Note: The notation is the same as that of the subalphabet product as the language product is an extension of the definition of subalphabet product and could only be defined after formal language was defined. Definition: Language power: Let Σ be an alphabet and let V be a formal languages over Σ. The nth power of V , where n ∈ N0 , is denoted V n and defined inductively: { {λ} if n = 0 n V = n−1 V V if n > 0 Note: The notation is the same as that of the subalphabet power as the language power is an extension of the definition of subalphabet power and could only be defined after formal language was defined. Definition: Linguistic structure: Let Σ be an alphabet, V be a formal language over Σ and ◦ denote concatenation. (V, ◦) is a linguistic structure iff ∀(x, y) ∈ V × V : x ◦ y ∈ V . That is, V is closed under ◦. Note: This definition was invented for this text. Every linguistic structure is an algebraic structure hence the name. Theorem: [The empty word is unique]: Let Σ be an alphabet and let λ and λ0 be words over Σ of length 0. Then λ = λ0 . Hence we can speak of the empty word. Proof: From the definition of word equality, we have len(λ) = len(λ0 ) and ∀i : λi = λ0i holds vacuously. Hence the result. ¥ Theorem: [Length of concatenation]: Let Σ be an alphabet, x and y be words over Σ and ◦ denote concatenation. Then len(x ◦ y) = len(x) + len(y). Proof: If x = λ or y = λ then the result follows immediately from the definition of concatenation with the empty word and the definition of word length. Otherwise, from the definition of concatenation: x ◦ y is a mapping from [1.. len(x) + len(y)] to Σ. From the definition of the length of a word: len(x ◦ y) is the cardinality of the domain of x ◦ y when viewed as a sequence. Hence len(x ◦ y) = len(x) + len(y). ¥
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