user-1: I was reading the SEP article on the two truths
According to the abhidharmakosa:

An entity, the cognition of which does not arise when it is destroyed
  and, mentally divided, is conventionally existent like a pot and
  water. Ultimate existence is otherwise

The article adds that:

For a thing to be ultimately real is for that thing to be
  “foundationally existent” (dravya-sat / rdzas yod)

Does any Buddhist school claim that what arises when it is divided or destroyed, whether or not that exists etc., is the self?

user6994: According to Tibetan system of tenets, the Vaibhāṣikas so-called 'Hinayana school of tenets' defines an ultimte truth as:

A phenomenon such that an awareness apprehending it is not cancelled
  if it is broken up or mentally separated into its individual parts.

Illustrations of ultimate truths are: 

directionally partless particles, temporally partless [moments of]
  consciousness, uncompounded phenomena, any of the five aggregates.


Mahayana tenets assert that Vaïbashikas do not posit selflessness of phenomena, but merely [a grosser] selflessness of persons. Jetsün Chökyi Gyaltsen writes:

A selflessness of phenomena is not accepted because it is accepted
  that if it is an established base it is necessarily a self of
  phenomena.


-1: <p>I was reading the <a href="http://plato.stanford.edu/entries/twotruths-india/#AbhConTru" rel="nofollow">SEP</a> article on the two truths</p>

<p>According to the abhidharmakosa:</p>

<blockquote>
  <p>An entity, the cognition of which does not arise when it is destroyed
  and, mentally divided, is conventionally existent like a pot and
  water. Ultimate existence is otherwise</p>
</blockquote>

<p>The article adds that:</p>

<blockquote>
  <p>For a thing to be ultimately real is for that thing to be
  “foundationally existent” (dravya-sat / rdzas yod)</p>
</blockquote>

<p>Does any Buddhist school claim that what arises when it is divided or destroyed, whether or not that exists etc., is the self?</p>
 6994: <p>According to Tibetan system of tenets, the Vaibhāṣikas so-called 'Hinayana school of tenets' defines an ultimte truth as:</p>

<blockquote>
  <p>A phenomenon such that an awareness apprehending it is not cancelled
  if it is broken up or mentally separated into its individual parts.</p>
</blockquote>

<p>Illustrations of ultimate truths are: </p>

<blockquote>
  <p>directionally partless particles, temporally partless [moments of]
  consciousness, uncompounded phenomena, any of the five aggregates.</p>
</blockquote>

<p><br>
Mahayana tenets assert that Vaïbashikas do not posit selflessness of phenomena, but merely [a grosser] selflessness of persons. Jetsün Chökyi Gyaltsen writes:</p>

<blockquote>
  <p>A selflessness of phenomena is not accepted because it is accepted
  that if it is an established base it is necessarily a self of
  phenomena.</p>
</blockquote>
