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Let S be a semigroup and let
w1 = w1(x1,⋯,xt), w2 = w2(x1,⋯,xt) be two words in the variables x1,⋯,xt. By a
solution of the word equation {w1,w2} in S, we mean a1,⋯,at ∈ S such that
w1(a1,⋯,at) = w2(a1,⋯,at). Let ℱR 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 Rn or the semigroup of designs around the unit disc, then any
solution of {w1,w2} in Y can be derived from a solution of {w1,w2} in ℱR. This
answers affirmatively a problem posed in Word equations of paths by Putcha. Word
equations in ℱR are studied in §1. Using these results, it is shown that any solution
in Y of {w1,w2} 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 {w1,w2} has only
trivial solutions in any free semigroup, then it has only trivial solutions in
Y .
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