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For an algebra = (A , f 1 , ... , f n) the algebra = (A , t 1 , ... , t k) is called a functional reduct if each t j is a term function of . We classify the functional reducts of the countable atomless Boolean algebra up to first-order interdefinability. That is, we consider two functional reducts the "same" if their group of automorphisms is the same. We show that there are 13 such reducts and describe their structures and group of automorphisms. [ABSTRACT FROM AUTHOR]

In order to understand the structure of the "typical" element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood; see, e.g., the works of Glasner and Weiss, and Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behaviour of the homeomorphisms is entirely different from the generic case. For Homeo+([0, 1]) we describe the non-Haar null conjugacy classes and also show that their union is co-Haar null, for Homeo+(S1) we describe the non-Haar null conjugacy classes, and for U(l2) we show that, apart from the classes of the multishifts, all conjugacy classes are Haar null. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set. [ABSTRACT FROM AUTHOR]

In order to understand the structure of the “typical” element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski, we investigate the measure theoretic dual of this problem, using Christensen’s notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper we generalize the theorems of Dougherty and Mycielski about S∞ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the cofinal strong amalgamation property. As an application, we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set. [ABSTRACT FROM AUTHOR]

Letdenote the countably infinite dimensional vector space over the two element field andGL(ω,2) its automorphism group. Letdenote the symmetric group acting on the elements ofandSpdenote the symplectic group. The closed intermediate subgroupsG, such thatare characterized. Asendowed with the standard symplectic form is anω-categorical structure, these groups correspond to its first order definable reducts. These reducts are also analyzed. [ABSTRACT FROM PUBLISHER]