(I have in mind here, something similar to "uniqueness" proofs in math. I think if we want to name something, we should show both that it exists and is unique.)
(By "divine attributes", I mean attributes like omnipotent, omniscient, omnibenevolent, all-loving, eternal, unchanging, necessarily existing, being worthy of worship, being the first cause or being the cause/ground of everything else, and so on. Of course, some theists of different stripes may disagree about whether God possesses all or only some of these attributes, and exactly how each attribute should be defined.)
(By "God-like entity", I mean something that possesses some of the divine attributes.)
Some background: I was discussing with a friend yesterday about various arguments for the existence of God. We agreed that each of the arguments, if successful, establishes the existence of an entity which possesses some of the divine attributes. For example, we might say that some versions of the cosmological argument establish the existence of an entity that necessarily exists, is immaterial, and acts as some kind of first cause or ground of material or contingent beings. Likewise, some versions of the design argument might establish the existence of an intelligence that "guides" or plans the state and laws of the physical world. And the moral argument, if successful, shows the existence of an entity that is the ground of moral truths and goodness. (Whatever, the exact arguments and what they show aren't particularly relevant to the main question that I'm about to ask. Just that they purport to show the existence of an entity that possesses some of the divine attributes.) But how might we go about showing that all of these entities that each argument argues for, are in fact the same one entity? Or, how might we try to show that there can be at most one God-like entity?
Another weaker way to phrase it: Are there any arguments that try to show that for some of the divine attributes, there can't be distinct entities that possess some combination of those attributes. For example, arguments that show things like for any x and any y, if x is omnipotent and y is omniscient, then x must be identical with y. An example I kind of sketched to my friend, but don't know how to argue for properly, is that there cannot be two distinct omnipotent beings. Intuitively, I feel that they would "encroach" upon each others omnipotence, but I don't know how to make this point precise.