Fully-Fuzzy Linear Systems (FFLS)

• Fully-Fuzzy Linear Systems (FFLS) are defined as linear systems where coefficients, unknowns, and right-hand sides are fuzzy, introducing unique computational challenges.
• Embedding methods and parametric representations convert fuzzy problems into structured crisp systems, enabling rigorous solvability analysis and practical computation.
• Dynamic and Gaussian-PDMF frameworks extend FFLS theory by providing explicit solution formulas and stability criteria for systems with fully fuzzy data.

Searching arXiv for the cited FFLS and closely related fuzzy linear systems papers.

A Fully-Fuzzy Linear System (FFLS) is commonly understood as a linear system in which the coefficient matrix, the unknown vector, and the right-hand side are fuzzy-valued, typically written as

$\tilde A \tilde x = \tilde b.$

Within the literature, however, a substantial part of the theoretical development has proceeded through closely related but narrower models of the form

$A\tilde x=\tilde b,$

where $A$ is crisp and only the unknowns and right-hand side are fuzzy. This distinction is central: several influential solvability, embedding, and geometric results concern the crisp-coefficient subcase rather than FFLS in the strict sense, whereas more recent work develops genuinely fully fuzzy algebraic frameworks and dynamic fully fuzzy models (Zheng, 21 Jul 2025).

1. Definition and scope of the subject

In the common strict sense, an FFLS has fuzzy coefficients, fuzzy unknowns, and a fuzzy right-hand side. The 2025 Gaussian-PDMF formulation states the system explicitly as

$\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$

with $$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$, $\bm{\tilde x}\in\mathcal X^n$, and $\bm{\tilde b}\in\mathcal X^m$, where all entries belong to the Gaussian probability density membership function space $\mathcal X$ (Zheng, 21 Jul 2025). By contrast, several foundational papers study the special case

$Ax=b$

or

$A\tilde x=\tilde b,$

with a crisp real matrix $A\tilde x=\tilde b,$0 and fuzzy data on the right-hand side, and they state explicitly that this is not a general FFLS, but a special or restricted case of it (Amrahov et al., 2011).

This distinction is not terminological only. In the crisp-coefficient setting, sign-splitting and block embedding are available because each scalar coefficient has a fixed sign. In a genuine FFLS, fuzzy coefficients generally do not admit a single unambiguous sign, and multiplication of fuzzy coefficients by fuzzy unknowns is no longer representable by the same fixed linear block system. The literature therefore separates two broad strands: one centered on semi-fuzzy or standard fuzzy linear systems with crisp $A\tilde x=\tilde b,$1, and another centered on genuinely fuzzy coefficient matrices, either in algebraic settings such as Gaussian-PDMF space or in dynamical systems of the form

$A\tilde x=\tilde b,$2

with fuzzy matrix $A\tilde x=\tilde b,$3 and fuzzy state $A\tilde x=\tilde b,$4 (Salkuyeh, 2014).

A recurring theme across both strands is that the mathematical status of the “solution” is nontrivial. Some papers seek a vector of fuzzy numbers, some distinguish strong and weak fuzzy solutions, and others argue that the more faithful object is a fuzzy set of crisp vectors in $A\tilde x=\tilde b,$5. This plurality of solution concepts is one of the defining features of FFLS-related research (0911.0790).

2. Parametric representations and embedded formulations

A standard representation of a fuzzy number in this literature is the parametric form

$A\tilde x=\tilde b,$6

with $A\tilde x=\tilde b,$7 bounded and monotone increasing, $A\tilde x=\tilde b,$8 bounded and monotone decreasing, and $A\tilde x=\tilde b,$9. For triangular fuzzy numbers $A$0, one has

$A$1

or, in an equivalent notation,

$A$2

for $A$3 (Amrahov et al., 2011).

In the crisp-coefficient subcase, the most influential algebraic device is the embedding of the fuzzy system into a $A$4 crisp system. Writing

$A$5

where $A$6 contains the positive entries of $A$7 and $A$8 contains the absolute values of the negative entries, one obtains the associated block matrix

$A$9

The unknown vector stacks lower endpoints and negatives of upper endpoints, and the fuzzy system becomes a crisp linear system

$\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$0

This construction appears repeatedly in the FSLE literature and is foundational for solvability analysis, matrix-structure arguments, and generalized inverse methods (Gao et al., 2021).

The same embedding supports several later refinements. One line studies structural matrix classes such as H-matrices, M-matrices, and strictly diagonally dominant matrices. There the system

$\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$1

is analyzed via comparison matrices and generalized strict diagonal dominance, and it is shown that if $\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$2 is an H-matrix, then there exists a permutation matrix $\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$3 such that $\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$4 is also an H-matrix (Salkuyeh, 2014). Another line modifies Ezzati’s method by solving two $\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$5 crisp systems, one for the difference

$\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$6

and one for the sum

$\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$7

thus avoiding the full $\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$8 solve when early feasibility failure can already be detected (Mikaeilvand et al., 2018).

These embedded constructions are not themselves a full FFLS theory. Their importance lies in establishing a technical template: decompose sign contributions, convert the fuzzy problem into a structured crisp or interval-like system, and then verify whether the recovered endpoints actually define fuzzy numbers. This suggests a general methodological pattern, even where direct transfer to fuzzy coefficient matrices is not available.

3. Strong solutions, weak solutions, and exact solvability conditions

A central distinction in the embedded theory is that between algebraic solvability of the auxiliary crisp system and validity of the corresponding fuzzy solution. If the solution of

$\bm{\tilde A}\bm{\tilde x}=\bm{\tilde b},$9

yields components $$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$0, then a strong solution requires

$$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$1

for every component; if for some $$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$2 one has $$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$3, the result is only a weak solution (Amrahov et al., 2011). This distinction isolates a fundamental issue in FFLS-related analysis: invertibility of an embedded matrix is not enough.

For systems with crisp $$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$4 and fuzzy $$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$5, an exact existence-and-uniqueness criterion for a unique strong solution was given in terms of both the coefficient matrix and the actual right-hand side. With

$$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$6

if $$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$7 is nonsingular, then strong solvability is equivalent to

$$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$8

Combining this with the nonsingularity characterization of $$\bm{\tilde A}=(\tilde a_{ij})\in\mathcal X^{m\times n}$$9 yields the criterion that the system has a unique strong solution if and only if $\bm{\tilde x}\in\mathcal X^n$0 and $\bm{\tilde x}\in\mathcal X^n$1 are both nonsingular, and

$\bm{\tilde x}\in\mathcal X^n$2

holds componentwise (Amrahov et al., 2011).

This theorem generalized an earlier right-hand-side-independent criterion requiring

$\bm{\tilde x}\in\mathcal X^n$3

Because $\bm{\tilde x}\in\mathcal X^n$4 is itself nonnegative, requiring both $\bm{\tilde x}\in\mathcal X^n$5 and $\bm{\tilde x}\in\mathcal X^n$6 forces $\bm{\tilde x}\in\mathcal X^n$7 to be a generalized permutation matrix, equivalently a product of a nonsingular diagonal matrix and a permutation matrix. The authors interpret this as meaning that the older theorem applies only to a narrow class of systems, “only when the system consists of equations, each of which has exactly one variable” (Amrahov et al., 2011). A plausible implication is that purely matrix-based solvability theorems in broader FFLS settings may also become highly restrictive unless they are allowed to depend on the actual fuzzy data.

Later work sharpened the computational side of this distinction. In the modified Ezzati framework, the first system solved is

$\bm{\tilde x}\in\mathcal X^n$8

If any component of $\bm{\tilde x}\in\mathcal X^n$9 is negative, then the system has no fuzzy number vector solution. Only if this nonnegativity test passes does one solve

$\bm{\tilde b}\in\mathcal X^m$0

and recover

$\bm{\tilde b}\in\mathcal X^m$1

The explicit purpose is early detection of impossibility of a strong fuzzy-number solution and a reduction in arithmetic operations relative to Friedman’s and Ezzati’s earlier schemes (Mikaeilvand et al., 2018).

A related but algebraically different development uses the core-EP inverse of the associated matrix $\bm{\tilde b}\in\mathcal X^m$2. For the crisp-coefficient fuzzy linear system $\bm{\tilde b}\in\mathcal X^m$3, the paper defines consistency by the rank condition for $\bm{\tilde b}\in\mathcal X^m$4, proves a block formula for $\bm{\tilde b}\in\mathcal X^m$5, and shows that

$\bm{\tilde b}\in\mathcal X^m$6

is a solution of

$\bm{\tilde b}\in\mathcal X^m$7

if and only if

$\bm{\tilde b}\in\mathcal X^m$8

where $\bm{\tilde b}\in\mathcal X^m$9. In inconsistent cases, the same expression $\mathcal X$0 is used as a generalized solution to the modified consistent systems

$\mathcal X$1

or

$\mathcal X$2

(Gao et al., 2021).

4. Geometric and set-valued interpretations

A different line of research rejects the assumption that the solution should necessarily be a vector of fuzzy numbers. For square systems with crisp $\mathcal X$3 and triangular fuzzy right-hand side, the fuzzy right-hand side is represented as a rectangular prism in $\mathcal X$4, and because the image of a parallelepiped is also a parallelepiped under a linear transformation, the solution set is obtained as the inverse image

$\mathcal X$5

where $\mathcal X$6 is the crisp center solution and $\mathcal X$7 is the uncertainty prism (0910.4049).

In this framework the solution is a fuzzy set of real vectors rather than a vector of fuzzy numbers. Membership is assigned through the possibility of the corresponding point in right-hand-side space; in support-vector form one obtains

$\mathcal X$8

For triangular fuzzy right-hand sides, the $\mathcal X$9-cuts satisfy

$Ax=b$0

The same paper states a sharp characterization: the system has a solution in the form of a vector of fuzzy numbers if and only if

$Ax=b$1

where $Ax=b$2 is diagonal and $Ax=b$3 is a permutation matrix, that is, when $Ax=b$4 is a generalized permutation matrix (0910.4049).

This geometric conclusion aligns with the matrix-theoretic restriction identified in strong-solution theory. In both cases, generalized permutation structure marks the exceptional situation in which axis-aligned fuzzy-number structure is preserved under the inverse transformation. This suggests that the demand for componentwise fuzzy-number solutions is structurally narrow, even before coefficient fuzziness is introduced.

The geometric viewpoint was extended to non-square systems by defining the solution not as a fuzzy vector but as a fuzzy set of crisp vectors in $Ax=b$5. For $Ax=b$6, the solution set is a parallelepiped with explicit $Ax=b$7-cuts; for $Ax=b$8, the support is a convex polyhedron obtained as the intersection of equation bands; for $Ax=b$9, the solution combines a fuzzy particular part with homogeneous contributions from free variables (0911.0790). In that setting the membership of a vector $A\tilde x=\tilde b,$0 is

$A\tilde x=\tilde b,$1

where each $A\tilde x=\tilde b,$2 evaluates how well the $A\tilde x=\tilde b,$3-th left-hand side matches the $A\tilde x=\tilde b,$4-th fuzzy number on the right-hand side (0911.0790).

This set-valued interpretation is highly relevant to FFLS because it shows that the “solution as fuzzy vector” paradigm may be too restrictive or even misleading. That claim is explicit in the geometric papers, and it becomes more salient when fuzzy coefficients are also present, since coordinate coupling then arises from the transformation itself rather than only from its action on a fuzzy right-hand side.

5. Matrix classes, structural solvability, and dynamic fully fuzzy systems

For the crisp-coefficient fuzzy system $A\tilde x=\tilde b,$5, matrix structure has been used to guarantee existence and uniqueness of the embedded crisp solution and, in stronger cases, validity as a fuzzy vector. If $A\tilde x=\tilde b,$6 is an H-matrix, then there exists a permutation matrix $A\tilde x=\tilde b,$7 such that $A\tilde x=\tilde b,$8 is also an H-matrix; hence $A\tilde x=\tilde b,$9 is nonsingular and the embedded system has a unique crisp solution. If $A\tilde x=\tilde b,$00 is an M-matrix, then $A\tilde x=\tilde b,$01 is also an M-matrix, so in particular

$A\tilde x=\tilde b,$02

which guarantees that the unique solution is a valid fuzzy vector for arbitrary fuzzy right-hand side (Salkuyeh, 2014).

This yields a hierarchy. H-matrix structure suffices for uniqueness of the embedded crisp solution. M-matrix structure gives more: uniqueness plus fuzzy validity for arbitrary $A\tilde x=\tilde b,$03. Strict diagonal dominance appears as a special case through generalized strict diagonal dominance. The paper also gives a practical procedure: construct $A\tilde x=\tilde b,$04, possibly construct a permutation matrix $A\tilde x=\tilde b,$05 if zero diagonal entries arise, solve

$A\tilde x=\tilde b,$06

and use LU factorization because H-matrices admit LU factorization (Salkuyeh, 2014).

A distinct but related branch concerns dynamic fully fuzzy systems. In the linear stationary discrete-time model

$A\tilde x=\tilde b,$07

the matrix $A\tilde x=\tilde b,$08 is fuzzy-valued and $A\tilde x=\tilde b,$09, so this is a fully fuzzy linear discrete-time system rather than a static algebraic FFLS (Oliva et al., 2011). The central device is to analyze each $A\tilde x=\tilde b,$10-cut through the interval-matrix inclusion

$A\tilde x=\tilde b,$11

The paper proves that if the support-level inclusion is stable, then the whole fuzzy difference inclusion is stable. For the stationary case it gives sufficient criteria based on stability of every $A\tilde x=\tilde b,$12, Gershgorin-type conditions, and interval eigenvalue bounds (Oliva et al., 2011).

For positive systems, the paper gives an especially explicit representation. If

$A\tilde x=\tilde b,$13

then for each $A\tilde x=\tilde b,$14,

$A\tilde x=\tilde b,$15

Equivalently,

$A\tilde x=\tilde b,$16

This is one of the clearest examples in the literature where positivity turns a fully fuzzy evolution problem into two monotone crisp recursions at each $A\tilde x=\tilde b,$17-level (Oliva et al., 2011).

6. Gaussian-PDMF FFLS and explicit algebraic solution theory

A recent development provides an explicit algebraic FFLS theory in the Gaussian probability density membership function space $A\tilde x=\tilde b,$18. In this framework each fuzzy number is parameterized as

$A\tilde x=\tilde b,$19

and $A\tilde x=\tilde b,$20 is shown to be both a linear space over $A\tilde x=\tilde b,$21 and a commutative ring with identity (Zheng, 21 Jul 2025). A standard ordered basis is

$A\tilde x=\tilde b,$22

with coordinate vector

$A\tilde x=\tilde b,$23

The paper explicitly defines addition, scalar multiplication, subtraction, and multiplication of fuzzy numbers in $A\tilde x=\tilde b,$24, making matrix multiplication with fuzzy entries algebraically well-defined (Zheng, 21 Jul 2025).

The main FFLS theorem is formulated for fuzzy reduced row echelon form. If

$A\tilde x=\tilde b,$25

then for the homogeneous system

$A\tilde x=\tilde b,$26

the solution set is

$A\tilde x=\tilde b,$27

and for the nonhomogeneous system

$A\tilde x=\tilde b,$28

if $A\tilde x=\tilde b,$29 is one particular solution, then

$A\tilde x=\tilde b,$30

There are exactly

$A\tilde x=\tilde b,$31

real free parameters, because each free fuzzy variable has five real coordinates in the basis $A\tilde x=\tilde b,$32 (Zheng, 21 Jul 2025).

The paper adapts Gaussian elimination to fuzzy matrices by restricting pivot normalization to the unit group

$A\tilde x=\tilde b,$33

Elementary row operations are row interchange, multiplication by a unit, and adding $A\tilde x=\tilde b,$34 times one row to another. It is then proved that row-equivalent augmented matrices have the same solution set (Zheng, 21 Jul 2025).

An important qualification is that $A\tilde x=\tilde b,$35 is a ring with zero divisors, not a field. Accordingly, the paper states that Cramer’s rule is available for the semi-fuzzy system with crisp square $A\tilde x=\tilde b,$36, but not generally for FFLS. The determinant-adjugate identity

$A\tilde x=\tilde b,$37

does not suffice for invertibility because $A\tilde x=\tilde b,$38 may fail to be a unit (Zheng, 21 Jul 2025). The same paper identifies a subset

$A\tilde x=\tilde b,$39

which is a field, and proves that if all entries of $A\tilde x=\tilde b,$40 lie in $A\tilde x=\tilde b,$41, then the FFLS reduces exactly to a semi-fuzzy linear system with a crisp coefficient matrix $A\tilde x=\tilde b,$42 and the same solution set (Zheng, 21 Jul 2025).

Taken together, these developments position FFLS as a field defined less by a single canonical method than by a set of recurring technical issues: how to represent fuzzy numbers, how to interpret the solution object, how to separate algebraic solvability from fuzzy validity, and how much matrix structure is required before explicit computation becomes possible. The crisp-coefficient literature provides foundational embedding, strong-solution, and geometric results; the dynamic literature shows how full coefficient fuzziness can be treated through $A\tilde x=\tilde b,$43-cut inclusions; and the Gaussian-PDMF framework offers an explicit algebraic realization of static FFLS in which fuzzy row reduction and parametric solution formulas become available (Zheng, 21 Jul 2025).