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I'm currently looking into the representation theory of Scott domains. In his paper "dI-Domains as prime information systems" (1992), Guo-Qiang Zhang uses prime information systems to represent dI-domains. A prime information system is an information system where additionally we have linearity and finiteness of entailment, in the following sense:

  1. $X \vdash b$ implies $\{a\} \vdash b$, for some $a \in X$, and
  2. the deductive closures $\overline{X} = \{b \mid X \vdash b\}$ are finite.

If $A$ is a (prime) information system, write $Pt(A)$ for the points (also "elements", "ideals") of $A$, that is, for the usual consistent and deductively closed (possibly infinite) sets of tokens in $A$; for the finite ones write $Pt_f(A)$. Then, a stable approximable mapping between two prime information systems with carriers $A$ and $B$, according to Zhang, is a relation $r \subseteq Pt_f(A) \times B$, such that

  1. if $(a_i,p_i) \in r$, for $i = 1, \ldots, m$ and $\bigcup_i a_i \in Con_A$, then $\{p_1, \ldots, p_m\} \in Con_B$,
  2. if $a \cup b \in Pt_f(A)$, and $(a, p) \in r$, $(b, p) \in r$, then $a=b$, and
  3. if $(a, p) \in r$ and $\{p\} \vdash_B q$, then $(b, q) \in r$, for some $b \subseteq a$.

Now in Definition 5.4 he writes:

Let $A = (A, Con_A, \vdash_A)$ and $B = (B, Con_B, \vdash_B)$ be prime information systems. A molecule $m$ is a finite stable approximable mapping, such that for some $(a , p) \in m$, $b \subseteq a$ and $\{p\} \vdash q$, for any other $(b , q)$ in $m$.

He then uses molecules as tokens of an appropriate function space and goes on to show cartesian closure of the category of prime information systems with stable approximable mappings. My problem is that my intuition is too low to allow me to even parse the definition.

My questions: What is the defining formula of a molecule, and in particular, how are these $b$'s quantified? What would be your intuition of a molecule?

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Restating the definition to make the quantifiers easier to understand:

A molecule is a finite stable approximable mapping, such that there exists a largest pair $(a,p) \in m$, such that for all other pairs $(b,q) \in m$ we have $b \subseteq a$ and $\{p\} \vdash q$.

In other words, a molecule is a mapping that closes off neatly, having a largest pair.

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  • $\begingroup$ I see! So, a fortiori, a molecule has no inconsistencies on the left whatsoever. Nice, it makes perfect sense now. $\endgroup$ – Basil Jan 11 '15 at 10:46
  • $\begingroup$ (I wouldn't say "largest" though; we have both left and right maximality for these pairs, but they live in a contravariant space.) $\endgroup$ – Basil Jan 11 '15 at 10:49

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