Heritability of types of a pregeometry relative to a family of relational structures
DOI:
https://doi.org/10.31489/2026m1/169-178Keywords:
pregeometry, Boolean algebra, degeneracy, modularity, local finiteness, algebraic closure, relational structure, intersection of structures, union of structuresAbstract
A series of geometrical and topological properties induced by structures, including degeneration, modularity, local modularity, projectivity, local finiteness, etc., play an important role in clarifying structural relationships and in classifying basic and derivative semantical and syntactical objects related to a given class of structures and their valuable characteristics. It is natural to turn to the family of all structures on a given finite or infinite universe, which allows us to represent all possible structures of a given cardinality up to the definability and to describe relationships, possibilities of preserving and violating structural properties during enrichments and restrictions of structures within the framework of the chosen family. This paper studies the behavior of pregeometry types (degenerate, locally finite, modular) within the Boolean algebra B(M) of regular expansions and reducts of a relational structure M. We establish criteria for the inheritance of pregeometry properties under Boolean operations, proving that degeneracy and local finiteness are preserved under intersections. In contrast, we show through counterexamples that modularity generally fails to be preserved, as does local finiteness under unions. We formulate a sufficient condition of linear growth of the closure operator under which the union of locally finite structures remains locally finite. These results reveal a fundamental asymmetry between intersection and union operations, contributing to geometric stability theory by delineating the preservation boundaries of pregeometries in Boolean families of structures.
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