wavelength difference it can tell apart but that do not say anything of how it experiences the information.
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