EQUATIONS O N PARTIAL WO RDS

I t is we ll- k n own t h at some of t h e most b asic p rop e rt ies ofword s, like t h e commu t at iv ity (xy= yx) an d t h e c on ju gacy ( xz= zy),can b e e x p ressed as solu t ion s of word eq u at ion s. A n im p ort ant p rob lemis to decide wh ether or n ot a given eq uation on words h as a solution.For instance, the equationxmyn= zphas only p erio dic solutions ina free m on oid , t h at is, if xmyn= zph old s wit h int egers m, n, p ≥ 2,then there exist s a word w su ch t h at x, y , z are p owers of w .Thisresu lt, which received a lot of attention, was fi rst p roved by Lyndon andSch ¨u t zenb erger for free grou p s. I n t h is p ap er, we invest igat e e q u at ion sonpa rti a l w ords . Partial words are sequen ces over a finite alphab et thatmay contain a numb er of “do not k now” sy mb ols. When we sp eakab ou t e q u at ion s on p art ial word s , we r ep lace t h e n ot ion of e q u ality( = ) wit h c omp at ib ility ( ↑). Amon g other equation s, we solvexy↑ yx,xz↑ zy, an d sp ecial c ases of xmyn↑ zpfor int egers m, n, p ≥ 2