The Word Problem for Semigroups 1 Semigroup Equations

is not always true for square integer matricesM and N . The most basic example of a semigroup is the set, Σ∗, of strings over an alphabet, Σ, where the binary operation is concatenation. In fact, we take the associativity of concatenation so much for granted that we typically write xyz for the concatenation of x, y, z ∈ Σ∗, without bothering to indicate whether they were concatenated as (x ◦ y) ◦ z or as x ◦ (y ◦ z), since of course it doesn’t make any difference. When Σ∗ is regarded as a semigroup under concatenation, it is traditional to refer to the strings in Σ∗ as words.