PRIMITIVE PROPERTIES OF WORDS IN {up, uq}

In 2002, Gheorghe Păun and Nicolae Son proved if u is a non-empty word and a, b are two distinct letters, then at least one of ua and ub is a primitive word. We want to discuss if the suffix words a and b in ua and ub, respectively, are not letters, then what happens. So we first prove if u is a non-empty word, p is the empty word and q is a non-empty word whose length is less than 4, then at least one of u and uq is a primitive word when uq ≠ qu. Then, we prove if u is a non-empty word and p, q are two distinct non-empty words whose lengths are less than 3, then at least one of up and uq is a primitive word when pq ≠ qp. At last, we discuss the primitive properties of words in {up, uq}, when u is a non-empty word whose length is odd or even, separately.