A Free Monoid Generated By Prefix Codes And The Set Of The Powers Of D-primitive Words

In 1972,Perrin,D.showed that the irreducible generating set of prefix codes P((?))is a code.In 1998,Shyr,H.J.and Tsai,Y.S.showed that the set of languages Pf((?))U{Q(i)|i≥2} is a code,where Q is the collection of all primitive words on X and Q(i)={fi|f∈Q}.In 2019,Liu Qin proved that the set of languages P((?))∪{Q(i)|i≥2} is a code.LetQd be the collection of all d-primitive words on X and Qd(i)={fi|f∈Qd,i≥2}.In this paper,we prove that the set of languages P((?))∪{Qd(i))|i≥2} is a code.So we find a free monoid which properly contains the free monoid of prefix codes and the set of powers of d-primitive words.Thus,this paper enriches the research of prefix codes and free monoids.In 1989,Shyr,H.J.proved that the collection of all d-primitive words is a disjunctive language.Basis on this,we construct some special d-primitive words and show that Qd(i)∩Ai;Qd(i)∩Aj,i<j and i divides j;(Q\Qd)n,r;QdcQdc;Qp(i)\Qd are disjunctive languages,where Ai={u∈X+|ua=ub+i}.