In this paper wo continue the study of the $\mathcal P$-funtions started in [CN], [CFP] and [N$_1$], in particular, for the property $\mathcal P$ = Tychonoff. We describe a simple tecnique to obtain a Tychonoff compactification of any Tychonoff function and we study the set of all the Tychonoff compactifications of a Tychonoff function, showing that if is a complete upper semilattice.