By Nichols E. L.
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Extra resources for Note on the Phosphorescence of Uranyl Salts (1916)(en)(6s)
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Proof. Let q be the number of equivalence classes of L intersecting D. Let N be the width of L. Let u = u1 · · · un ∈ D∗ , with u1 , . . , un ∈ D. By a general result on congruences, [u1 ] · · · [un ] ⊂ [u] If n > N , then u is the equivalence class of words that are not factors of L. Otherwise, [u] contains at least one of the q + q 2 + · · · q N products of equivalence classes. Thus the number of equivalence classes of L intersecting D∗ is bounded by this number. The proposition is false if the width is unbounded.
Vn ∈ D. There exists a unique word X1 · · · Xn ∈ V ∗ such that ∗ ∗ S −→ gX1 · · · Xn d for some words g, d and some axiom S, and Xi −→ vi . We denote this word X1 · · · Xn by X(u). Deﬁne an equivalence relation on words in D∗ by u ∼ v if and only if X(u) ≡RX,a X(v) for all X ∈ V and a ∈ A. Here ≡RX,a is the syntactic congruence of the language RX,a . Since the sets RX,a are regular, there are only ﬁnitely many equivalence class for ∼. We show that u ∼ v implies u ≡L v. This shows that the set of Dyck words that are factors of words in L are contained in a ﬁnite number of classes for ≡L .