Let be a disjoint decomposition of and let be a vector field on , defined to be linear on each cell of the decomposition . Under some natural assumptions, we show how to associate a semiflow to and prove that such semiflow belongs to the o-minimal structure . In particular, when is a continuous vector field and is an invariant subset of , our result implies that if is non-spiralling then the Poincaré first return map associated is also in .
We prove that the automorphism group of the random lattice is not amenable, and we identify the universal minimal flow for the automorphism group of the random distributive lattice.
We prove that a type-definable Lascar strong type has finite diameter. We also answer some other questions from [1] on Lascar strong types. We give some applications on subgroups of type-definable groups.
We prove that, for any Hausdorff continuum X, if dim X ≥ 2 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then either dim C(X) = 2 or C(X) is not a C-space. This generalizes some results known for metric continua.
By a ternary structure we mean an ordered pair , where is a finite nonempty set and is a ternary relation on . A ternary structure is called here a directed geodetic structure if there exists a strong digraph with the properties that and for all , where denotes the (directed) distance function in . It is proved in this paper that there exists no sentence of the language of the first-order logic such that a ternary structure is a directed geodetic structure if and only if it satisfies...
Let K be a subclass of Mod() which is closed under isomorphism. Vaught showed that K is (respectively, ) in the Borel hierarchy iff K is axiomatized by an infinitary (respectively, ) sentence. We prove a generalization of Vaught’s theorem for the effective Borel hierarchy, i.e. the Borel sets formed by union and complementation over c.e. sets. This result says that we can axiomatize an effective or effective Borel set with a computable infinitary sentence of the same complexity. This result...
Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.