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

Suppose that G is a definably connected, definable group in an ominimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism p : G −→ G is a locally definable covering homomorphism and π 1 (G) is isomorphic to the o-minimal fundamental group π(G) of G defined using locally definable covering homomorphisms. * With partial support from the FCT (Fundação para a Ciência e Tecnologia), program POCTI (Portugal/FEDER-EU).

We examine the notion of bisimulation and its ramifications, in the context of the family of Heyting-valued modal languages introduced by M. Fitting. Each modal language in this family 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 accessibility relation. We define two notions of bisimulation that allow us to obtain truth invariance results. We provide game semantics and, for the more interesting and complicated notion, we are able to provide characteristic formulae and prove a Hennessy-Milner type theorem. If the underlying algebra H is finite, Heyting-valued modal models can be equivalently reformulated to a form relevant to epistemic situations with many interrelated experts. Our definitions and results draw inspiration from this formulation, which is of independent interest to Knowledge Representation applications.

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