Word equations in some geometric semigroups
Mohan S. Putcha · Pacific Journal of Mathematics · 1978
Let 5 be a semigroup and let HΊ = w } (x u ,Jt,), w 2 = w 2 (xu' m ',Xt) be two words in the variables x u ,x t .By a solution of the word equation {w u w 2 } in 5, we mean a u , a t E S such that w λ {a u , a t ) = w 2 (a u , a t ).Let ^R denote the free product of t copies of positive reals under addition.In §3 and §5 we show that if Y is either the semigroup of certain paths in R" or the semigroup of designs around the unit disc, then any solution of {wi,w 2 } in Y can be derived from a solution of {wi, w 2 } in ^R.This answers affirmatively a problem posed in Word equations of paths by Putcha.Word equations in 3* R are studied in §1.Using these results, it is shown that any solution in Y of {wi, w 2 } can be approximated by a solution which is derived from a solution in a free semigroup.There are two books by Hmelevskii and Lentin on word equations in free semigroups.We also show that if {HΊ, W 2 } has only trivial solutions in any free semigroup, then it has only trivial solutions in Y.