**Lemma 2.** *Assume u*, *v*, *w* ∈ I*.*

(i) *If u* + *v* + *w* > 2*, then* (*u* ⊗*<sup>L</sup> v*) ⊗*<sup>L</sup> w* = *u* + *v* + *w* − 2 > 0*,*

(ii) *If u* + *v* + *w* ≤ 2*, then* (*u* ⊗*<sup>L</sup> v*) ⊗*<sup>L</sup> w* = 0*.*

**Proof.** Let *u* + *v* + *w* > 2. Then *u* + *v* > 2 − *w* ≥ 2 − 1 = 1, since 0 ≤ *w* ≤ 1. Hence, *u* ⊗*<sup>L</sup> v* = *u* + *v* − 1 > 0, and (*u* ⊗*<sup>L</sup> v*) ⊗*<sup>L</sup> w* = (*u* + *v* − 1) + *w* − 1 = *u* + *v* + *w* − 2 > 0.

On the other hand, if *u* + *v* + *w* ≤ 2, then (*u* + *v* − 1)+(*w* − 1) ≤ 0. We consider two subcases.

Subcase 1. Suppose *u* + *v* − 1 > 0. Then *u* ⊗*<sup>L</sup> v* = *u* + *v* − 1 and *u* ⊗*<sup>L</sup> v* + *w* − 1 ≤ 0. Hence, (*u* ⊗*<sup>L</sup> v*) ⊗*<sup>L</sup> w* = 0.

Subcase 2. Suppose *u* + *v* − 1 ≤ 0. Then *u* ⊗*<sup>L</sup> v* = 0 and *u* ⊗*<sup>L</sup> v* + *w* − 1 = 0 + *w* − 1 ≤ 0, since *w* ≤ 1. That is, (*u* ⊗*<sup>L</sup> v*) ⊗*<sup>L</sup> w* = 0.

**Remark 6.** *It is easy to see directly from the definition that the Łukasiewicz conjunction* ⊗*<sup>L</sup> is commutative. As a consequence of Lemma 2,* ⊗*<sup>L</sup> is associative, as well.*

*Namely due to Lemma 2(i), we have, for any u*, *v*, *w* ∈ I *with u* + *v* + *w* > 2*, that* (*u* ⊗*<sup>L</sup> v*) ⊗*<sup>L</sup> w* = *u* + *v* + *w* − 2*, and, by the commutative law, u* ⊗*<sup>L</sup>* (*v* ⊗*<sup>L</sup> w*)=(*v* ⊗*<sup>L</sup> w*) ⊗*<sup>L</sup> u* = *v* + *w* + *u* − 2*. That is,* (*u* ⊗*<sup>L</sup> v*) ⊗*<sup>L</sup> w* = *u* ⊗*<sup>L</sup>* (*v* ⊗*<sup>L</sup> w*)*. Similar reasoning is used when u* + *v* + *w* ≤ 2*.*

To recognize the existence of a certificate *A* ∈ **A** satisfying the conditions (15) from Theorem 4, the unknown *A* will be written as a max-Łuk linear combination of generators *A*˜(*ij*) as in Lemma 1 (ii).

The coefficients in the linear combination will be found as the solution to a system of max-Łuk linear equations with parameter *λ*, and the variables *α*(*ij*) in the bounds *aij* = *α*(*ij*) ≤ *α*(*ij*) ≤ *α*(*ij*) = *aij*. The form of the system will require that every solution of the system, for some parameter value *λ*, gives coefficients for such a max-Łuk linear combination of generators which is a *λ*-certificate matrix for the given instance. Then the recognition of strong tolerability is equivalent to the recognition of whether there is a value *λ* for which the system is solvable. On the other hand, if the system is unsolvable for every *λ* ∈ I, then no certificate exists for the given instance.

Formally, we consider the bounded max-Łuk linear system

$$
\vec{\mathcal{L}} \otimes\_L \mathfrak{a} = \lambda \otimes\_L \mathfrak{b} \tag{17}
$$

$$
\underline{u} \le \underline{u} \le \overline{u} \tag{18}
$$

with parameter *<sup>λ</sup>* ∈ I, where the columns *<sup>C</sup>*˜(*ij*) of *<sup>C</sup>*˜ ∈ I(*n*2, *<sup>n</sup>*2) are constructed blockwise from *<sup>A</sup>*˜(*ij*) <sup>⊗</sup> *<sup>x</sup>*˜(*k*), *<sup>k</sup>* <sup>∈</sup> *<sup>N</sup>*. The right-hand side vector ˜ *<sup>b</sup>* ∈ I(*n*2) is constructed blockwise from the generators *<sup>x</sup>*˜(*k*) for *<sup>k</sup>* <sup>∈</sup> *<sup>N</sup>*, and the bounds *<sup>α</sup>*, *<sup>α</sup>* ∈ I(*n*2) for the variable vector *<sup>α</sup>* ∈ I(*n*2) are constructed from the columns of *A*, *A*, according to Lemma 1 (ii). That is, we have

$$
\tilde{\mathcal{L}}^{(ij)} = \begin{pmatrix}
\tilde{A}^{(ij)} \otimes\_L \tilde{\mathbf{x}}^{(1)} \\
\tilde{A}^{(ij)} \otimes\_L \tilde{\mathbf{x}}^{(2)} \\
\vdots \\
\tilde{A}^{(ij)} \otimes\_L \tilde{\mathbf{x}}^{(n)}
\end{pmatrix}, \quad \tilde{b} = \begin{pmatrix}
\tilde{\mathbf{x}}^{(1)} \\
\tilde{\mathbf{x}}^{(2)} \\
\vdots \\
\tilde{\mathbf{x}}^{(n)}
\end{pmatrix}, \tag{19}
$$

$$
\underline{a}\_{(ij)} = \underline{a}\_{i\bar{j}} - \overline{a}\_{i\bar{j}} + 1 \le a\_{(i\bar{j})} \le 1 = \overline{a}\_{(i\bar{j})}.\tag{20}
$$

**Theorem 5.** *The interval vector* **X** = [*x*, *x*] *is a strongly tolerable eigenvector of the interval matrix* **A** = [*A*, *A*] *if and only if there is a <sup>λ</sup>* ∈ I *such that the linear system* (17) *and* (18)*, has a solution <sup>α</sup>* ∈ I(*n*2)*. In the positive case, the max-Łuk linear combination*

$$A = \bigoplus\_{(ij)\in N\times N} \mathfrak{a}\_{(ij)} \otimes\_L \breve{A}^{(ij)} \tag{21}$$

*is a λ-certificate for the given instance.*

**Proof.** Assume that there exists a *λ* ∈ I such that *α* satisfies (17), (18) with (19), (20). Then, *A* ∈ I(*n*, *n*) as defined in (21) belongs to [*A*, *A*], in view of Lemma 1(ii). Moreover, we have the following block equations, for every *k* ∈ *N*

$$\bigoplus\_{i,j \in N} \left( \vec{A}^{(ij)} \otimes\_L \vec{\mathfrak{x}}^{(k)} \right) \otimes\_L \mathfrak{a}\_{(ij)} = \lambda \otimes\_L \mathfrak{x}^{(k)},\tag{22}$$

$$\left(\bigoplus\_{i,j\in N} \mathfrak{a}\_{(ij)} \otimes\_L \vec{A}^{(ij)}\right) \otimes\_L \mathfrak{x}^{(k)} = \lambda \otimes\_L \mathfrak{x}^{(k)},\tag{23}$$

$$A \circledast\_L \mathfrak{F}^{(k)} = \lambda \circledast\_L \mathfrak{F}^{(k)}.\tag{24}$$

We will prove that the block Equations (22) and (23) are equivalent. In particular, we show that the left-hand sides of (22) and (23) in every row *h* and in every block row *k* are equal.

Assume *k*, *h* ∈ *N* are fixed. Then

$$\left(\bigoplus\_{i,j\in N} \left(\vec{A}^{(ij)}\otimes\_L \vec{\mathbf{x}}^{(k)}\right)\otimes\_L \mathfrak{a}\_{(ij)}\right)\_h = \bigoplus\_{i,j\in N} \left(\vec{A}^{(ij)}\otimes\_L \vec{\mathbf{x}}^{(k)}\right)\_h \otimes\_L \mathfrak{a}\_{(ij)}\tag{25}$$

$$=\bigoplus\_{i,j\in\mathcal{N}}\left(\bigoplus\_{\mathfrak{F}\in\mathcal{N}}A^{(ij)}\_{\mathfrak{h}\mathfrak{g}}\otimes\_{L}\mathfrak{z}^{(k)}\_{\mathfrak{k}}\right)\otimes\_{L}\mathfrak{a}\_{\mathfrak{(ij)}}=\bigoplus\_{i,j\in\mathcal{N}}\left(\bigoplus\_{\mathfrak{g}\in\mathcal{N}}\left(\vec{A}^{(ij)}\_{\mathfrak{h}\mathfrak{g}}\otimes\_{L}\vec{\mathfrak{z}}^{(k)}\_{\mathfrak{g}}\right)\otimes\_{L}\mathfrak{a}\_{\mathfrak{(ij)}}\right)\_{\mathfrak{k}}\tag{26}$$

$$=\bigoplus\_{i,j\in\mathcal{N}}\left(\bigoplus\_{\mathcal{S}\in\mathcal{N}}a\_{(ij)}\otimes\_{L}\left(\boldsymbol{A}\_{\mathrm{hg}}^{(ij)}\otimes\_{L}\tilde{\mathbf{x}}\_{\mathcal{S}}^{(k)}\right)\right)=\bigoplus\_{i,j\in\mathcal{N}}\left(\bigoplus\_{\mathcal{S}\in\mathcal{N}}\left(a\_{(ij)}\otimes\_{L}\boldsymbol{A}\_{\mathrm{hg}}^{(ij)}\right)\otimes\_{L}\tilde{\mathbf{x}}\_{\mathcal{S}}^{(k)}\right)\tag{27}$$

$$=\bigoplus\_{i,j\in N} \left(\mathfrak{a}\_{(ij)}\otimes\_L \vec{A}^{(ij)}\right)\_h \otimes\_L \mathfrak{x}^{(k)} = \left(\bigoplus\_{i,j\in N} \left(\mathfrak{a}\_{(ij)}\otimes\_L \vec{A}^{(ij)}\right)\otimes\_L \mathfrak{x}^{(k)}\right)\_h\tag{28}$$

Please note that the associative law has been used in (27). That is,

$$
\mathfrak{a}\_{(ij)} \otimes\_L \left( \check{\mathcal{A}}\_{h\emptyset}^{(ij)} \otimes\_L \mathfrak{x}\_{\mathcal{S}}^{(k)} \right) = \left( \mathfrak{a}\_{(ij)} \circledcirc\_L \check{\mathcal{A}}\_{h\emptyset}^{(ij)} \right) \circledcirc\_L \mathfrak{x}\_{\mathcal{S}}^{(k)} \swarrow$$

according to Remark 6. The remaining equalities (25), (26) and (28) are consequences of standard arithmetic rules in max-Łuk algebras.

Now, in view of the fact that (22) means that *<sup>α</sup>* ∈ I(*n*2) is a solution of (19), while (23) says that (21) satisfies (15), we obtain, due to Theorem 4 that **X** is a strongly tolerable eigenvector of **A**. The converse implication follows from the converse implication in Theorem 4.

Theorem 5 reduces the recognition problem of whether **X** is a strongly tolerable eigenvector of **<sup>A</sup>** to the solvability problem of the bounded parametric system (17), (18) with dimension *<sup>n</sup>*<sup>2</sup> <sup>×</sup> *<sup>n</sup>*<sup>2</sup> for some *λ* ∈ I. The latter problem is a particular case of the bounded parametric solvability problem with general dimension *m* × *n*. The recognition algorithm can be briefly described by the following steps (for details and notation, see [28]):

(i) permute the equations in the system so that the right-hand side will be decreasing, that is

$$0 \le 1 - b\_1 \le 1 - b\_2 \le \cdots \le 1 - b\_m \le 1,\tag{29}$$


**Theorem 6.** *The recognition problem of whether a given interval vector* **X** *is a strongly tolerable eigenvector of a given interval matrix* **A** *in a max-Łuk algebra is solvable in O*(*n*6) *time.*

**Proof.** According to [28], the parametric solvability problem with dimension *m* × *n* has the computational complexity *O*(*m n*2). Therefore, the computational complexity of the strong tolerance problem with dimension *<sup>n</sup>*<sup>2</sup> <sup>×</sup> *<sup>n</sup>*<sup>2</sup> is *<sup>O</sup>* - (*n*2)<sup>3</sup> = *O*(*n*6).

**Example 4.** *(Numerical illustration: Computing a certificate)*

*Assume that the lower and upper bounds for A* ∈ [*A*, *A*] *and x* ∈ [*x*, *x*] *in the interval eigenproblem are*

$$\underline{A} = \begin{pmatrix} 0.9 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.8 & 0.9 \end{pmatrix}, \qquad \overline{A} = \begin{pmatrix} 1 & 0.8 & 0.8 \\ 0.7 & 0.9 & 0.8 \\ 1 & 0.9 & 1 \end{pmatrix}.$$

$$\underline{\underline{x}} = \begin{pmatrix} 0.7 \\ 0.8 \\ 0.9 \end{pmatrix}, \qquad \overline{\underline{x}} = \begin{pmatrix} 0.9 \\ 0.8 \\ 0.9 \end{pmatrix}.$$

*If we wish to recognize whether* **X** *is a strongly tolerable max-Łuk eigenvector of* **A***, then, according to Theorem 4, we must recognize the existence of a λ-certificate for* (**A**, **X**)*. In view of Theorem 5, we must recognize the solvability of the max-Łuk linear system <sup>C</sup>*˜ <sup>⊗</sup>*<sup>L</sup> <sup>α</sup>* <sup>=</sup> *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup>* ˜ *b with bounds α* ≤ *α* ≤ *α, for some λ* ∈ I*. The vector (matrix) generators are*

$$\begin{aligned} \tilde{\mathbf{x}}^{(1)} &= \begin{pmatrix} 0.9 \\ 0.8 \\ 0.9 \end{pmatrix}, & \tilde{\mathbf{x}}^{(2)} &= \begin{pmatrix} 0.7 \\ 0.8 \\ 0.9 \end{pmatrix}, & \tilde{\mathbf{x}}^{(3)} &= \begin{pmatrix} 0.7 \\ 0.8 \\ 0.9 \end{pmatrix}, \\\ \tilde{A}^{(11)} &= \begin{pmatrix} 1 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.8 & 0.9 \end{pmatrix}, & \tilde{A}^{(12)} &= \begin{pmatrix} 0.9 & 0.8 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.8 & 0.9 \end{pmatrix}, \\\ \tilde{A}^{(13)} &= \begin{pmatrix} 0.9 & 0.7 & 0.8 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.8 & 0.9 \end{pmatrix}, & \tilde{A}^{(21)} &= \begin{pmatrix} 0.9 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.8 & 0.9 \end{pmatrix}, \end{aligned}$$

$$\begin{aligned} \tilde{A}^{(22)} &= \begin{pmatrix} 0.9 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.8 & 0.9 \end{pmatrix}, & \tilde{A}^{(23)} &= \begin{pmatrix} 0.9 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.8 \\ 0.8 & 0.8 & 0.9 \end{pmatrix}, \\\ \tilde{A}^{(31)} &= \begin{pmatrix} 0.9 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 1 & 0.8 & 0.9 \end{pmatrix}, & \tilde{A}^{(32)} &= \begin{pmatrix} 0.9 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.9 & 0.9 \end{pmatrix}, \\\ \tilde{A}^{(33)} &= \begin{pmatrix} 0.9 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.8 & 1 \end{pmatrix}. \end{aligned}$$

*The columns of the matrix <sup>C</sup>*˜ ∈ I(9, 9) *and the right-hand side vector* ˜ *b* ∈ I(9) *are computed blockwise according to* (19)*, as follows.*

*C*˜(11) = ⎛ ⎜⎝ *<sup>A</sup>*˜(11) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(1) *<sup>A</sup>*˜(11) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(2) *<sup>A</sup>*˜(11) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(3) ⎞ ⎟⎠ <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0.9 0.7 0.8 0.7 0.7 0.8 0.7 0.7 0.8 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , *C*˜(12) = ⎛ ⎜⎝ *<sup>A</sup>*˜(12) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(1) *<sup>A</sup>*˜(12) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(2) *<sup>A</sup>*˜(12) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(3) ⎞ ⎟⎠ <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0.8 0.7 0.8 0.6 0.7 0.8 0.6 0.7 0.8 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , *C*˜(13) = ⎛ ⎜⎝ *<sup>A</sup>*˜(13) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(1) *<sup>A</sup>*˜(13) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(2) *<sup>A</sup>*˜(13) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(3) ⎞ ⎟⎠ <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0.8 0.7 0.8 0.7 0.7 0.8 0.7 0.7 0.8 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , *C*˜(21) = ⎛ ⎜⎝ *<sup>A</sup>*˜(21) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(1) *<sup>A</sup>*˜(21) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(2) *<sup>A</sup>*˜(21) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(3) ⎞ ⎟⎠ <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0.8 0.7 0.8 0.6 0.7 0.8 0.6 0.7 0.8 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , *C*˜(22) = ⎛ ⎜⎝ *<sup>A</sup>*˜(22) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(1) *<sup>A</sup>*˜(22) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(2) *<sup>A</sup>*˜(22) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(3) ⎞ ⎟⎠ <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0.8 0.7 0.8 0.6 0.7 0.8 0.6 0.7 0.8 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , *C*˜(23) = ⎛ ⎜⎝ *<sup>A</sup>*˜(23) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(1) *<sup>A</sup>*˜(23) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(2) *<sup>A</sup>*˜(23) <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(3) ⎞ ⎟⎠ <sup>=</sup> ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 0.8 0.7 0.8 0.6 0.7 0.8 0.6 0.7 0.8 ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ,

$$
\begin{split}
\tilde{\mathcal{L}}^{(31)} &= \begin{pmatrix}
\tilde{A}^{(31)} \otimes\_{L} \tilde{\pi}^{(1)} \\
\tilde{A}^{(31)} \otimes\_{L} \tilde{\pi}^{(2)} \\
\tilde{A}^{(31)} \otimes\_{L} \tilde{\pi}^{(3)}
\end{pmatrix} = \begin{pmatrix}
0.8 \\
0.7 \\
0.9 \\
0.6 \\
0.7 \\
0.8 \\
0.6 \\
0.7 \\
0.8
\end{pmatrix}, \quad \tilde{\mathcal{L}}^{(32)} = \begin{pmatrix}
\tilde{A}^{(32)} \otimes\_{L} \tilde{\pi}^{(1)} \\
\tilde{A}^{(32)} \otimes\_{L} \tilde{\pi}^{(2)} \\
\tilde{A}^{(32)} \otimes\_{L} \tilde{\pi}^{(3)}
\end{pmatrix} = \begin{pmatrix}
0.8 \\
0.7 \\
0.8 \\
0.6 \\
0.7 \\
0.8 \\
0.8 \\
0.7 \\
0.7 \\
0.8
\end{pmatrix}, \\
\tilde{\mathcal{L}}^{(33)} &= \begin{pmatrix}
\tilde{A}^{(33)} \otimes\_{L} \mathcal{L}^{(1)} \\
\tilde{A}^{(33)} \otimes\_{L} \tilde{\pi}^{(2)} \\
\tilde{A}^{(31)} \otimes\_{L} \tilde{\pi}^{(3)}
\end{pmatrix} = \begin{pmatrix}
0.8 \\
0.7 \\
0.6 \\
0.6 \\
0.7 \\
0.7 \\
\tilde{\pi}^{(2)} \\
0.8 \\
0.7 \\
0.9
\end{pmatrix}, \quad \tilde{\mathcal{B}} = \begin{pmatrix}
0.9 \\
0.8 \\
0.8 \\
0.7 \\
0.8 \\
0.9 \\
0.7 \\
0.7 \\
0.8 \\
0.8
\end{pmatrix}.
\end{split}
$$

*Hence, we wish to recognize the solvability of the system*

$$
\tilde{\mathcal{L}} \odot\_L \mathbf{a} = \begin{pmatrix} 0.9 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8\\ 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7\\ 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.9 & 0.8 & 0.9\\ 0.7 & 0.6 & 0.7 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6\\ 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7\\ 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.9\\ 0.7 & 0.6 & 0.7 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6 & 0.6\\ 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7 & 0.7\\ 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.8 & 0.9 \end{pmatrix} \odot\_L \begin{pmatrix} a\_1 \\ a\_2 \\ a\_3 \\ a\_4 \\ a\_5 \\ a\_6 \\ a\_7 \\ a\_8 \\ a\_9 \end{pmatrix} = \lambda \odot\_L \begin{pmatrix} 0.9 \\ 0.8 \\ 0.9 \\ 0.7 \\ 0.8 \\ 0.9 \\ 0.7 \\ 0.8 \\ 0.9 \end{pmatrix}
$$

*The problem is a particular case of the bounded parametric solvability problem, with dimension <sup>n</sup>*<sup>2</sup> <sup>×</sup> *<sup>n</sup>*2*, and can be solved by the algorithm suggested in [28] (see also a brief description in this paper, before Theorem 6).*

*Depending on the permuted entries of* ˜ *b, we distinguish the following four cases: (a) λ* ∈ (0.3, 1*, (b) λ* ∈ (0.1, 0.2*, <sup>λ</sup>* <sup>∈</sup> (0.2, 0.3 *and (c) <sup>λ</sup>* <sup>∈</sup> (0, 0.1*. We can verify that for <sup>λ</sup>* <sup>=</sup> 0.9 *the system <sup>C</sup>*˜ <sup>⊗</sup>*<sup>L</sup> <sup>α</sup>* <sup>=</sup> *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup>* ˜ *b has a solution <sup>α</sup>* = (0.9, 0.7, 0.9, 0.8, 0.9, 0.9, 0.8, 0.8, 1)*<sup>T</sup> fulfilling the inequalities aij* <sup>−</sup> *aij* <sup>+</sup> <sup>1</sup> <sup>≤</sup> *<sup>α</sup>*(*ij*) <sup>≤</sup> 1, *for every* (*i*, *j*) ∈ *N* × *N*.

*Using the coefficients α*(*ij*) *we get, by Theorem 5, that* **X** *is a strongly tolerable eigenvector of* **A***, with certificate*

$$A = \bigoplus\_{(ij)\in N\times N} \mathfrak{a}\_{(ij)} \otimes\_L \vec{A}^{(ij)} = \begin{pmatrix} 0.9 & 0.7 & 0.7 \\ 0.7 & 0.9 & 0.7 \\ 0.8 & 0.8 & 1 \end{pmatrix}.$$

**Example 5.** *(Numerical illustration: Computing a certificate - no certificate exists)*

*We assume the same lower and upper bounds for A* ∈ [*A*, *A*] *as in Example 4 and take different bounds for x* ∈ [*x*, *x*]*.*

$$
\Delta = \begin{pmatrix} 0.9 & 0.7 & 0.6 \\ 0.7 & 0.9 & 0.6 \\ 0.8 & 0.8 & 0.9 \end{pmatrix}, \qquad \overline{A} = \begin{pmatrix} 1 & 0.8 & 0.8 \\ 0.7 & 0.9 & 0.8 \\ 1 & 0.9 & 1 \end{pmatrix},
$$

$$
\underline{\mathbf{x}} = \begin{pmatrix} 0.7 \\ 0.7 \\ 0.7 \end{pmatrix}, \qquad \underline{\mathbf{x}} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.
$$

*Then the generators of A stay the same and*

$$\mathfrak{x}^{(1)} = \begin{pmatrix} 1 \\ 0.7 \\ 0.7 \end{pmatrix}, \qquad \mathfrak{x}^{(2)} = \begin{pmatrix} 0.7 \\ 1 \\ 0.7 \end{pmatrix}, \qquad \mathfrak{x}^{(3)} = \begin{pmatrix} 0.7 \\ 0.7 \\ 1 \end{pmatrix}.$$

*The matrix C and the right-hand side* ˜ ˜ *b then are*


*Similarly as in the previous example we distinguish, depending on the permuted entries of* ˜ *b, the following three cases: (a) <sup>λ</sup>* <sup>∈</sup> (0.3, 1*, (b) <sup>λ</sup>* <sup>∈</sup> (0, 0.3 *and (c) <sup>λ</sup>* <sup>=</sup> <sup>0</sup>*. It can be verified that the system <sup>C</sup>*˜ <sup>⊗</sup>*<sup>L</sup> <sup>α</sup>* <sup>=</sup> *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup>* ˜ *b has no solution in any of these cases.*

*Consequently, the considered system is not solvable for any value of λ. That is, no certificate for strong tolerability exists and the given* **X** *is not a strongly tolerable eigenvector of* **A***.*

#### **5. Strongly Universal Interval Eigenvectors in a Max-Łuk Algebra**

In this section, we present two necessary and sufficient conditions for characterizing a strongly universal eigenvector. The first condition is based on the generators of **A**, while the second one uses the lower bound and the upper bound of **A**.

**Theorem 7.** *Let* **A** *and* **X** *be given such that aij* = *aij for some i*, *j* ∈ *N. The interval vector* **X** *is a strongly universal max-Łuk eigenvector of the interval matrix* **A** = [*A*, *A*] *if and only if there exists an x* ∈ **X** *and a λ* ∈ I *such that*

$$\vec{A}^{(ij)} \otimes\_L \mathbf{x} = \lambda \otimes\_L \mathbf{x} \quad \text{for every } i, j \in N. \tag{30}$$

**Proof.** Let us assume that there are *<sup>x</sup>* <sup>∈</sup> **<sup>X</sup>** and *<sup>λ</sup>* ∈ I fulfilling condition (30). If *<sup>A</sup>* ∈ I(*n*2) is an arbitrary matrix in **A**, then *A* is a max-Łuk linear combination *A* = ( *ij*∈*<sup>N</sup> <sup>α</sup>ij* <sup>⊗</sup>*<sup>L</sup> <sup>A</sup>*˜(*ij*) for some coefficients *αij* ∈ I, *i*, *j* ∈ *N* with *aij* − *aij* + 1 ≤ *αij* ≤ 1. According to Lemma 1(ii),

$$\begin{aligned} A \otimes\_L x &= \left( \bigoplus\_{ij \in N} \mathfrak{a}\_{ij} \otimes\_L \vec{A}^{(ij)} \right) \otimes\_L x = \bigoplus\_{ij \in N} \mathfrak{a}\_{ij} \otimes\_L \left( \vec{A}^{(ij)} \otimes\_L x \right) \\ &= \bigoplus\_{ij \in N} \mathfrak{a}\_{ij} \otimes\_L \left( \lambda \otimes\_L x \right) = \left( \bigoplus\_{ij \in N} \mathfrak{a}\_{ij} \otimes\_L \lambda \right) \otimes\_L x = \lambda \otimes\_L x \end{aligned}$$

because *aij* = *aij* for some *i*, *j* ∈ *N* implies ( *ij*∈*<sup>N</sup> <sup>α</sup>ij* = 1 and ( *ij*∈*<sup>N</sup> <sup>α</sup>ij* ⊗*<sup>L</sup> <sup>λ</sup>* = *<sup>λ</sup>*. By Definition 1, **X** is a strongly universal eigenvector of **A**. The converse implication follows immediately.

**Theorem 8.** *Suppose given an interval matrix* **A** = [*A*, *A*] *and interval vector* **X** *with bounds x*, *x. Then* **X** *is a strongly universal eigenvector of* **A** *if and only if there are λ* ∈ I *and x* ∈ **X** *such that A* ⊗*<sup>L</sup> x* = *λ* ⊗*<sup>L</sup> x and A* ⊗*<sup>L</sup> x* = *λ* ⊗*<sup>L</sup> x*.

**Proof.** Let us suppose that there are *λ* and *x* ∈ **X** such that *A* ⊗*<sup>L</sup> x* = *A* ⊗ *x* = *λ* ⊗ *x*. From the monotonicity of the operations ⊕ and ⊗*<sup>L</sup>* we get *λ* ⊗*<sup>L</sup> x* = *A* ⊗*<sup>L</sup> x* ≤ *A* ⊗*<sup>L</sup> x* ≤ *A* ⊗*<sup>L</sup> x* = *λ* ⊗*<sup>L</sup> x* for every *A* ∈ **A**. The converse implication is trivial.

The condition described in Theorem 8 can be verified by solving a two-sided max-Łuk system defined as follows. Define the block matrices *C* ∈ I(2*n*, *n*), *D* ∈ I(2*n*, *n*) by

$$\mathbb{C} = \left( \begin{array}{c} \underline{\mathbb{A}} \otimes\_{L} \tilde{\mathfrak{x}}^{(1)} \dots \underline{\mathbb{A}} \otimes\_{L} \tilde{\mathfrak{x}}^{(n)} \\\overline{\mathbb{A}} \otimes\_{L} \tilde{\mathfrak{x}}^{(1)} \dots \overline{\mathbb{A}} \otimes\_{L} \tilde{\mathfrak{x}}^{(n)} \end{array} \right), \qquad D = \left( \begin{array}{c} \tilde{\mathfrak{x}}^{(1)} \quad \dots \quad \tilde{\mathfrak{x}}^{(n)} \\\end{array} \right). \tag{31}$$

**Theorem 9.** *Assume that an interval matrix* **A** = [*A*, *A*] *and an interval vector* **X** = [*x*, *x*] *are given. Then* **X** *is a strongly universal eigenvector of* **A** *if and only if the bounded two-sided max-Łuk linear system with variable β* ∈ I(*n*)

$$\mathcal{C}\circledast\_L \mathcal{B} = \lambda \circledast\_L D \circledast\_L \mathcal{B} \tag{32}$$

$$
\mathbb{x} - \overline{\mathbb{x}} + 1 \le \beta \le 1 \tag{33}
$$

*is solvable for some value of the parameter <sup>λ</sup>* ∈ I*. If <sup>β</sup>* ∈ I(*n*) *is a solution to the system, then <sup>x</sup>* <sup>=</sup> (*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *<sup>β</sup><sup>i</sup>* <sup>⊗</sup> *<sup>x</sup>*˜*<sup>i</sup> satisfies the condition in Theorem 8.*

**Proof.** Let us suppose that there is a *λ* such that the two-sided system *C* ⊗ *β* = *λ* ⊗ *D* ⊗ *β* has a solution *β*. Put *x* = (*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *<sup>β</sup><sup>i</sup>* <sup>⊗</sup> *<sup>x</sup>*˜*<sup>i</sup>* . Then the following formulas are equivalent

$$
\mathbb{C} \otimes \mathcal{B} = \lambda \otimes D \otimes \mathcal{B} \tag{34}
$$

$$\bigoplus\_{i=1}^{n} \underline{\mathbf{A}} \otimes \mathfrak{x}^{(i)} \otimes \beta\_{i} = \lambda \otimes \bigoplus\_{i=1}^{n} \mathfrak{x}^{(i)} \otimes \beta\_{i} \quad \text{and} \quad \bigoplus\_{i=1}^{n} \overline{\mathbf{A}} \otimes \mathfrak{x}^{(i)} \otimes \beta\_{i} = \lambda \otimes \bigoplus\_{i=1}^{n} \mathfrak{x}^{(i)} \otimes \beta\_{i} \tag{35}$$

$$\bigoplus\_{i=1}^{n} \Delta \otimes \tilde{\mathbf{x}}^{(i)} \otimes \beta\_{i} = \lambda \otimes \bigoplus\_{i=1}^{n} \tilde{\mathbf{x}}^{(i)} \otimes \beta\_{i} \quad \text{and} \quad \bigoplus\_{i=1}^{n} \overline{\mathbf{A}} \otimes \tilde{\mathbf{x}}^{(i)} \otimes \beta\_{i} = \lambda \otimes \bigoplus\_{i=1}^{n} \tilde{\mathbf{x}}^{(i)} \otimes \beta\_{i} \tag{36}$$

$$\underline{\mathbf{A}} \otimes \bigoplus\_{i=1}^{n} \beta\_{i} \otimes \tilde{\mathbf{x}}^{(i)} = \boldsymbol{\lambda} \otimes \bigoplus\_{i=1}^{n} \beta\_{i} \otimes \tilde{\mathbf{x}}^{(i)} \quad \text{and} \quad \overline{\boldsymbol{\mathcal{A}}} \otimes \bigoplus\_{i=1}^{n} \beta\_{i} \otimes \tilde{\mathbf{x}}^{(i)} = \boldsymbol{\lambda} \otimes \bigoplus\_{i=1}^{n} \beta\_{i} \otimes \tilde{\mathbf{x}}^{(i)} \tag{37}$$

$$
\underline{\mathbf{A}} \otimes \mathbf{x} = \lambda \otimes \mathbf{x} \quad \text{and} \quad \overline{\mathbf{A}} \otimes \mathbf{x} = \lambda \otimes \mathbf{x}.\tag{38}
$$

The assertion follows by Theorem 8.

By Theorem 9, the verification of whether **X** is a strongly universal eigenvector of **A** is reduced to the verification of the solvability of the system *C* ⊗*<sup>L</sup> x* = *λ* ⊗*<sup>L</sup> D* ⊗*<sup>L</sup> x*. A similar situation in max-min algebra is solved by a polynomial algorithm for the solvability of such a system, with a complexity equal to *O*(*n*3) [29].

In max-plus algebra, the solvability of the considered system has been generally shown to be polynomially equivalent to solving a mean-payoff game [30]. That is, there exist efficient pseudopolynomial algorithms for this problem. On the other hand, the existence of a polynomial algorithm is a long-standing open question. Similarly, for a max-Łuk algebra, the existence of a polynomial algorithm for the solvability recognition problem remains open.

**Example 6.** *(Numerical illustration: Strongly universal interval eigenvector)*

*Assume that A* ∈ [*A*, *A*] *and x* ∈ [*x*, *x*]

$$
\underline{A} = \begin{pmatrix} 0.6 & 0.3 & 0.4 \\ 0.1 & 0 & 0.2 \\ 0.2 & 0.2 & 0.6 \end{pmatrix}, \qquad \overline{A} = \begin{pmatrix} 0.6 & 0.7 & 0.5 \\ 0.4 & 0.6 & 0.5 \\ 0.5 & 0.7 & 0.6 \end{pmatrix}.
$$

$$
\underline{x} = \begin{pmatrix} 0.1 \\ 0.2 \\ 0.1 \end{pmatrix}, \qquad \overline{x} = \begin{pmatrix} 0.7 \\ 0.6 \\ 0.7 \end{pmatrix}.
$$

*It is easy to verify that for a given <sup>λ</sup>* <sup>=</sup> 0.4*, <sup>x</sup>* = (0.1, 0.2, 0.1)*<sup>T</sup> and for every matrix generator <sup>A</sup>*˜*ij, <sup>i</sup>*, *<sup>j</sup>* <sup>∈</sup> *N,*

$$A^{ij}\otimes \mathbf{x} = \mathcal{A}^{ij}\otimes \begin{pmatrix} 0.1\\0.2\\0.1 \end{pmatrix} = \begin{pmatrix} 0\\0\\0 \end{pmatrix} = 0.4\otimes \begin{pmatrix} 0.1\\0.2\\0.1 \end{pmatrix} = \lambda\otimes\_L \mathbf{x}.$$

*Hence, in view of Theorem 7, the interval vector* **X** *is a strongly universal max-Łuk eigenvector of the interval matrix* **A***.*

*Theorem 9 offers a more systematic approach: find a solution to the two-sided system* (32) *and* (33) *with unknown coefficients β, for some λ* ∈ I*:*

$$
\begin{pmatrix} 0.3 & 0 & 0.1 \\ 0 & 0 & 0 \\ 0 & 0 & 0.3 \\ 0.3 & 0.3 & 0.2 \\ 0.1 & 0.2 & 0.2 \\ 0.2 & 0.3 & 0.3 \end{pmatrix} \otimes\_L \begin{pmatrix} \beta\_1 \\ \beta\_2 \\ \beta\_3 \end{pmatrix} = \lambda' \otimes\_L \begin{pmatrix} 0.7 & 0.1 & 0.1 \\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.1 & 0.7 \\ 0.7 & 0.1 & 0.1 \\ 0.2 & 0.6 & 0.2 \\ 0.1 & 0.1 & 0.7 \end{pmatrix} \otimes\_L \begin{pmatrix} \beta\_1 \\ \beta\_2 \\ \beta\_3 \end{pmatrix},
$$

$$
\begin{pmatrix} 0.4 \\ 0.6 \\ 0.6 \\ 0.4 \end{pmatrix} \le \begin{pmatrix} \beta\_1 \\ \beta\_2 \\ \beta\_3 \end{pmatrix} \le \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.
$$

$$
\mathbf{x}^T$$

*In this particular instance, it is easy to verify that β* = (0.9, 0.8, 0.8)*<sup>T</sup> is a solution to the system with <sup>λ</sup>* <sup>=</sup> 0.6*. Then the corresponding linear combination of generators, <sup>x</sup>* = (0.9 <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(1)) <sup>⊕</sup> (0.8 <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(2)) <sup>⊕</sup> (0.8 <sup>⊗</sup>*<sup>L</sup> <sup>x</sup>*˜(3))=(0.6, 0.4, 0.5)*T, satisfies the condition from Theorem 8. It is worth noticing that we have found two different universal eigenvectors x and x for two different values λ and λ in this example. Hence, we have shown that neither the "strongly universal" eigenvalue nor the strongly universal eigenvector are uniquely determined.*

#### **6. Conclusions**

Strong versions of the notion of an interval eigenvector of an interval matrix in a max-Łuk algebra have been investigated in this paper. The steady states of a given discrete events system (DES) correspond to eigenvectors of the transition matrix of the system under consideration. When the entries of the state vectors and transition matrix are supposed to be contained in some intervals, then several types of interval eigenvector can be defined, according to the choice of the quantifiers used in the definition. Three of the main important types of interval eigenvectors of a given interval matrix in a max-Łuk algebra have been studied: the strong eigenvector, the strongly tolerable eigenvector, and the strongly universal eigenvector.

Using vector generators and matrix generators belonging to given intervals, the structure of the eigenspace for each of the above mentioned types has been described, and necessary and sufficient conditions for the existence of an interval eigenvector have been formulated. Moreover, recognition algorithms have been suggested for the recognition of these conditions for the first two types: strong and strongly tolerable eigenvector. The existence of an efficient recognition algorithm for the strongly universal type has not been shown. This question remains as a challenge for future research.

These results can be useful in practical applications aimed at the construction of real DES working with Łukasiewicz fuzzy logic. The results have been illustrated by numerical examples.

**Author Contributions:** Investigation, M.G., Z.N. and J.P.; Writing–review and editing, M.G., Z.N. and J.P. All authors contributed equally to this manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Czech Science Foundation (GACR) #18-01246S and by the ˇ Faculty of Informatics and Management UHK, specific research project 2107 Computer Networks for Cloud, Distributed Computing, and Internet of Things III.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

### **References**


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