The Structure Of Some Kinds Of Primitive Words And Non-Primitive Words

In2008, Tetsuo Moriya shaw that if lg(p)=lg(q) for two distinct primitive words p, q, then pnqm is a primitive word for any n, m≥1and (n, m)≠(1,1). In Chapter2of this thesis, we prove that:(1) If lg(q)|lg(p) and lg(p)≤mlg(q) for two distinct primitive words p, q, then pnqm is a primitive word for any n, m≥1and (n,m)≠(1,1);(2) If lg(q)|lg(p) and lg(p)≤jlg(g), then piqj(prqj)m is a primitive word for any m, r,j≥1,i≥0, i≠r,(r,j)≠(1,1);(3) If uv, u are non-primitive words and lg(v)|lg(u), then u∈v+. In2001, H.J.Shyr in his book (Free Monoids and Languages. Taiwan:Hon Min Book Company,2001) proved that if uv is a primitive word, then {u, v} is a code. But the converse is not true. In Chapter3, we show that let u is a non-primitive word and v is a d-primitive word, then {u, v} is a code if and only if uv is a primitive word. It is an supplement to the statement proved by H.J.Shyr. Since p-primitive words and U-primitive words are special primitive words. In Chapter4, we prove that:(1) Let a, b be distinct letters and x1,x2,...,xn be distinct words. If lg(x1)≤lg(x2)≤...≤lg(xn)≤m, where m≥1, and x1is not a proper prefix of x2, then abm x1abm x2...abmxn is a p-primitive word;(2) For any two distinct primitive word p, q, at least one of the word pqm and pqm+1is a U-primitive word;(3) If p is a U-primitive word and p1is a proper prefix of p, then pkp1is also a U-primitive word for any k≥2;(4) If pqm for m≥1is not a primitive word then pqm+k is a U-primitive word for any k≥1.