Abstract Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. As an application, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures.

Abstract We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.

Abstract. We define notions of bisimulation for the family of Heyting-valued modal logics introduced by M. Fitting. In this family of logics, each modal language is built on an underlying space of truth values, a Heyting algebra H. All the truth values are directly represented in the language, which is interpreted on relational frames with an H-valued ac-cessibility relation. We investigate the correct notion of bisimulation in this context: we define two variants of bisimulation relations and derive relative (to a truth value) modal equivalence results for bisimilar states. We further investigate game semantics for our bisimulation, Hennessy-Milner classes and other relevant properties. If the underlying algebra H is finite, Heyting-valued modal models can be equivalently reformu-lated to a form relevant to epistemic situations with many interrelated experts. Our definitions and results draw from this formulation, which is of independent interest to Knowledge Representation applications.

Abstract Let = ⟨M, +, <, 0, S⟩ be a linear o-minimal expansion of an ordered group, and G = ⟨G, ⊕,eG) an n-dimensional group definable in . We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L. for some convex ∨-definable subgroup U of ⟨Mn, +⟩ and a lattice L of rank equal to the dimension of the ‘compact part’ of G.

Abstract Let M = 〈M, +, <, 0, {λ}λЄD〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a ‘definable quotient group’ U/L, for some convex V-definable subgroup U of 〈Mn, +〉 and a lattice L of rank n. As two consequences, we derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L.