| Abstract: |
Omega-algebras are algebraic structures enriched with a generalized, lattice-valued equality, extending the classical notion of equality, introduced in [5]. In other words, Omega-algebra is an algebraic system containing an algebraic structure endowed with an Omega-equality, enabling the interpretation of classical identities as lattice-valued formulas. Quotients defined via level cuts of equality yield classical algebras that satisfy the same identities in the traditional sense.
In recent years, various types of Omega-algebraic structures have been developed and investigated [1-4,6]. The distinction between classical fuzzy algebras and the broader, more flexible framework of Omega-algebras is that in classical fuzzy algebras, the construction begins with a set endowed with a fixed algebraic structure, like a classical group, and fuzziness is introduced through membership functions (obtaining, e.g., fuzzy groups, or subgroups). In contrast, in Omega-algebras we do not assume a predefined algebraic structure. While operations exist, they need not satisfy classical algebraic laws, allowing for a more adaptable modeling approach—particularly useful in contexts involving uncertainty or imprecise data. A key feature of this framework is the equivalence relation E connected to Omega-sets, which acts as a congruence in the algebraic context. When elements are indistinguishable within a certain p-cut, E identifies them as equivalent, facilitating a tolerant interpretation of data. This capability is essential in applications where exact precision must yield to the realities of incomplete or noisy information.
In this context, we introduced and applied equations and systems of equations in various algebraic structures. While a quasigroup is the most general algebraic structure allowing unique solving of equations with one binary operation, fields are such structures with equations with two binary operations. Vector spaces are preferable structures for solving systems of linear equations. We developed the theory of solving equations in a lattice-valued setting with Omega-quasigroups, Omega-fields, and Omega-vector spaces [1-4].
In this lecture, the results obtained in different articles and in different teams will be presented and generalized in a unified and uniform manner, allowing for approximate solutions of various types of equations and systems of equations in this context.
References.
1. Patricia Ferrero, Jorge Jiménez, María Luisa Serrano, Branimir Šešelja and Andreja Tepavčević. Omega vector spaces, submitted.
2. Jorge Jimenez, María Luisa Serrano, Branimir Šešelja, and Andreja Tepavčević. Omega ideals in omega rings and systems of linear equations over omega fields. Axioms, 12(8):757, 2023.
3. Jorge Jimenez, María Luisa Serrano, Branimir Šešelja, and Andreja Tepavčević. Omega-rings. Fuzzy Sets and Systems, 455:183–197, 2023.
4. Aleksandar Krapež, Branimir Šešelja, and Andreja Tepavčević. Solving linear equations by fuzzy quasigroups techniques. Information Sciences, 491:179–189, 2019.
5. Branimir Šešelja, and Andreja Tepavčević. Fuzzy identities. In 2009 IEEE International Conference on Fuzzy Systems, pages 1660–1664, 2009.
6. Branimir Šešelja and Andreja Tepavčević. Omega-groups in the language of omega-groupoids. Fuzzy Sets and Systems, 397:152–167, 2020 |