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Let two nonperiodic binary morphisms ... be given. A word w is called a solution of g and h if g(w) = h(w). We say that a solution w is simple if whenever ... are prefixes of wu such that ... for some word z, then |u| = |u'| = k|w|, for some k ∈ N. In this paper we will study simple solutions and show that if a word w is a simple solution containing at least nine occurrences of the letter a and at least nine occurrences of the letter b, then either w = (ab)ia, or w = ajbk with gcd (j, k) = 1, up to the exchange of letters a and b. [ABSTRACT FROM AUTHOR]

Recent developments of Baxter algebras have led to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first and the second kinds, partitions and multinomial coefficients. This allows us to apply congruences from number theory to obtain congruences in Baxter algebras. [ABSTRACT FROM AUTHOR]