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We introduce a method for constructing Boolean algebras from trees which preserves some of the trees' properties. The method is used to produce a very simple construction for rigid Boolean algebras and to construct a weakly homogeneous Boolean algebra without homogeneous factors. 0. Introduction. In this paper we present two constructions: a weakly homogeneous Boolean algebra that answers a question of Rubin, and what we feel is the simplest construction of a rigid Boolean algebra. A rigid Boolean algebra is one which has no nonidentity automorphism. We begin our construction of a rigid Boolean algebra by taking a tree T satisfying (a) the height of T is w; (b) T has a single root; (c) for all distinct a, ,3 E T, the number of immediate successors to a is different from the number of immediate successors to ,8; and (d) the number of immediate successors is always regular. We form " wedges". Sa = (/3 E T: a < ,B}, for each a E T, and close the set of wedges under finite unions and complements. The result is our algebra. Many constructions of rigid Boolean, algebras have been given. In [vDMR] there is a short history. van Douwen, Monk and Rubin ask for a "natural" construction of a rigid Boolean algebra. We offer our construction as a candidate. van Douwen, in [vD] has a similar construction. He begins with a tree T satisfying (a) and (c) and with the property that for all a E T, icK, the number of immediate successors to a, satisfies KNO = K.. He then topologizes the tree by taking certain infinite unions of wedges. The result is a rigid 0-dimensional compact space. The algebra of closed-and-open sets is the desired rigid Boolean algebra. A Boolean algebra B is homogeneous if for any nonzero b E B, the set (a E B: a < b} viewed as a Boolean algebra is isomorphic to B. It is weakly homogeneous if for all distinct nonzero a, b E B there exist nonzero b, < b, a, < a such that the algebras consisting of {c: c < b,} and (c: c < a,} are isomorphic. We answer a question of Rubin in [R] by constructing a weakly homogeneous Boolean algebra that has no homogeneous factors. Received by the editors January 27, 1982 and, in revised form, October 25, 1982. 1980 Mathematics Subject Classificatiotn. Primary 06E99. 'These results are contained in Chapter 2 of the author's Ph. D. dissertation, prepared under the direction of J. D. Monk at the University of Colorado. The author expresses his gratitude to Professor Monk. 2This paper is dedicated to Merry Havens. ((1983 American Mathenatical Socicty 0002-9939/82/(XXX)-0385/$02.(X)

A simple construction for rigid and weakly homogeneous Boolean algebras answering a question of Rubin