On Regularity and Roots of Strong Codes (Logic, Algebraic system, Language and Related Areas in Computer Science)

Deletion and insertion are interesting and common operations which often appear in text editing. A language L ⊂ A* closed under the both operations forms a free submonoid of A*. Its base C is called a strong code, that is, L = C*. The language L is regular if and only if its base C is regular. Then, we prove in another way that the syntactic monoid of L becomes a finite group. This gives us many examples of regular strong codes. We also investigate the relation between strong codes and groups

On Commutativity of Extractable Codes (Algebraic system, Logic, Language and Related Areas in Computer Sciences II)

Deletion and insertion are interesting and common operations which often appear in string rewriting systems. Extractable submonoids and insertable submonoids of free monoids generated by finite alphabets allow to perform deletion operations and insertion operations, respectively. A submonoid N ⊂ A* is called extractable (resp. insertable) if x, uxv ∈ N implies uv ∈ N(resp. x, uv ∈ N implies uxv E N). The code C is called extractable (resp. insertable) if the submonoid C* is extractable (resp. insertable)[7]. Both extractable and insertable codes are identical to well-known strong codes, which is deeply related to the identities of syntactic monoids of languages. This paper deals with the commutativity of extractable codes. After the preliminaries in the first section, we summarize the fundamental properties of codes above in the second section. In the last section, we deal with commutative extractable codes. At first the language operators S and Q, which make languages commutative, are introduced. We show that a commutative extractable code is finite