On the subject of "self" as memory, I think this is an oversimplification. The khandha/cetasika perception or sanna, according to abhidhamma, is what is resposible for memory and is only one component of the functional construction of "self-notion." From Cetasikas
by Nina van Gorkon:
Perception has the characteristic of perceiving by on act of general inclusion, and the function of making marks as a condition for repeated perception (for recognizing or remembering) (I am using the translation of the ven. Nyanaponika, Abhidhamma Studies, page 69, BPS, Kandy, 1976), as when woodcutters 'perceive' logs and so forth. Its manifestation is the action of interpreting by means of the sign as apprehended.
Sanna is not the same as citta which is the 'leader' in cognizing an object. As we have seen, sanna recognizes the object and it 'marks' it so that it can be recognized again. This is explained by way of a simile: carpenters put tags or signs on logs so that they can recognize them at once by means of these marks. This simile can help us to understand the complex process of recognizing or remembering. What we in conventional language call "remembering" consists of many different moments of citta and each of these moments of citta is accompanied by sanna which connects past experiences with the present one and conditions again recognition in the future. This connecting function is represented by the words 'recognition' and 'marking' (1 See Abhidhamma Studies, by the Ven. Nyanaponika, 1976, page 70, where it is explained that the making of marks and remembering is included in every act of perception.)
As the Ven . Dr. Walpole Rahula explains in What the Buddha taught, "self" is a functional creation of the khandhas so the khandas operate together. Otherwise it would be like five horses attempting to pull a cart in five different directions.
So defining "self" as memory would be the same as defining it as a function of sanna, which I think would be incomplete and unsupported by sutta or abhidhamma.
Yes, self is not just memory, but memory is the "rupa" of self. Ego is made of memory and leaded by memory, conditioned by memory and die with memory. What is conditon to this memory (consciosness, body etc), is an another question, but self-rupa, is memory.
I cant find "my self" out of my memory, and i cant concider consciosness like part of "my self".
Imagine, you have a party, you drunk a lot
, you wake up
, and you friend tel you see that, it's you who does it
. You tell that it's not you, you dont remember that you does it.
So you have not memory about it, but if we return in the time, at the moment when that accident was done, you have all, consciosness, citta, sanna every you want, but this information is not a part of your memory, it's gone, so you cant say that it was you, this accident is not a part of your "I".
So if the memory is the "self"? - yes.
If this definition of self must be completed to work with it? - No. Perharps some one have to know much more details about "self-nature" to stop concider himself like a self, to stop apropriate this "self", and finaly be liberated from it, but i think is not the essential information to be freed from "self-view".
PS: ""self" is a functional creation of the khandhas so the khandas operate together." - it's sounds to complicate, i dont understant it. Dhamma is simple, dry and dirrect, can be seen right now, and actualy, with all respect to the Ven . Dr. Walpole Rahula, i cant see what he said. If some one can traduse it in sample language i will be gentil.