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Consistent posets.

Ivan Chajda1, Helmut Länger1,2

  • 1Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic.

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|November 8, 2021
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Summary
This summary is machine-generated.

We introduce consistent posets, a type of bounded poset with an antitone involution. These posets can be represented algebraically and form the basis for non-classical logic with unsharp conjunction and implication.

Keywords:
AdjointnessAntitone involutionCommutative meet-directoidConsistent posetDedekind–MacNeille completionDistributive posetResiduationStrongly modular poset

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Area of Science:

  • * Mathematical Logic
  • * Lattice Theory
  • * Abstract Algebra

Background:

  • * Explores the structure of bounded posets with antitone involutions.
  • * Investigates properties related to lower cones and specific element exclusions (0, 1).

Purpose of the Study:

  • * To define and introduce the concept of "consistent posets."
  • * To establish algebraic representations for these posets.
  • * To explore their connection to non-classical logic and lattice theory.

Main Methods:

  • * Definition of consistent posets based on specific properties of antitone involutions and lower cones.
  • * Representation of consistent posets using commutative meet-directoids with antitone involutions.
  • * Conversion of finite distributive or strongly modular consistent posets into residuated structures.
  • * Application of Dedekind-MacNeille completion to consistent posets.

Main Results:

  • * Consistent posets are characterized by coinciding lower cones under specific conditions.
  • * These posets admit representation via commutative meet-directoids.
  • * Finite distributive or strongly modular consistent posets can be converted into residuated structures, serving as algebraic semantics for non-classical logics.
  • * The Dedekind-MacNeille completion of a consistent poset results in a consistent lattice.

Conclusions:

  • * Consistent posets offer a novel structure with connections to abstract algebra and logic.
  • * The algebraic representations provide a framework for understanding these posets.
  • * The link to non-classical logic highlights their potential applications in formal systems.
  • * Consistent lattices emerge as a significant outcome of the completion process.