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For a more detailed presentation of "unit", read the Ayn Rand Lexicon entry on unit. Objectivism does not generally use or rely on the term "referent", which is used in other approaches, and which is not well defined. As long as you don't import anything philosophically sketchy from the term "referent" besides "that which a thing refers to", then it's okay to talk about a "referent". "Unit" does not imply any act of referring, but concepts do refer, to units (which are existents). The label (word) attached to a concept refers to those existents. To give a concrete example, dog #1 is an existent, and it is a unit, but the dog does not refer to itself – it is itself. The dog's name, such as "Poika", refers to the specific existent, and the word "dog" refers to that existent, as well as many others.

A proper noun such as "Joe the Horse" is not an invalid concept. This is because it is not even a concept. Also, the alternative of valid or invalid does not apply to names. "Presidency in Saudi Arabia" - "Presidency" is an abstract high level concept in the area of politics. As a concept of method it would be a valid concept even if there were no presidents because there were presidents in the past and could be presidents in the future. "Presidency in Saudi Arabia" can be used validly when advocating a change in the method of governance of Saudi Arabia, even though it is true that there are no current or past presidents in Saudi Arabia. "The President of Saudi Arabia" could be used validly in a conditional or future tense, but would be nonsensical in the context of current events or the history of S.A. Context matters. We have more concepts than we have words for them. A single word can refer to several concepts and the ambiguity is usually resolved by the context and careful writing or speaking. Actual concretes can be the referents of many different valid concepts. Concepts of concepts can divided up in several alternative yet valid ways as well. There is no one-to-one correspondence between words and concepts or between concepts and referents. An invalid concept is still referred to as a concept, because if it wasn't a concept at all no rules would apply and there would be no justification to judge it as valid or invalid. So in the sense of badly formed and thus invalid concepts there can be a concept with zero referents.