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Seok Zun Song1, Hee Sik Kim2, Young Bae Jun3

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This study introduces integer-soft semigroups and their ideals, exploring their properties and characterizations. The findings confirm that the soft intersection of these structures preserves their properties, offering new insights into semigroup theory.

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Area of Science:

  • Abstract Algebra
  • Fuzzy Set Theory
  • Mathematical Structures

Background:

  • Existing research in semigroup theory lacks comprehensive frameworks for handling uncertainty.
  • The integration of soft set theory with algebraic structures is an emerging area of interest.
  • Need for robust mathematical tools to analyze complex algebraic systems with imprecise data.

Purpose of the Study:

  • To introduce and define the concepts of integer-soft semigroups and integer-soft left/right ideals.
  • To investigate the fundamental properties of these newly defined structures.
  • To establish characterizations of subsemigroups, left/right ideals, and regular semigroups using these concepts.

Main Methods:

  • Introduction of int-soft semigroups and int-soft left/right ideals.
  • Utilizing the notion of inclusive sets for characterizations.
  • Application of int-soft products for further characterizations.
  • Demonstration of closure properties under soft intersection.

Main Results:

  • Characterizations of subsemigroups and left/right ideals are derived using int-soft set theory.
  • Int-soft semigroups and int-soft left/right ideals are characterized via int-soft products.
  • The soft intersection of int-soft left/right ideals and int-soft semigroups is proven to be closed.
  • The concept of int-soft quasi-ideals is introduced, leading to the characterization of regular semigroups.

Conclusions:

  • The study successfully extends semigroup theory with the novel concept of integer-soft structures.
  • The established characterizations provide a deeper understanding of the algebraic properties of these soft structures.
  • The results highlight the utility of soft set theory in analyzing algebraic objects and their ideals.