min C
$$\left.\frac{C^{\mathcal{I}}(i)}{D^{\mathcal{I}}(i)},\frac{D^{\mathcal{I}}(i)}{C^{\mathcal{I}}(i)}\right\}\,\leq\,C^{\mathcal{I}}(i)\Rightarrow D^{\mathcal{I}}(i)\,\leq\,1,$$
it follows that limi→∞(C
I(i) ⇒ DI(i**)) = 1. Thus, we know that the set**
I = {i ∈ N : C
I(i) ⇒ DI(i) ≤ C
I(i0) ⇒ DI(i0)} **is finite. Since** I is finite it is obvious that there is some i ∈ I **such that (**C ⊑ D)
I = C
I(i) ⇒
DI(i). **Q.E.D.**

## 4 Discussion

The aim of this section is to discuss some interesting remarks of the failure of the FMP exposed in Theorems 3.3 and 3.7.

The first remark is that the failure of the FMP is essentially a consequence of the fact that the TBox is cyclic**. As we will see, in the case of Lukasiewicz** this situation does not happen if we consider only unfoldable KBs.

A TBox T is acyclic **iff it verifies the following three constraints:**

1. Every axiom in T is either of the form hA ⊑ C ≥ αi or A ≡ C**, where** A
is an atomic concept.6
2. There is no concept A **such that it appears more than once on the left**
hand side of some axiom in T .

3. No cyclic definitions are present in T . We will say that A **directly uses**
primitive concept B in T , if there is some axiom τ ∈ T such that A **is on** the left hand side of τ and B occurs in the right hand side of τ. Let **uses** be the transitive closure of the relation directly uses in T . T **is cyclic iff** there is A such that A **uses** A in T .
Analogously, a fuzzy KB is said to be acyclic **when its TBox is acyclic.**
In the crisp case, acyclic TBoxes can be eliminated through an **expansion**
process **[15] which only takes a finite number of steps. The fact that this process** can create an exponential growth of the KB has motivated the introduction of the lazy expansion **optimization technique [3], very useful in practice.**
In the fuzzy case, acyclic TBoxes cannot be completely removed in general due to the presence of degrees α, and we need to require some additional property in order to do so. We say that a fuzzy TBox is unfoldable **if it is an acyclic**
TBox which only contains axioms of the form hA ⊑ C ≥ 1i, and A ≡ C **(that** is, an acyclic TBox where every fuzzy concept inclusion axiom is of the **form**
hA ⊑ C ≥ αi with α **= 1) (cf. [19, Section 3.3]). Analogously, a fuzzy KB is said** to be unfoldable **when its TBox is unfoldable.**
In case we allow to use truth constants α **as concept constructors, then it is**
trivially true for all left-continuous t-norms that hC ⊑ D ≥ αi **is equisatisfiable**
6 In particular this forbids concept ⊤ **on the left hand side of axioms.**