Applications and Computational Varieties of Free Soft Trioids
DOI:
https://doi.org/10.32350/sir.101.03Keywords:
computational applications, free soft trioid, soft set, soft trioidAbstract
Integrating soft set theory with multi-operational structures poses a foundational challenge In parameterized universal algebra,. In this paper, the theory of free soft trioids were developed establishing the existence and construction of free objects in the category SoftTrioid. Also we construct a non-restrictive definition of soft trioids equipped with pointwise operations with genuine context sensitivity. The free soft trioid being realized as a two-sorted term algebra prove the universal property and show that the word problem is decidable in linear time via a confluent rewriting system as the algorithmic implementations were supported by empirical validation. The relevance of our study were demonstrated by its applications in context-dependent type systems, formal verification, knowledge representation and access control thus situating soft trioid theory as a robust link between algebraic foundations and computational practice.
Downloads
References
[1] Mac Lane S. Categories for the Working Mathematician. New York, NY: Springer-Verlag; 1971. https://doi.org/10.1007/978-1-4757-4721-8
[2] Borceux F. Handbook of Categorical Algebra. Cambridge, UK: Cambridge University Press; 1994.
[3] Kelly GM. Basic concepts of enriched category theory. Repr Theory Appl Categ. 2005;10:1-136. https://doi.org/10.1023/A:1023124524132
[4] Beck J. Triples, algebras and cohomology. PhD thesis. Columbia University; 1967.
[5] Molodtsov DA. Soft set theory—First results. Comput Math Appl. 1999;37(4-5):19-31. https://doi.org/10.1016/S0898-1221(99)00020-8
[6] Zadeh LA. Fuzzy sets. Inform Control. 1965;8:338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
[7] Pawlak Z. Rough sets. Int J Comput Inform Sci. 1982;11:341-356. https://doi.org/10.1007/BF01001956
[8] Feng F, Jun YB, Zhao X. Soft semirings. Comput Math Appl. 2008;56:2621-2628. https://doi.org/10.1016/j.camwa.2008.07.018
[9] Shabir M, Naz M. On soft topological spaces. Comput Math Appl. 2011;61:1786-1799. https://doi.org/10.1016/j.camwa.2010.07.046
[10] Maji PK, Biswas R, Roy AR. Soft set theory. Comput Math Appl. 2003;45:555-562. https://doi.org/10.1016/S0898-1221(03)00016-6
[11] Maji PK, Roy AR, Biswas R. An application of soft sets in a decision making problem. Comput Math Appl. 2002;44:1077-1083. https://doi.org/10.1016/S0898-1221(02)00216-X
[12] Loday J-L. Dialgebras. In: Dialgebras and Related Operads. Berlin, Germany: Springer; 2001:7-66. https://doi.org/10.1007/978-3-662-04546-4
[13] Zhuchok AV. Dimonoids. Algebra Logic. 2011;50:323-340.
[14] Loday J-L, Ronco MO. Trialgebras and families of polytopes. Contemp Math. 2004;346:369-398.
[15] Zhuchok AV. Trioids. Asian-Eur J Math. 2015;8(4):1550089. https://doi.org/10.1142/S1793557115500896
[16] Zhuchok AV. Semiretractions of trioids. Ukr Math J. 2014;66:218-231.
[17] Usenko VM. Semiretractions of monoids. Proc Inst Appl Math Mech. 2000;5:155-164.
[18] Zhuchok AV. Commutative dimonoids. Algebra Discrete Math. 2009;2:116-127.
[19] Novelli J-C, Thibon J-Y. Construction of dendriform trialgebras. C R Math Acad Sci Paris. 2006;342:365-369. https://doi.org/10.1016/j.crma.2005.12.007
[20] Casas JM. Trialgebras and Leibniz 3-algebras. Bol Soc Mat Mex. 2006;12:165-178.
[21] Zhuchok AV. Some congruences on trioids. J Math Sci. 2012;187:138-145. https://doi.org/10.1007/s10958-012-0938-9
[22] Zhuchok AV. Semilattice decompositions of trioids. Bull Acad Sci Repub Mold Math. 2013;71:130-134.
[23] Oguz G. Some notes on soft dimonoids. Stoch Model Comput Sci. 2023;3:45-62.
[24] Oguz G. On actions of soft digroups. J Intell Fuzzy Syst. 2025;49:1529-1544. https://doi.org/10.3233/JIFS-XXXXX
[25] Oguz G. On characterizations of soft dirings. Turk J Math Comput Sci. 2025;17:512-518.
[26] Burris S, Sankappanavar HP. A Course in Universal Algebra. New York, NY: Springer-Verlag; 1981.
[27] Baader F, Nipkow T. Term Rewriting and All That. Cambridge, UK: Cambridge University Press; 1998.
[28] Rutten JJM. Universal coalgebra: a theory of systems. Theoret Comput Sci. 2000;249:3-80. https://doi.org/10.1016/S0304-3975(00)00056-6
[29] Appel AW. Modern Compiler Implementation in ML. Cambridge, UK: Cambridge University Press; 1998.
[30] Mickens RE. Nonstandard Finite Difference Models of Differential Equations. Singapore: World Scientific; 2000.
[31] Khan AR, Raja MAZ, Ahmad I. J Appl Math Comput. 2021;65:709-734. https://doi.org/10.1007/s12190-021-01534-2
[32] Raja MAZ, Tabassum R, Ahmad I. Math Methods Appl Sci. 2021;44:12545-12566. https://doi.org/10.1002/mma.7201
[33] Aslam NA, Raja MAZ, Alshomrani AS. Chaos Solitons Fractals. 2022;158:112034. https://doi.org/10.1016/j.chaos.2022.112034
[34] Raja MAZ, Alassar RS, Alshomrani AS. Int J Biomath. 2023;16:2250097. https://doi.org/10.1142/S1793524522500970
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Gülay Oğuz, David A. Oluyori
This work is licensed under a Creative Commons Attribution 4.0 International License.
