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Every concept name $C \in \mathcal{N}_C$ is a concept.
If $C$ and $D$ are concepts, and $R \in \mathcal{N}_R$ is a role, then the following expressions are concepts as well: $(\neg C)$, $(C \sqcap D)$, $(C \sqcup D)$, $(\exists R.C)$, and $(\forall R.C)$.
Nothing else is a concept. ■

The set of concepts is the same as for the language $\mathcal{ALC}$. If a concept starts with “(”, we call it a compound concept, otherwise a concept name or atomic concept. Brackets may be omitted for the sake of readability if the concept is still uniquely parsable.

We use the following abbreviations: if $R_1, \dots, R_n$ are roles, and $C$ is a concept, then we define $(\forall R_1 \sqcup \dots \sqcup R_n.C) ={def} (\forall R_1.C) \sqcap \dots \sqcap (\forall R_n.C)$ and $\exists R_1 \sqcup \dots \sqcup R_n.C ={def} (\exists R_1.C) \sqcup \dots \sqcup (\exists R_n.C)$. Additionally, for some $CN \in \mathcal{N}C$ we define $\top ={def} CN \sqcup \neg CN$ and $\bot =_{def} CN \sqcap \neg CN$ (therefore, $\top^\mathcal{T} = \Delta^\mathcal{T}, \bot^\mathcal{T} = \emptyset$).

The set of roles being used within a concept term $C$ is defined:

Definition 2 (Used Roles, roles($C$))

$\text{roles}(C) =_{def} \left\{ \begin{array}{ll} \emptyset & \text{if } C \in \mathcal{N}_C \\ \text{roles}(D) & \text{if } C = (\neg D) \\ \text{roles}(D) \sqcup \text{roles}(E) & \text{if } C = (D \sqcap E) \\ \text{or } C = (D \sqcup E) \\ \{R\} \sqcup \text{roles}(D) & \text{if } C = (\exists R.D) \\ \text{or } C = (\forall R.D) & \hfill \blacksquare \end{array} \right.$

For example, $\text{roles}(\forall R.\exists S.SC \sqcap \exists T.D) = {R, S, T}$.

As already noted, $\mathcal{ALC}_{RA\odot}$ provides role axioms of the form $S \circ T \subseteq R_1 \sqcup \dots \sqcup R_n$. More formally, the syntax of these role axioms is as follows:

Definition 3 (Role Axioms, Role Box) If $S, T, R_1, \dots, R_n \in \mathcal{N}R$, then the expression $S \circ T \subseteq R_1 \sqcup \dots \sqcup R_n$, $n \ge 1$, is called a role axiom. If $ra = S \circ T \subseteq R_1 \sqcup \dots \sqcup R_n$, then $\text{pre}(ra) ={def} (S, T)$ and $\text{con}(ra) ={def} {R_1, \dots, R_n}$. A finite set $\mathfrak{R}$ of role axioms is called a role box. Let $\text{roles}(ra) ={def} {S, T, R_1, \dots, R_n}$, and $\text{roles}(\mathfrak{R}) ={def} \bigcup{ra \in \mathfrak{R}} \text{roles}(ra)$. $\square$

Additionally, a set of global concept inclusion axioms (GCIs) can be specified. A set of these GCIs is called a free TBox: