Locally projective graphs and their densely embedded subgraphs

The article contributes to the classification project of locally projective graphs and their locally projective groups of automorphisms outlined in Chapter 10 of Ivanov (The Mathieu Groups, Cambridge University Press, Cambridge, 2018). We prove that a simply connected locally projective graph $$\Gamma $$ Γ of type (n, 3) for $$n \ge 3$$ n ≥ 3 contains a densely embedded subtree provided (a) it contains a (simply connected) geometric subgraph at level 2 whose stabiliser acts on this subgraph as the universal completion of the Goldschmidt amalgam $$G_3^1\cong \{S_4 \times 2,S_4 \times 2\}$$ G 3 1 ≅ { S 4 × 2 , S 4 × 2 } having $$S_6$$ S 6 as another completion, (b) for a vertex x of $$\Gamma $$ Γ the group $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) which stabilizes every line passing through x induces on the neighbourhood $$\Gamma (x)$$ Γ ( x ) of x the (dual) natural module $$2^n$$ 2 n of $$G(x)/G_{\frac{1}{2}}(x) \cong L_n(2)$$ G ( x ) / G 1 2 ( x ) ≅ L n ( 2 ) , (c) G(x) splits over $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) , (d) the vertex-wise stabilizer $$G_1(x)$$ G 1 ( x ) of the neighbourhood of x is a non-trivial group, and (e) $$n \ne 4$$ n ≠ 4 .

Неограниченная порядковая сходимость и теорема Гордона

The celebrated Gordons theorem is a natural tool for dealing with universal completions of Archimedean vector lattices. Gordons theorem allows us to clarify some recent results on unbounded order convergence. Applying the Gordon theorem, we demonstrate several facts on order convergence of sequences in Archimedean vector lattices. We present an elementary Boolean-Valued proof of the Gao--Grobler--Troitsky--Xanthos theorem saying that a sequence x_n in an Archimedean vector lattice X is uo-null (uo-Cauchy) in X if and only if x_n is o-null (o-convergent) in Xu. We also give elementary proof of the theorem, which is a result of contributions of several authors, saying that an Archimedean vector lattice is sequentially uo-complete if and only if it is sigma-universally complete. Furthermore, we provide a comprehensive solution to Azouzis problem on characterization of an Archimedean vector lattice in which every uo-Cauchy net is o-convergent in its universal completion.

Completions of Non-Symmetric Metric Spaces Via Enriched Categories

It is known from [Lawvere, Repr. Theory Appl. Categ. 1: 1–37 2002] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0, ∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0, ∞] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the so-called dcpo completion.

INTRANSITIVE GEOMETRIES

A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples.