Counterexample. Guarantee kernel does not distribute over union [ltp-000O]

(from The Infinite, in Finite Time)

Take the properties $P = \{ \sigma \mid \texttt {a}\ \text {occurs infinitely often in}\ \sigma \}$ and $R = \{ \sigma \mid \texttt {a}\ \text {occurs finitely many times in}\ \sigma \}$ . Both have an empty guarantee kernel, i.e. $\underline {P} = \underline {R} = \emptyset$ , however $P \cup R = \Sigma ^\omega$ so the guarantee kernel of the union is $\Sigma ^\omega$ , not $\emptyset$ .