Study On The Free Monoid Containing All Strongly Bi-singular Languages And Non-primitive Words

In 1977,Lassez and Shyr has been shown that the conclusion that the union of language set of all the irreducible prefix codes and all primitive words is not a code,and {A,Q} is a code for every language A in the irreducible prefix codes in "Factorization in the Monoid of languages".Thus,in the "Free Submonoids in the Monoid of Languages" published in 1998,Shyr and Tsai founded a free monoid containing all the finite prefix codes in order to study the freedom problem of prefix codes.The main idea is to prove that the union of language set of all the finite irreducible prefix codes and the set of ith powers of primitive words is a code,where i?2.But unfortunately,it is not promoted to a free monoid containing all prefix codes.In this paper,the freedom problem of several monoid of languages involving strongly bi-singular languages.We prove that if we start with the basis of strongly bi-singular languages and add all primitive words,then the resulting family of languages is not a code,and {A,Q} is a code for every language A in the irreducible strongly bi-singular languages.We shaw that if we start with the basis of strongly bi-singular languages and add the set of ith powers of primitive words,where i?2,then the resulting family of languages is a code.So we find a free monoid which properly contains the free monoid of all strongly bi-singular languages and the set of powers of primitive words.Thus,this paper enriches the scope of research on free monoid.