On 2 of Lemma 4 and 4 of Proposition 2 in “ * - Continuous Idempotent Semirings and Their Ideal Completion ”

2 of Lemma 4 and 4 of Proposition 2 in our paper [1] entitled “*Continuous Idempotent Semirings and Their Ideal Completion” are wrong. This note shows counter examples of them. The counter examples are given in linguistic models of *-Continuous Idempotent Semirings. 1 Monodic Tree Languages A ranked set is a couple (Σ, r) of a set Σ of (function) symbols and a mapping r from Σ to the set N of natural numbers. We often abbreviate (Σ, r) to Σ. Σn denotes the set {a ∈ Σ | r(a) = n} of symbols of rank n. The set TΣ of terms over Σ is defined by the following induction: • ¤ ∈ TΣ (¤ 6∈ Σ), • Σ0 ⊆ TΣ, • for a ∈ Σn and t1, · · · , tn ∈ TΣ, a(t1, · · · , tn) ∈ TΣ. A subset of TΣ is called monodic tree language over Σ. Obviously, (℘(TΣ),∪, ∅) is an idempotent commutative monoid. Remark 1.1. In the case of Σ = Σ1, TΣ is isomorphic to the set Σ∗ of finite strings over Σ. For t ∈ TΣ and L ⊆ TΣ, we define t · L ⊆ TΣ as follows: • if t = ¤, then t · L = L, • if t ∈ Σ0, then t · L = {t}, • if t = a(t1, · · · , tn), then t · L = {a(u1, · · · , un) | ui ∈ ti · L}. ∗Department of Mathematics and Computer Science, Kagoshima University. [email protected]