Vol. 75, No. 1, 1978

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Word equations in some geometric semigroups

Mohan S. Putcha

Vol. 75 (1978), No. 1, 243–266

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 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 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 R. This answers affirmatively a problem posed in Word equations of paths by Putcha. Word equations in R 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 .

Mathematical Subject Classification 2000
Primary: 20M99
Received: 28 December 1976
Published: 1 March 1978
Mohan S. Putcha