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A set with an arbitrary binary operation is called a magma.

A set of dots with a set of arrows between them is called a quiver.

A category is a quiver with a binary operation on the arrows obeying some conditions.

Is there a name for quivers with an arbitrary binary operation on the arrows (that doesn't have to obey any conditions)?

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Do you have any evidence that this concept is meaningful enough to deserve a name? –  Rasmus Oct 2 '12 at 14:34

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