On complementation of prime semilattices (Q1095163)

The authors give a negative answer to the following question: ``Is it true for any element a of an infinite prime semilattice S with 0 and 1, there exists a sequence of elements \(\{t_ i\}\), \(i=1,2,...,n\) in \(<a,0>\) such that \(a\vee t_ 1\vee...\vee t_ n=1\) implies the complementation of S?'' This question was asked by \textit{K. P. Shum}, \textit{M. W. Chan}, \textit{C. K. Lai}, and \textit{K. Y. So} [in Can. J. Math. 37, 1059-1073 (1985; Zbl 0571.06005)]. Some relationships between ultra filters and complementation of semilattices are studied as well.