disjoint union, we know that ∆ 201 ∩ ∆ 202 = ∅ . If I ∈ N O is a nominal, then the definition of the disjoint union implies that f 20 = f 20 , U f 2 , Since nominals are interpreted by singleton sets in M 1 and M 2 , and since the domains of M 1 and M 2 are disjoint, this implies that f 20 is a set of cardinality 2. Consequently, 20 cannot correspond to an ADM induced by a frame for a DL with nominals, since such frames interpret nominals by singleton sets.

- U niversal role. Again, consider the disjoint union 20 of the ADMs 201 and 202, and assume that M 1 and M 2 correspond to frames of a DL with the universal role. Let U
denote the universal role, i.e., a role name for which the interpretation is restricted to the binary relation relating each pair of individuals of the domain. By the definition of the disjoint union, we have U 201 = U 201 U 2012 = ∀ 201 × ∀ 201 ∪ 2012 × ∀ 2012 ≠ ∀ 201 × ∀ 201 Consequently, 20 cannot correspond to an ADM induced by a frame for a DL with universal role, since such a frame would interpret U by ∆ ² ⁷ × ∆ ² ⁷

- Role complement. Again, consider the disjoint union 20 of the ADMs 201 and 202, and assume that 201 and 202 correspond to frames of a DL with role negation. For an arbitrary role name R, we have R 20 = R 20 ; ∪ R 20 ; = ( Δ 20 ; × Δ 20 ; / R 20 ; ) ∪ ( Δ 20 ; ×
Δ 2012 \ R 2012 ) ̸= (Δ 2011 ∪ Δ 2012 ) \ (R 2011 ∪ R 2012 ) = Δ 2011 \ R 2011
It remains to prove the "if" direction. Assume that L is one of the DLs introduced in Section 2 that does not allow for nominals, role complements, and the universal role. Let ( 
ADMs corresponding to them. By definition, A s r = W p for all p E P . Assume that the domains ( W p ) pEP are pairwise disjoint. We must show that the disjoint union of ( W p ) pEP
also corresponds to an L -frame. To this purpose, we define the frame ¥ = ( ∆ ¥, ¥) as follows:

$$\bullet\;\;\Delta^{\mathfrak{F}}:=\bigcup_{p\in P}\Delta^{\mathfrak{F}p}\;\;\mathrm{and}$$
$$\bullet\ R^{\hat{\mathfrak{T}}}:=\bigcup_{p\in P}\ R^{\hat{\mathfrak{T}}p}\ \mathrm{for\all}\ R\in N_{R}.$$

Let W = (W, F W, R W) E M be the ADM corresponding to $. By Definition 12 (corresponding ADM), we have W = U p E P W p and R 2W = U p E P R 2W p for all R E N R . By induction on the structure of complex roles, it is easy to show that this also holds for all R E R i.e., all complex role descriptions. For example, consider the role description R 1 ◦ R 2 .  By induction, we know that R ™ = U p EP R 1 ™ p and R 2 ™ = U p EP R 2 ™ p .  Since the sets ( Wp )pEp are pairwise disjoint,

$$(R_{1}\circ R_{2})^{\mathfrak{M}}=R_{1}^{\mathfrak{M}}\circ R_{2}^{\mathfrak{M}}=\bigcup_{p\in P}R_{1}^{\mathfrak{M}_{p}}\circ\bigcup_{p\in P}R_{2}^{\mathfrak{M}_{p}}=\bigcup_{p\in P}R_{1}^{\mathfrak{M}_{p}}\circ R_{2}^{\mathfrak{M}_{p}}=\bigcup_{p\in P}(R_{1}\circ R_{2})^{\mathfrak{M}_{p}}.$$

Since R 20 p = R δ p for all R ∈ R and p ∈ P p we obtain the following fact:
(*) For all p E P, a E A 3 P, and role descriptions R E R, the following holds: R 3 ( a ) =

R δ ν (α); in particular, R δ (α) ⊆ Δ δ ν .