Definition. Guarantee kernel [ltp-000I]

(from The Infinite, in Finite Time)

Given a property $P \subseteq \Sigma ^\omega$ , the guarantee kernel of $P$ , written $\underline {P}$ , is the largest subset of $P$ which is a guarantee property. That is, $\underline {P}$ is the set of all infinite extensions of the good prefixes of $P$ .

It follows that $P$ is a guarantee property iff $\underline {P} = P$ .

This is a kernel operator, so it is co-extensive ( $\underline {P} \subseteq P$ ), idempotent ( $\underline {\underline {P}} = \underline {P}$ ), and monotonic ( $P \subseteq R$ implies $\underline {P} \subseteq \underline {R}$ ).

Topologically speaking, the guarantee kernel is the interior operator.

Intuitively, the guarantee kernel of $P$ is all behaviours for which some finite prefix extended in any way will get into $P$ .