Abstract. We introduce a class of finite configurations, which we call combinatorial Grassmannians, and which generalize the Desargues configuration. Fundamental geometric properties of them are established, in particular we determine their automorphisms, correlations, mutual embedability, and prove that no one of them contains a Pascal or Pappus figure.

Finite generalized Veronese spaces of special type (associated with single-line geometry) are studied, and problems concerning embeddability of the resulting configurations are discussed. (2000). Primary 51E14, Secondary 51A45, 51A20. Mathematics Subject Classification

We prove that Euclidean geometry is interpretable in terms of orthogonality of its k-subspaces, and thus it can be formalized as a theory of such an orthogonality. (2000). 51F20, 51M04. Mathematics Subject Classification

We define the operation of convolution, which associates with a partial Steiner triple system and an abelian group a new partial Steiner triple system, and we determine conditions under which some fundamental geometric properties remain invariant under the operation of convolution.