Signature

A set $\mathbb{S}$ of sets of arguments is realisable under a semantics $\sigma$ iff there is an AF $F$ with $\sigma(F) = \mathbb{S}$. The signature of $\sigma$ is $\mathsf{sig}(\sigma) = \{\mathbb{S} \mid \mathbb{S}$ is realisable under $\sigma\}$.

Properties of sets of extensions

For a set $\mathbb{S}$ of sets of arguments, let $\mathrm{Args}_\mathbb{S} = \bigcup_{S \in \mathbb{S}} S$, $\mathrm{Pairs}_\mathbb{S} = \{(a,b) \mid \{a,b\} \subseteq S$ for some $S \in \mathbb{S}\}$ and $\mathrm{dcl}_\mathbb{S} = \{S' \mid S' \subseteq S$ for some $S \in \mathbb{S}\}$ (Dunne et al., 2015). Then $\mathbb{S}$ is

• downward-closed iff $\mathbb{S} = \mathrm{dcl}_\mathbb{S}$ (see Downward Closure),
• incomparable iff for all $S, S' \in \mathbb{S}$, $S \subseteq S'$ implies $S = S'$ (see I-Maximality),
• tight iff for all $S \in \mathbb{S}$ and $a \in \mathrm{Args}_\mathbb{S}$ with $S \cup \{a\} \notin \mathbb{S}$ there is $b \in S$ with $(a,b) \notin \mathrm{Pairs}_\mathbb{S}$ (see Tightness),
• conflict-sensitive iff for all $S, S' \in \mathbb{S}$ with $S \cup S' \notin \mathbb{S}$ there are $a, b \in S \cup S'$ with $(a,b) \notin \mathrm{Pairs}_\mathbb{S}$ (see Conflict-Sensitiveness),
• com-closed iff for every $\mathbb{T} \subseteq \mathbb{S}$ with $(a,b) \in \mathrm{Pairs}_\mathbb{S}$ for all $a, b \in \mathrm{Args}_\mathbb{T}$, $\mathrm{Args}_\mathbb{T}$ has exactly one $\subseteq$-minimal superset in $\mathbb{S}$ (see Com-Closure).

Known Signatures