Ordered Fuzzy Inner Product

• Ordered fuzzy inner product is a rigorous extension of classical inner products, incorporating ordered intervals to embed fuzziness into similarity and geometric properties.
• It preserves critical axioms such as positive definiteness, conjugate symmetry, and fuzzy Cauchy–Schwarz through two-sided interval bounds.
• This construct underpins new approaches in fuzzy rough set theory and operator characterization, enhancing analysis in uncertain, graded environments.

The ordered fuzzy inner product is a mathematically rigorous construct extending the classical inner product to fuzzy contexts via ordered intervals and general fuzzy algebraic systems. It refines notions of similarity, norm, and geometric properties for vectors and sets by embedding fuzziness into the fundamental structure, thereby enabling robust analysis in uncertain and graded environments. This concept is central in recent axiomatic developments of fuzzy inner product spaces and L-valued rough set theory, introducing a new class of functional forms that preserve essential geometric inequalities and offer generalization over earlier models (Daraby et al., 24 Dec 2025, Li et al., 2024).

1. Formal Definition and Algebraic Foundation

Let $a, b \in \mathbb{R}$. An ordered interval $[a,b]_o$ is defined by: $$[a,b]_o = \begin{cases} \{x\mid a\le x\le b\}, & a\le b, \ \{x\mid b\le x\le a\}, & a>b. \end{cases}$$ The set of all such intervals is $$O(\mathbb{R}) = \{ [a,b]_o \mid a,b\in\mathbb{R} \}$$. On $O(\mathbb{R})$, coordinate‐wise operations are defined: $$\begin{aligned} [a,b]_o \oplus [c,d]_o &= [a+c,\, b+d]_o, \ [a,b]_o \ominus [c,d]_o &= [a-c,\, b-d]_o, \ [a,b]_o \odot [c,d]_o &= [a\,c,\, b\,d]_o, \ k\,[a,b]_o &= [k\,a,\, k\,b]_o \quad (k\in \mathbb{R}). \end{aligned}$$ A natural partial order is given by: $$[a,b]_o \succeq [c,d]_o \quad \Longleftrightarrow \quad \min\{a,b\}\ge\min\{c,d\},\ \max\{a,b\}\ge\max\{c,d\}.$$ Within an L-universe framework, with $X$ a nonempty crisp set and $(L,\wedge,\vee,\circledast,\to,0,1)$ a GL-quantale, for $L$-subsets $M,Q\in P(\mathbb{U})$, the ordered fuzzy inner product is: $$\mathbb{I}(M,Q) = \bigvee_{x\in X} \left[M(x)\circledast\left(\mathbb{U}(x)\to Q(x)\right)\right] \in L,$$ where $\mathbb{U}: X\to L$ is the universe's local capacity and $$P(\mathbb{U})=\{W:X\to L \mid W(x) \leq \mathbb{U}(x) \ \forall x\}$$ (Li et al., 2024).

2. Ordered Fuzzy Inner Product on Vector Spaces

For a complex vector space $X$, a fuzzy number $\mathcal{K}:\mathbb{R}\to[0,1]$, and classical inner products $$\langle\cdot,\cdot\rangle', \langle\cdot,\cdot\rangle''$$ on $X$, define: $$\langle\cdot,\cdot\rangle_\alpha : (0,1] \times X \times X \to \mathbb{C},$$ to be an ordered fuzzy inner product if for all $x, y \in X$ and $\alpha \in (0,1]$,

$$\mathcal{K}(|\langle x, y\rangle_\alpha|) \ge \alpha \Longleftrightarrow |\langle x, y\rangle_\alpha| \in [A_\alpha |\langle x, y\rangle'|,\, B_\alpha |\langle x, y\rangle''|]_o$$

with $0 < A_\alpha \leq B_\alpha < \infty$ constants. A single inner product variant is frequently used: $$A_\alpha |\langle x, y\rangle'| \leq |\langle x, y\rangle_\alpha| \leq B_\alpha |\langle x, y\rangle'|,$$ maintaining the interval bounds (Daraby et al., 24 Dec 2025).

3. Fundamental Properties and Geometric Inequalities

The ordered fuzzy inner product satisfies a set of axiomatic “quasi-inner-product” properties:

• Positive Definiteness:

$$\langle x, x\rangle_\alpha \succeq 0;\quad \langle x, x\rangle_\alpha = 0 \iff x=0.$$

• Conjugate Symmetry:

$$\langle y, x\rangle_\alpha = \overline{\langle x, y\rangle_\alpha}$$

• Quasi-homogeneity:

$$\frac{|k|A_\alpha}{B_\alpha} |\langle x,y\rangle_\alpha| \leq |\langle kx,y\rangle_\alpha| \leq \frac{|k|B_\alpha}{A_\alpha} |\langle x,y\rangle_\alpha|$$

• Quasi-additivity:

$$|\langle x+z, y\rangle_\alpha| \leq \frac{B_\alpha}{A_\alpha}(|\langle x, y\rangle_\alpha| + |\langle z, y\rangle_\alpha|).$$

When $A_\alpha = B_\alpha = 1$, these reduce to the classical axioms. Several important inequalities follow:

Inequality	Formula	Context
Fuzzy Cauchy–Schwarz	$$\left|\langle x,y\rangle_\alpha\right|^2 \leq \left(\frac{B_\alpha}{A_\alpha}\right)^2 \left|\|x\|_\alpha \|y\|_\alpha\right|^2$$	General $X$
Fuzzy Parallelogram Law	$$2A_\alpha B_\alpha(|\|x\|_\alpha|^2+|\|y\|_\alpha|^2) \leq |\|x+y\|_\alpha|^2+|\|x-y\|_\alpha|^2 \leq 2B_\alpha A_\alpha(|\|x\|_\alpha|^2+|\|y\|_\alpha|^2)$$	General $X$
Fuzzy Bessel’s Inequality	$$\sum_{i=1}^\infty |\langle x, e_i\rangle_\alpha|^2 \leq \left(\frac{B_\alpha}{A_\alpha}\right)^2|\|x\|_\alpha|^2$$	Orthonormal set

These interval-valued bounds preserve geometric properties essential to inner product spaces (Daraby et al., 24 Dec 2025).

4. Comparative Analysis with Classical and Fuzzy Models

When $A_\alpha = B_\alpha = 1$ and $\mathcal{K}$ is crisp-cut, the ordered fuzzy inner product collapses to the classic form. In contrast, classical fuzzy inner products, often defined by a t-norm, either lose fundamental inequalities like Cauchy–Schwarz or degenerate to a real-valued inner product. The ordered interval approach, complemented by two-sided bounds, retains both the “genuine fuzziness” and the full geometric strength, making it suitable for rigorous analysis and generalization (Daraby et al., 24 Dec 2025).

In L-universe settings, when $\mathbb{U}(x) \equiv 1$, the construct recovers the traditional fuzzy intersection operator: $$\mathbb{I}(M,Q) = \bigvee_x [M(x)\circledast Q(x)].$$ Ordered versions further generalize by incorporating universe capacity normalization (Li et al., 2024).

5. Applications in Fuzzy Rough Set Theory and Operator Characterization

The ordered fuzzy inner product serves as the foundation for new axiomatic approaches in L-valued rough sets. It is instrumental in defining the upper inverse of rough approximation operators: $$\mathbb{H}^{-1}(Q)(x) = \mathbb{I}\bigl(\mathbb{U}(x)\wedge\mathbb{H}(\mathbb{U}_{\{x\}}), Q\bigr),$$ and features in the single-axiom characterizations: $$\forall\,M,Q\in P(\mathbb{U}):\quad \mathbb{I}(Q, \mathbb{H}(M)) = \mathbb{I}(M, \mathbb{H}^{-1}(Q)).$$ Structural properties such as reflexivity, transitivity, symmetry, Euclidean, and median relations are incorporated by strengthening this axiom, allowing for comprehensive and unified characterization of $L$-valued rough approximations. These results tie the algebraic structure of ordered fuzzy inner products directly to the theory of fuzzy rough sets (Li et al., 2024).

6. Induced Fuzzy Norms and Example Computations

For each $\alpha$, the induced fuzzy norm is

$\|x\|_\alpha = \sqrt{\langle x, x\rangle_\alpha},$

and it satisfies interval-valued norm conditions: $$\|\!x\|_\alpha \in [C_\alpha\,\|x\|', D_\alpha\,\|x\|'']_o,$$ where $C_\alpha = \sqrt{A_\alpha}$ and $D_\alpha = \sqrt{B_\alpha}$. For explicit computation, consider $X = \mathbb{R}^2$, $\alpha = 0.5$, $A_{0.5} = 1$, $B_{0.5} = 2$, and

$$\langle x, y\rangle_{0.5} = 1.5 \langle x, y\rangle',\quad x, y \in \mathbb{R}^2.$$

For $x = (1,2)$, $y = (2,1)$: $$\langle x, y\rangle' = 4;\quad \langle x, y\rangle_{0.5} = 6;\quad 6 \in [4, 8]_o.$$ Analogously, in L-universe contexts, computations leverage GL-quantale t-norms and residua for $L$-subset overlaps:

• Gödel t-norm: $\mathbb{I}(M,Q) = \max\{0.2, 0.5, 0.3, 0.5\} = 0.5$.
• Łukasiewicz t-norm: $\mathbb{I}(M,Q) = \max\{0.2, 0, 0, 0.1\} = 0.2$ (Daraby et al., 24 Dec 2025, Li et al., 2024).

7. Mathematical and Methodological Significance

The ordered fuzzy inner product generalizes classical constructs, preserves critical geometric inequalities, and enables precise characterizations across fuzzy and rough set theoretical frameworks. Its introduction is pivotal for a “third axiomatic approach” to L-valued rough sets, supporting robust, interval-valued reasoning in complex fuzzy contexts and subsuming earlier approaches as special cases. This suggests broader applications in functional analysis, fuzzy information processing, and uncertainty quantification, with implications for future research in generalized operator algebras and fuzzy model theory (Daraby et al., 24 Dec 2025, Li et al., 2024).