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A module is weakly minimal if and only if every pp-definable subgroup is either finite or of finite index. We study weakly minimal modules over several classes of rings, including valuation domains, Prufer domains and integral group rings. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Discrete weakly o-minimal structures, although not so stimulating as their dense counterparts, do exhibit a certain wealth of examples and pathologies. For instance they lack prime models and monotonicity for definable functions, and are not preserved by elementary equivalence. First we exhibit these features. Then we consider a countable theory of weakly o-minimal structures with infinite definable discrete (convex) subsets and we study the Boolean algebra of definable sets of its countable models. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study ω-categorical weakly o-minimal expansions of Boolean lattices. We show that a structure = (A,≤, ℐ) expanding a Boolean lattice (A,≤) by a finite sequence I of ideals of A closed under the usual Heyting algebra operations is weakly o-minimal if and only if it is ω-categorical, and hence if and only if A/I has only finitely many atoms for every I ∈ ℐ. We propose other related examples of weakly o-minimal ω-categorical models in this framework, and we examine the internal structure of these models.

We consider R-torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T(RG) of all R-torsionfree RG-modules and the theory T0(RG) of RG-lattices (i. e. finitely generated R-torsionfree RG-modules), and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG-lattices are of finite, or wild representation type.

Let G be the direct sum of the noncyclic groupof order four and a cyclic groupwhoseorderisthe power pn of some prime p. We show that ℤ2G-lattices have a decidable theory when the cyclotomic polynomia (x) is irreducible modulo 2ℤ for every j ≤ n. More generally we discuss the decision problem for ℤ2G-lattices when G is a finite group whose Sylow 2-subgroups are isomorphic to the noncyclic group of order four.

We propose a definition of weak o-minimality for structures expanding a Boolean algebra. We study this notion, in particular we show that there exist weakly o-minimal non o-minimal examples in this setting.