Application of Höhle’s square roots on Hoop algebras

Document Type : Research Paper

Authors

Department of Mathematics, Semnan University, Semnan, Iran

Abstract

In this article, we use the square root as a tool to study hoop algebras. To do so, we define square root and make the first attempt to explore the significance properties of this concept in this setting. Then, due to the key role of square roots in obtaining new hoop algebras, we apply them to the filters of hoop algebras, and show that the formation of square roots on quotient structures of hoop algebras by their filters is well-behaved. In addition, a new class of hoop algebras having square roots, so-called good hoop algebras, is introduced, and some relationships with other classes of ordered algebras such as Boolean algebras and Gödel algebras are explored. Several examples are provided as well. Ultimately, it is shown that the class of all (good) bounded V-hoop algebras with square roots is a variety.

Keywords

Main Subjects

References

[1] Agliano, P., Ferreirim, I. M. A., & Montagna, F. (2007). Basic hoops: an algebraic study of continuous t-norms, Studia Logica, 87(1), 73-98. https://doi.org/10.1007/s11225-007-9078-1
[2] Ambrosio, R. (1999). Strict MV -algebras, J. Math. Anal. Appl., 237(1), 320-326. https://doi.org/10.1006/jmaa.1999.6482
[3] Berman, J., & Blok, W. J. (2004). Free Lukasiewicz and Hoop Residuation Algebras. Studia Logica 77, 153{180. https://doi.org/10.1023/B:STUD.0000037125.49866.50
[4] Blok, W. J., & Ferririm, I. M. A. (1993). Hoops and their implicational reducts, Algebraic methods in logic and computer Science, Banach center Publications, 28(1), (1993), 219-230. http://eudml.org/doc/262577
[5] Blok, W., & Ferreirim, I. (2000). On the structure of hoops. Algebra univers. 43, 233{257. https://doi.org/10.1007/s000120050156
[6] Borzooei, R. A., & Aaly Kologani, M. (2014). Filter theory of hoop-algebras, Journal of advanced research in pure mathematics, 6(4), 72{86.
[7] Borzooei, R. A., & Aaly Kologani, M. (2020). Result on hoops. J. Algebr. Hyperstruct. Log. Algebras, 1(1), 61-77. https://doi.org/10.29252/hatef.jahla.1.1.5
[8] Bosbach, B. (1969). Komplementare Halbgruppen. Axiomatik und Arithmetik, Fund. Math. 64, 257-287. http://eudml.org/doc/214075.
[9] Bosbach, B. (1970). Komplementare Halbgruppen. Kongruenzen und Quotienten, Fund. Math. 69, 1-14. http://eudml.org/doc/214248
[10] Buchi, J. R., & Owenz, T. M., Complemented monoids and hoops, unpublished manuscript.
[11] Chen, W., & Dudek, W.A. (2015). The representation of square root quasi-pseudo-MV algebras. Soft. Comput. 19, 269{282. https://doi.org/10.1007/s00500-014-1466-7
[12] Dvurecenskij, A., & Zahiri, O. (2023). Some results on pseudo MV -algebras with square roots, Fuzzy Sets and Systems, 465, 108527. https://doi.org/10.1016/j.fss.2023.108527
[13] Georgescu, G., Leustean, L., & Preoteasa, V. (2005). Pseudo-hoops, Mult.-Valued Logic Soft Comput., 11(1), 153-184.
[14] U. Hohle, Commutative, residuated `-monoids. In: U. Hohle., E.P. Klement (eds.) Non-Classical Logics and their Applications to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory, 32, Springer, Dordrecht, (1995), 53-106.
[15] Kondo, M., (2011). Some types of  lters in hoops, 41st IEEE International Symposium on Multiple-Valued Logic, (2011), 50-53. https://doi.org/10.1109/ISMVL.2011.9.
[16] Namdar, A, Borzooei R. A., Borumand Saeid, A., & Aaly Kologani, M., (2017). Some results in hoop algebras. Journal of Intelligent and Fuzzy Systems: Applications in Engineering and Technology, 32(3), 1805-1813. https://doi.org/10.3233/JIFS-152553
[17] D. Piciu, Algebras of Fuzzy Logic, Editura Universtaria, Craiova, 2007. 

Files

History

  • Receive Date: 04 November 2025
  • Revise Date: 16 February 2026
  • Accept Date: 17 March 2026

Share

How to cite

Statistics