On Nachbin's characterization of a Boolean lattice.

A classical theorem of L. Nachbin [6] characterizes Boolean lattices as those bounded distributive lattices in which each prime ideal is maximal. This result has been generalized and applied to non-bounded distributive lattices by G. Gratzer and E. T. Schmidt, see [3], especially p. 276. Recently, D. Adams ([1], Theorem 1) has given a version of Nachbin's theorem for bounded non-distributive lattices. The object of this note is to give a transparent alternative proof of Gratzer and Schmidt's generalization and also to establish a theorem akin to that of Adams. The notation and terminology follows that of [2] and Stone's Theorem ([2], Theorem 15, p. 74) will be used freely. Incidentally, a proof of Nachbin's Theorem is given in [2], Theorem 22, p. 76; it is a simplication (possibly due to boundedness) of the proof in [3]. For elements x and y of a lattice £, let (x,y) {z e L: x AZ ̂ y}. When L is distributive, (x9y) is an ideal. For a detailed account of such ideals, see Mandelker [5], The following lemma is an extension of [4], Lemma 12.