Fuzzy Ideals in Algebraic Structures

Fuzzy ideals are an extension of classical ideal theory in algebra whereby each element of an algebraic structure is assigned a degree of membership between zero and one. This generalisation accommodates uncertainty, vagueness and partial truth in settings ranging from rings and semirings to semigroups and other universal algebras. The fundamental notion of a fuzzy ideal retains the closure properties and absorption laws of crisp ideals while allowing the flexibility of membership functions. In recent years, this theory has amalgamated with rough set approximations, bipolar and multipolar fuzzy sets, soft set theory and higher-arity operations such as ternary semirings. These hybrid approaches yield a unified framework for modelling imprecision in algebraic systems and have found applications in decision support, information retrieval and the structural analysis of complex networks. By exploring relations, homomorphisms and congruences in fuzzy and rough contexts, researchers have established a rich web of interconnections that broadens both the theoretical foundations and the scope of practical applications.

Research from Nature Portfolio

Recent studies have advanced the theory of rough fuzzy ideals by extending it to three-dimensional algebraic structures. One notable investigation has introduced the concept of roughness for (∈,∈∨q)-fuzzy ideals in a ternary semiring setting. By employing set-valued and strong set-valued homomorphisms, this work defines lower and upper approximations of fuzzy semiprime and prime ideals under ternary multiplication, demonstrating that these approximations themselves satisfy the defining conditions of fuzzy ideals. This generalisation not only preserves ideal-like properties in higher-arity operations but also bridges traditional binary algebra with emerging multi-operator frameworks.

Research from all publishers

A complementary line of inquiry has examined rough bipolar fuzzy ideals in semigroups through the lens of congruence relations. This approach defines rough approximations of bipolar fuzzy subsemigroups and extends to left, right, two-sided, interior and bi-ideals, illuminating structural properties such as stability under complete congruences and interactions between roughness and bipolarity.

Another significant contribution introduces hybrid ideals in near-subtraction semigroups by merging fuzzy set theory with soft set theory. This research characterises hybrid left and right ideals, explores their intersection and product structures, and analyses the behaviour of these ideals under homomorphic preimages, thereby offering a versatile toolkit for handling asymmetric and uncertain information.

A further development in ordered semigroups defines fuzzy bipolar soft semiprime ideals, combining order relations, bipolar fuzzy sets and soft set parameters. This framework yields characterisations of ordered semigroup classes via fuzzy bipolar soft semiprimality and studies Cartesian products of semiprime and prime ideals, furnishing concrete examples that demonstrate the applicability of these concepts.

Fuzzy Ideals in Algebraic Structures publication trend

The graph below shows the total number of articles in fuzzy ideals in algebraic structures across all publications each year (not limited to Nature Index journals).

Technical terms

Fuzzy set: A collection in which each element has a grade of membership ranging between zero and one, representing degrees of truth or uncertainty.

Ideal: A subset of an algebraic structure that is closed under the structure’s operations and absorbs multiplication by elements of the structure.

Rough approximation: A method to model vagueness by assigning lower and upper bounds to fuzzy sets using equivalence or congruence relations.

Bipolar fuzzy set: A fuzzy set that separately captures positive and negative membership degrees to represent bipolar information.

Soft set: A parameterised family of subsets used for handling uncertainty without requiring a membership function.

Ternary semiring: An algebraic structure with a ternary (three-argument) associative multiplication and a binary addition satisfying distributive properties.

References

1. Rough bipolar fuzzy ideals in semigroups. Complex & Intelligent Systems (2023).
2. Generalized roughness of three dimensional (∈,∈∨q)-fuzzy ideals in terms of set-valued homomorphism. Scientific Reports (2024).
3. Fuzzy bipolar soft semiprime ideals in ordered semigroups. Heliyon (2021).
4. Hybrid ideals in near-subtraction semigroups. AIMS Mathematics (2022).