Lattices of relatively axiomatizable classes

  • Authors:

  • Dmitry E. Pal'chunov

  • Affiliations:

  • Institute of Mathematics, Siberian Branch of Russian Acad. Sci., Novosibirsk, Russia

  • Venue:
  • ICFCA'07 Proceedings of the 5th international conference on Formal concept analysis
  • Year:
  • 2007

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Abstract

In the paper we study lattices of axiomatizable classes and relatively axiomatizable classes. This study is based on Formal Concept Analysis [4,5]. The notion of a relatively axiomatizable class is a generalization of such concepts as variety, quasivariety, ∀-axiomatizable class, ∃-axiomatizable class, Πn0-axiomatizable class, Σn0-axiomatizable class and so on. Relatively axiomatizable classes were studied in [10]. It is proved in the paper that any finite lattice may be represented as the lattice of all relatively axiomatizable subclasses of the class of all models of a one-element signature with respect to some set of sentences. Also we prove that any finite or countable complete lattice is isomorphic to the lattice of all relatively axiomatizable subclasses of some class of models with respect to a proper set of sentences.