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<Paper uid="E99-1009">
  <Title>Geometry of Lexico-Syntactic Interaction</Title>
  <Section position="3" start_page="0" end_page="61" type="intro">
    <SectionTitle>
F::=A IFV rlF/FIF-F
..4 ::= S I N I CN I PP I ...
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    <Paragraph position="0"> The categories in A are referred to as atomic and correspond to the kinds of expressions which are considered to be &amp;quot;complete&amp;quot;. Fairly uncontroversially, this class may be taken to include at least sentences S and names N; what the class is exactly is not fixed by the formalism.</Paragraph>
    <Paragraph position="1"> Left division categories A~B ('A under B') are those of expressions (functors) which concatenate with (arguments) in A on the left to yield Bs. Right division categories B/A ('B over A') are those of expressions (functors) which concatenate with (arguments) in A on the right yielding Bs. Product categories A.B are those of expressions which are the result of concatenating an A with a B; products do not play a dominant role here.</Paragraph>
    <Paragraph position="2"> More precisely, let L be the set of strings (including the empty string e) over a finite vocabulary V and let + be the  operation of concatenation (i.e. (L, +, ~) is the free monoid generated by V) 1 . Each category formula A is interpreted as a subset \[\[A\]\] of L. When the interpretation of atomic categories has been fixed, that of complex categories is defined by (4).</Paragraph>
    <Paragraph position="3"> (4) \[\[AkB\]\] = {sl Vs'~ \[\[A\]\], s'+s~ \[\[B\]\] }</Paragraph>
    <Paragraph position="5"> In general, given some type assignments others may be inferred. Such reasoning is precisely formulated in the Lambek calculus L.</Paragraph>
  </Section>
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