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Implication in finite posets with pseudocomplemented sections.

Ivan Chajda1, Helmut Länger1,2

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

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Summary
This summary is machine-generated.

This study extends relative pseudocomplementation to posets, creating an algebraic semantics for a generalized intuitionistic logic. An "unsharp" implication and conjunction are introduced, recovering the poset

Keywords:
Intuitionistic implicationPosetPoset with pseudocomplemented sectionsRelative pseudocomplementSectionUnsharp conjunctionUnsharp implicationUnsharp residuation

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Area of Science:

  • Algebraic logic
  • Order theory
  • Universal algebra

Background:

  • Relatively pseudocomplemented lattices are algebraic semantics for intuitionistic logic.
  • Sectionally pseudocomplemented lattices extend this concept to non-distributive lattices.

Purpose of the Study:

  • To extend the concept of sectional pseudocomplementation to posets with a top element.
  • To establish these posets as algebraic semantics for a generalized intuitionistic logic.
  • To introduce and analyze an "unsharp" implication and conjunction.

Main Methods:

  • Generalization of sectionally pseudocomplemented lattices to posets.
  • Introduction of a novel "unsharp" implication operator.
  • Demonstration of the adjoint relationship between "unsharp" conjunction and implication.

Main Results:

  • The proposed poset structure serves as an algebraic semantics for a more general intuitionistic logic.
  • The "unsharp" implication, mapping to subsets, allows for the recovery of the original poset's order.
  • An "unsharp" conjunction operator is introduced, forming an "unsharp" residuated poset.

Conclusions:

  • The study successfully generalizes relative pseudocomplementation to posets.
  • The introduced "unsharp" logical operators provide a framework for a broader class of intuitionistic logics.
  • The findings contribute to the algebraic theory of non-classical logics.