TwoFold Algebra

 

The TwoFold Algebra, with its corresponding TwoFold Law, and their derivatives of Fuzzy (and all fuzzy-extensions) Two Fold Algebras and Laws were introduced by Smarandache [1] in 2024. Although very recent, it is has spread out very fast. Practical applications and several numerical examples were presented as well.

 

    Definition

Let's denote by fuzzy-extension theories, all types of fuzzy related theories:

Fuzzy-Extension Theories = { Fuzzy Set, Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set (Atanassov’s Intuitionistic Fuzzy Set of second-type), q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, Neutrosophic Set, Quadruple Neutrosophic Set, Quintuple Neutrosophic Set, n-Refined Neutrosophic Set, Plithogenic Set, and n-HyperSpherical Fuzzy Set, Regret Theory, Grey System Theory, Three-Ways Decision, etc.}

A TwoFold Algebra is formed by two types of algebras:

       (i) the first algebra is a classical algebra, with respect to the elements x belonging to a given set A;

(ii) and the second algebra is a fuzzy or fuzzy-extension algebra with respect to any fuzzy-extension components [regarding the degrees of membership/truth, and/or nonmembership/falsehood/, and/or indeterminacy, and/or other types of degree of belongingness of the elements x with respect to the set A].

This is a hybrid structure, since a classical algebraic operation is inter-related with a fuzzy or fuzzy-extension operation.

That's what happens in our real world.

    Practical Example:

Two employees, John and George, work for the same company. John's degree of work quality is 70% (or 0.7) and produces 40 pieces per hour, while George's degree of work quality is 50% (or 0.5) and produces 20 pieces per hour. The manager has observed that what a worker does not do well, the other does better, or the workers complement each other. As such, he had the idea to make them work together as a team, to improve the quality and increase the quantity by 10%.

Same thing happened to the other workers in the team, so he decided to organize them to work in teams to improve the quality.

          John0.7 George0.5 = TeamJGmax{0.7, 0.5}

     or     400.7 200.5        = (40 + 20)·(100% + 10%)max{0.7, 0.5} = 60 · (1.1)0.7 = 660.7

Therefore, the team John & George produces now 66 pieces of quality 70%, instead of 60 pieces with some of them of lower quality (50%).

 

    References

[1] Florentin Smarandache, Neutrosophic TwoFold Algebra, Plithogenic Logic and Computation, 1, 11-15, 2024, https://fs.unm.edu/NeutrosophicTwoFoldAlgebra.pdf 

[2] Abdallah Shihadeh, Khaled Ahmad Mohammad Matarneh, Raed Hatamleh, Randa Bashir Yousef Hijazeen, Mowafaq Omar Al-Qadri, Abdallah Al-HusbanAn Example of Two-Fold Fuzzy Algebras Based On Neutrosophic Real Numbers, Neutrosophic Sets and Systems, Vol. 67, 2024, pp. 169-178. DOI: 10.5281/zenodo.11151930, https://fs.unm.edu/TFA/TwoFoldFuzzy11.pdf

[3] Hazim M. Wali Al-TameemiFuzzy Metric Spaces Of The Two-Fold Fuzzy Algebra, Neutrosophic Sets and Systems, Vol. 67, 2024, pp. 105-133. DOI: 10.5281/zenodo.11123602, https://fs.unm.edu/TFA/FuzzyMetricSpaces7.pdf

[4] Abdallah Shihadeh, Khaled Ahmad Mohammad Matarneh, Raed Hatamleh, Mowafaq Omar Al-Qadri, Abdallah Al-HusbanOn The Two-Fold Fuzzy n-Refined Neutrosophic Rings For 2≤n≤3, Neutrosophic Sets and Systems, Vol. 68, 2024, pp. 8-25. DOI: 10.5281/zenodo.11406449, https://fs.unm.edu/TFA/TwoFold2.pdf

[5] Nabil Khuder Salman, On the Special Gamma Function Over the Complex Two-Fold Algebras, Neutrosophic Sets and Systems, Vol. 68, 2024, pp. 26-38. DOI: 10.5281/zenodo.11406461, https://fs.unm.edu/TFA/SpecialGamma3.pdf

[6] Nawar Hazim Mohammed, On The Solutions of Some Different Types of Two-Fold Fuzzy and Neutrosophic Differential Equations, Neutrosophic Sets and Systems, Vol. 70, 2024, pp. 359-368. DOI: 10.5281/zenodo.13177544, https://fs.unm.edu/TFA/TwoFold22.pdf

[7] Raed Hatamleh, Ayman HazaymehFinding Minimal Units In Several Two-Fold Fuzzy Finite Neutrosophic Rings, Neutrosophic Sets and Systems, Vol. 70, 2024, pp. 1-16. DOI: 10.5281/zenodo.13160839, https://fs.unm.edu/TFA/FindingMinimal1.pdf