We define rational expressions, their star height and rational identities. Section 4.1 studies the rational identity E* ≡ 1 + EE* ≡ 1 + E*E and its consequences and the operators a−1E. In Section 4.2, we show that, over a commutative ring, rational identities are all consequences of the previous identities. In Section 4.3, we show that, over a field, star height may be characterized through some minimal representation, and deduce that the star height of the star of a generic matrix of order n is n. In the last section, we see that the star height may decrease under field extension and show how to compute the absolute star height, which is the star height over the algebraic closure of the ground field.

Rational expressions

Let K be a commutative semiring and let A be an alphabet. We define below the semiring of rational expressions on A over K. This semiring, denoted ε, is defined as the union of an increasing sequence of subsemirings εn for n ≥ 0. Each such subsemiring is of the form εn = K ⟨An⟩ for some (in general infinite) alphabet An; moreover, there will be a semiring morphism E ↦ (E, 1), εn → K. We call (E, 1) the constant term of the rational expression E.

Now A0 = A, ε0 = K ⟨A⟩ and the constant term is the usual constant term.