This addendum provides the details of a computation not presented in [1] but needed for a complete proof that Foulis-Holland sets generate distributive sublattices. In [1] I called a nonempty subset S of an orthomodular lattice a Foulis-Holland set in case whenever x, y and z are distinct elements of S one of them commutes with the other two. I presented a proof that Foulis-Holland sets generate distributive sublattices. The purpose of this note is to provide a detailed proof that the function 4, defined in Lemma 2.2 of [1] is indeed onto. Throughout this paper let S = (s,, . . . , s,,t1, .... , t,} be a finite nonempty subset of an orthomodular lattice L such that si E C(SN{ti)) and t, E C(S\{s,)), for i = 1, .. . ., n, and let As = {xi A Axxnlxi E {sE , t)i ) \{O). Lemma 2.2 of [1] states that the power set P (As) of As is isomorphic to the sublattice of L generated by S in case, for each i = 1, 2, . .. ., n, si and ti are complements in L. The proof proceeds by defining 4,: 6-P(As) -* KS> by the rule 4,(M) = V M for M C As. A computation shows that M C N if and only if 4,(M) (or equivalently of L). Clearly 4, preserves joins. But it is not clear that 4, preserves meets. This fact is needed to get from S C image(4,) to C image(4,). I am indebted to Professor M. F. Janowitz for this observation. That 4, preserves meets is the content of the following proposition. We begin by reviewing some notation and making some observations. For M C As, define 8(M) = {X2 A AxIn for some xl E (s,, t,), xI A *..* Axn E M) and fory, E (s,, t,) let MYI = {x A* *.* Axn E Mlyi = x). Assume that si and ti are complements in L, i = 1,2, ... ., n. LEMMA. If M,N C As and x E (sI, t,), then (LI) VM = V Ms, V V M,,, Received by the editors September 1, 1978. AMS (MOS) subject classifications (1970). Primary 06A35, 06A25; Secondary 46C05, 46L10, 81A12. O 1979 American Mathematical Society 216 0002-9939/79/0000-0405/$01.75 This content downloaded from 207.46.13.134 on Sun, 25 Sep 2016 05:20:33 UTC All use subject to http://about.jstor.org/terms