We define an algebraic structure, Paired Complete Idempotent Semirings (pcis), which are appropriate for defining a denotational semantics for multiple context-free grammars of dimension 2 (2-mcfg). We demonstrate that homomorphisms of this structure will induce well-behaved morphisms of the grammar, and generalize the syntactic concept lattice from context-free grammars to the 2-mcfg case. We show that this lattice is the unique minimal structure that will interpret the grammar faithfully and that therefore 2-mcfgs without mergeable nonterminals will have nonterminals that correspond to elements of this structure.
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