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Sheffer operation in relational systems.

Ivan Chajda1, Helmut Länger1,2

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

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|January 21, 2022
PubMed
Summary
This summary is machine-generated.

This study extends the Sheffer operation concept to relational systems, establishing a link between Sheffer groupoids and these systems. This connection allows algebraic methods to analyze relational structures, revealing properties like antisymmetry and transitivity.

Keywords:
Directed relational systemInvolutionKleene relational systemRelational systemSheffer groupoidSheffer operationTwist product

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Area of Science:

  • Universal Algebra
  • Lattice Theory
  • Relational Systems

Background:

  • The Sheffer operation is a key concept in Boolean algebras and orthomodular lattices.
  • Relational systems with involution are fundamental structures in various mathematical fields.

Purpose of the Study:

  • To generalize the Sheffer operation to arbitrary directed relational systems with involution.
  • To establish a duality between Sheffer groupoids and directed relational systems with involution.
  • To explore algebraic properties of these systems and their relations.

Main Methods:

  • Extension of the Sheffer operation concept.
  • Construction of Sheffer groupoids from relational systems and vice versa.
  • Characterization of relational properties using algebraic identities and quasi-identities.
  • Introduction of twist products and Kleene relational systems.

Main Results:

  • A one-to-one correspondence is established between Sheffer groupoids and directed relational systems with involution.
  • Commutative Sheffer groupoids are shown to form a congruence distributive variety.
  • Symmetry, antisymmetry, and transitivity of binary relations are characterized algebraically.
  • Directed relational systems can be embedded into involutive systems via twist products.
  • Transitive relational systems can be embedded into Kleene relational systems.

Conclusions:

  • The study provides a powerful algebraic framework for investigating directed relational systems with involution.
  • The established duality simplifies the analysis of relational structures by leveraging groupoid theory.
  • New constructions like twist products offer methods for embedding and extending relational systems.