Disjoint and complete unions of incidence structures. (English) Zbl 0897.08002

Machala, František; Pomp, Marek

Math. Bohem. 122, No. 4, 365-374 (1997).

First the authors specify some important types of incidence structures (incidence structure = triplet \((A,B,C)\) where \(A,B,C\) are sets with \(C\subseteq A\times B\)) with respect to the operations of “full preimages” or “full images”, respectively: complete, open, trivial, regular or simple.
If \(\operatorname{Im} = (A,B,C)\) is an incidence structure, then decompositions \(\mathcal A\) of \(A\) and \(\mathcal B\) of \(B\) are investigated such that the corresponding couples of their blocks form incidence substructures of \(\operatorname{Im} \). These decompositions arise naturally by homomorphisms of incidence structures. Inspired by the notion of modularity of lattices, also modularity of incidence structures is defined [cf. F. Machala, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. Math. 34, 137-145 (1995; Zbl 0853.08001)] and its influence onto the decompositions mentioned above is described in the main Theorem 6.

Reviewer: V.Havel (Brno)

MSC:

08A02	Relational systems, laws of composition
08A35	Automorphisms and endomorphisms of algebraic structures
51G05	Ordered geometries (ordered incidence structures, etc.)

Keywords:

epimorphisms of incidence structures; unions of substructures

Citations:

Zbl 0853.08001