Sublattices of C(X) and Wallman rings (Q1060729)

This paper concerns the following two questions posed by \textit{H. L. Bentley} and \textit{B. J. Taylor} [J. Austral. Math. Soc., Ser. A 25, 215- 229, 230-240 (1978; Zbl 0387.54006 and Zbl 0387.54005)]: ''Is every Wallman ring on X equivalent to one which is a sublattice of C(X)? For an arbitrary Wallman ring, is there an equivalent Wallman ring which contains all real constant functions?''. In connection with them, we characterize for a given Wallman ring \({\mathcal A}\) on X the existence of an equivalent Wallman ring \({\mathcal B}\) containing \({\mathcal A}\) and being a sublattice of C(X). Also we give an example of a Wallman ring \({\mathcal A}\) on X such that if \({\mathcal B}\) is a Wallman ring on X which contains all real constant functions and is a sublattice of C(X), then \({\mathcal B}\) is not equivalent to \({\mathcal A}\).