By Frank Richter
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Extra resources for A Mathematical Formalism for Linguistic Theories with an Application in Head-Driven Phrase Structure Grammar
If Σ is an 87 signature, I write for the class of concrete feature structures under Σ. Note that the definition of concrete feature structures under Σ is independent of many details of Σ. For example, it is not affected by the presence or absence of an appropriateness function or a subsumption ordering on the set of sorts, S. The class of concrete feature structures under Σ could thus be defined relative to other signatures as well, as long as they provide two disjoint sets of symbols, A and S.
Other terminologies exist in the literature. 34 CHAPTER 2. FORMAL FOUNDATIONS I q0 = qˆ, for each j ∈ IN such that j < i, for some α ∈ A, ∆ qj , α is defined and ∆ qj , α = qj+1 , and qi = q, and Λ is a total function from Q to S. ✁ ✄✂ Σ If Σ is an 87 signature, I write for the class of concrete feature structures under Σ. Note that the definition of concrete feature structures under Σ is independent of many details of Σ. For example, it is not affected by the presence or absence of an appropriateness function or a subsumption ordering on the set of sorts, S.
In our Heyting algebras of 87 disjunctive feature structures, join corresponds to set is the Alexandrov topology of the dual preorder of ✁ ✄ Σ 87 , ≤ , and any Σ topology is a complete Heyting algebra. , infinitely many distinct elements an of ✁☎✄ Σ 87 such that a1 ≤ a2 , a2 ≤ a3 , a3 ≤ a4 , . , Σ then ✁ ✄ Σ 87 , ⊆ is isomorphic to the lattice of the set of subsets of ✁☎✄ 87 whose elements are pairwise subsumption-incomparable, ordered in such a way that a set, 1 , subsumes a set, 2 , iff each element of 1 subsumes some element of 2 , which is Pollard and Sag’s original characterization of the lattice of disjunctive feature structures (Pollard and Sag, 1987, p.