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there is no name given to these structures
True, no name is given, however it is often used to describe special kinds of addition in abstract algebra and other subjects. For example, in physics, relativistic velocity addition can be defined using it:
v ⊕ u = (v + u‖+ αvu⊥)/(1 - vu/c2)
where αv is the reciprocal of the Lorentz factor, u‖ is the parallel component of velocity and u⊥ is the perpendicular component of velocity.
In other, non-mathematical words, the circle plus is often used to signify a "fancy" type of addition.
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