**Proof.** (i) ⇒ (ii).

Assume (9)–(11). Then (14) and (15) are satisfied. In particular, considering only the rows *i*, *k* ∈ *H* ⊆ *M*, we get

$$\mathbb{C}\_{H} \otimes\_{L} \mathcal{Y} = \lambda \otimes\_{L} b\_{H}. \tag{25}$$

The inequalities *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup>* follow by assumption (i). Moreover, (24) implies *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup><sup>h</sup>* <sup>=</sup> *i*∈*M*\*H* - 1 − *Ci*). Summarizing, we have

$$\forall y \in \mathcal{S} \left( \mathcal{C}\_{H,\prime} \lambda \otimes\_L b\_H, \overline{y} \wedge \overline{y}^h, \underline{y} \right). \tag{26}$$

(ii) ⇒ (i). Conversely, assume (26). Then ( *<sup>j</sup>*∈*N*(*cij* ⊗*<sup>L</sup> yj*) = *<sup>λ</sup>* ⊗*<sup>L</sup> bi*, for every *<sup>i</sup>* ∈ *<sup>H</sup>*.

Moreover, for *i* ∈ *M* \ *H* and *j* ∈ *N*, we have *yj* ≤ 1 − *cij*, by assumption *y* ≤ *yH*. Therefore, *cij* + *yj* − 1 ≤ 0, which implies *cij* ⊗*<sup>L</sup> yj* = 0 = *λ* ⊗*<sup>L</sup> bi* since *i* ∈ *M* \ *H*, which gives *λ* + *bi* − 1 ≤ 0. The inequalities *y* ≤ *y* ≤ *y* follow immediately.

For *λ* ∈ I and for every *i* ∈ *M*, *j* ∈ *N*, we define

$$\mathcal{Y}\_{ij}^{\star}(\lambda) = \begin{cases} \lambda + d\_{ij} & \text{if } i \in H \\ 1 - c\_{ij} & \text{if } i \in M \backslash H. \end{cases} \tag{27}$$

Furthermore, we define *y*-(*λ*) ∈ I(*n*) by putting, for *j* ∈ *N*,

$$\mathcal{Y}\_{\vec{j}}^{\star}(\lambda) = \overline{y}\_{\vec{j}} \wedge \bigwedge\_{i \in H} \left(\lambda + d\_{i\vec{j}}\right) \wedge \bigwedge\_{i \in M \backslash H} \left(1 - c\_{i\vec{j}}\right). \tag{28}$$

**Lemma 3.** *Let C* ∈ I(*m*, *n*)*, b* ∈ I(*m*)*, λ* ∈ I *and y* ∈ I(*n*)*. Then*


*3. y*-(*λ*) *is the maximal vector in* I(*n*) *fulfilling conditions (i) and (ii).*

**Proof.** It is easy to verify, using the definition of <sup>⊗</sup>*L*, that *cij* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*- *ij*(*λ*) = *λ* ⊗*<sup>L</sup> bi* for every *i* ∈ *M*, *j* ∈ *N*. Then, assertions (i) and (ii) follow from the definition of *y*-(*λ*).

(iii) Assume *<sup>y</sup>* ∈ I(*n*) satisfies conditions (i) and (ii) with *<sup>y</sup>*-(*λ*) replaced by *y*. Let *i* ∈ *M*, *j* ∈ *N*. By (i) we have *cij* <sup>⊗</sup>*<sup>L</sup> yj* <sup>≤</sup> *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup> bi*. Suppose, by contradiction, that there is *<sup>j</sup>* <sup>∈</sup> *<sup>N</sup>* such that *yj* <sup>&</sup>gt; *<sup>y</sup>*- *<sup>j</sup>* (*λ*). Then, in view of (28), there is *<sup>i</sup>* <sup>∈</sup> *<sup>M</sup>* such that *yj* <sup>&</sup>gt; *<sup>y</sup>*- *ij*(*λ*). We consider two cases.

Case (a): *i* ∈ *H*. Then *yj* > *λ* + *bi* − *cij*, which implies *cij* + *yj* − 1 > *λ* + *bi* − 1. Thus, *cij* ⊗*<sup>L</sup> yj* > *λ* ⊗*<sup>L</sup> bi*, a contradiction.

Case (b): *i* ∈ *M* \ *H*. Then *yj* > 1 − *cij* implies *cij* + *yj* − 1 > 0. Thus, *cij* ⊗*<sup>L</sup> yj* > 0 = *λ* ⊗*<sup>L</sup> bi*, a contradiction.

Given that *<sup>i</sup>*, *<sup>j</sup>* are arbitrary, *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(*λ*) follows.

**Lemma 4.** *Let C* ∈ I(*m*, *n*)*, b* ∈ I(*m*)*, λ* ∈ I *and y* ∈ I(*n*)*. Then the following statements are equivalent.*

*1. S*(*C*, *λ* ⊗*<sup>L</sup> b*, *y*) = ∅*, 2. y*-(*λ*) ∈ *S*(*C*, *λ* ⊗*<sup>L</sup> b*, *y*)*.*

*If y* <sup>∈</sup> *<sup>S</sup>*(*C*, *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>b</sup>*, *<sup>y</sup>*)*, then y* <sup>≤</sup> *<sup>y</sup>*-(*λ*)*.*

**Proof.** The assertion of the lemma follows directly from Lemma 3.

**Lemma 5.** *Let C* ∈ I(*m*, *n*)*, b* ∈ I(*m*)*, λ* ∈ I *and let y*, *y* ∈ I(*n*) *with y* ≤ *y. Then the following statements are equivalent.*


*If y* <sup>∈</sup> *<sup>S</sup>*(*C*, *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>b</sup>*, *<sup>y</sup>*, *<sup>y</sup>*)*, then y* <sup>≤</sup> *<sup>y</sup>*-(*λ*)*.*

**Proof.** Assume that *y* ∈ *S*(*C*, *λ* ⊗*<sup>L</sup> b*, *y*, *y*). Then, in particular, *y* ∈ *S*(*C*, *λ* ⊗*<sup>L</sup> b*, *y*), which implies *y*-(*λ*) <sup>∈</sup> *<sup>S</sup>*(*C*, *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>b</sup>*, *<sup>y</sup>*) and *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(*λ*), in view of Lemma 3. Furthermore, *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(*λ*) implies *y*-(*λ*) ∈ *S*(*C*, *λ* ⊗*<sup>L</sup> b*, *y*). The converse implication is trivial.

**Remark 1.** *The assertions of Lemma <sup>5</sup> are often expressed by saying that for fixed <sup>λ</sup>* ∈ I*, <sup>y</sup>*-(*λ*) *is the maximal possible candidate for a solution of the system* (9) *and* (10)*.*

**Remark 2.** *By a standard definition, the minimum of the empty subset of* I *is the maximal element in* I*. Hence, if there is no i* ∈ *M with λ* ≤ 1 − *bi, then i*∈*M*\*H* - 1 − *cij* = *I, and*

$$y\_j^\star(\lambda) = \overline{y}\_j \land \bigwedge\_{i \in H} \left(\lambda + d\_{ij}\right).$$

*Similarly, if there is no i* ∈ *M with λ* > 1 − *bi, then i*∈*H* - *λ* + *dij* = *I, and*

$$y\_j^\star(\lambda) = \overline{y}\_j \land \bigwedge\_{i \in \mathcal{M} \backslash H} \left(1 - c\_{i\overline{j}}\right).$$

#### **4. Parametric Solvability Problem of Max-Łuk Linear Equations**

The main result of this paper is description of a recognition algorithm for the parametric solvability problem. The problems (9) and (10) will be discussed according to

$$h(\lambda) = \max\{i \in M; \lambda > 1 - b\_i\}.\tag{29}$$

Similarly to Remark 2, we put *h*(*λ*) = max ∅ = 0 if *λ* ≤ 1 − *bi* for all *i* ∈ *M*. For *j* ∈ *N* we also use the notation

$$g\_j^{\star\star}(\lambda) = \bigwedge\_{i \in H} (\lambda + d\_{ij}) \wedge \bigwedge\_{i \in M \backslash H} (1 - c\_{ij}).\tag{30}$$

We consider three cases: (A) *h*(*λ*) = *m*, (B) 0 < *h*(*λ*) < *m* and (C) *h*(*λ*) = 0. The solvability in case A is described by the following theorem, with the notation

$$\lambda\_{\text{max}}^{m} = 1 \land \bigwedge\_{j \in N} \bigvee\_{i \in M} (\overline{y}\_{j} - d\_{i\bar{j}}). \tag{31}$$

**Theorem 3.** *Case (A). Assume C* ∈ I(*m*, *n*) *and b* ∈ I(*m*) *with the monotonicity condition* (11)*. Then the following statements are equivalent.*


**Proof.** (i) ⇒ (ii) Assume *λ* ∈ I is given with 1 ≥ *λ* > 1 − *bm* and *S*(*C*, *λ* ⊗*<sup>L</sup> b*, *y*, *y*) = ∅.

We have 1 ≥ *λ* > 1 − *bm* ≥ 1 − *bi* for every *i* ∈ *M*. That is, *H* = *M* and *M* \ *H* = ∅. In view of Remark 2, for every *j* ∈ *N*

$$y\_j^\*(\lambda) = \overline{y}\_j \land \bigwedge\_{i \in M} (\lambda + d\_{i\overline{j}}).\tag{32}$$

The assumption *<sup>S</sup>*(*C*, *<sup>λ</sup>*⊗*L*, *<sup>y</sup>*, *<sup>y</sup>*) <sup>=</sup> <sup>∅</sup> implies *<sup>C</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*-(*λ*) = *λ* ⊗*<sup>L</sup> b*. That is, (10), (14) and (15) are satisfied with *yj* = *y*- *<sup>j</sup>* (*λ*).

Now we consider *κ* ∈ I with *λ* ≤ *κ* ≤ 1. We look for a necessary and sufficient condition such that (10), (14) and (15) hold for *yj* = *y*- *<sup>j</sup>* (*κ*).

Since *<sup>λ</sup>* <sup>≤</sup> *<sup>κ</sup>*, we have *<sup>y</sup>*- *<sup>j</sup>* (*λ*) <sup>≤</sup> *<sup>y</sup>*- *<sup>j</sup>* (*κ*), for every *<sup>j</sup>* <sup>∈</sup> *<sup>N</sup>*. Therefore, *yj* <sup>≤</sup> *<sup>y</sup>*- *<sup>j</sup>* (*λ*) implies *yj* <sup>≤</sup> *<sup>y</sup>*- *<sup>j</sup>* (*κ*). That is, *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(*κ*) for every *<sup>λ</sup>* <sup>≤</sup> *<sup>κ</sup>* <sup>≤</sup> 1. The conditions for the upper bound inequality *<sup>y</sup>*-(*κ*) ≤ *y* will be discussed later.

First, we verify conditions (14) and (15). In view of the assumption, we have 1 ≥ *κ* ≥ *λ* > 1 − *bm* ≥ 1 − *bi* for every *i* ∈ *M*. That is, *H* = *M* and *M* \ *H* = ∅. Then, for every *j* ∈ *N*

$$y\_j^{\star\star}(\lambda) = \bigwedge\_{i \in M} \left(\lambda + d\_{i\bar{j}}\right) = \lambda + \bigwedge\_{i \in M} d\_{i\bar{j}}.\tag{33}$$

Similarly, for every *j* ∈ *N*

$$y\_j^{\star\star}(\kappa) = \kappa + \bigwedge\_{i \in M} d\_{ij}. \tag{34}$$

It follows that the equalities

$$
\omega\_{ij} + y\_j^{\star \star}(\lambda) - 1 = \lambda + b\_i - 1,\tag{35}
$$

$$x\_{i\bar{j}} + y\_{\bar{j}}^{\star\star}(\kappa) - 1 = \kappa + b\_{\bar{i}} - 1\tag{36}$$

are equivalent. Similarly,

$$
\lambda c\_{i\bar{j}} + y\_{\bar{j}}^{\star \star}(\lambda) - 1 \le \lambda + b\_{\bar{i}} - 1,\tag{37}
$$

$$
\kappa\_{i\bar{j}} + y\_{\bar{j}}^{\star\star}(\kappa) - 1 \le \kappa + b\_{\bar{i}} - 1 \tag{38}
$$

are equivalent.

Therefore, the assumption *y*--(*λ*) <sup>∈</sup> *<sup>S</sup>*(*C*, *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>b</sup>*) is equivalent to *<sup>y</sup>*--(*κ*) ∈ *S*(*C*, *κ* ⊗*<sup>L</sup> b*), for every *κ* ∈ I with *λ* ≤ *κ* ≤ 1.

To achieve also the upper bound inequality *y*--(*κ*) ≤ *y*, further conditions must be imposed on *κ*. Namely, for every *j* ∈ *N* the condition

$$\|y\_j^{\star\star}(\kappa) = \kappa + \bigwedge\_{i \in M} d\_{ij} \le \overline{y}\_j \tag{39}$$

must be added. As a consequence, we get

$$\kappa \le \bigwedge\_{j \in \mathcal{N}} \bigvee\_{i \in \mathcal{M}} (\overline{y}\_j - d\_{i\bar{j}}).\tag{40}$$

Now, with the notation

$$\lambda\_{\text{max}}^{\text{m}} = 1 \land \bigwedge\_{j \in \mathcal{N}} \bigvee\_{i \in \mathcal{M}} (\overline{y}\_j - d\_{ij}) \tag{41}$$

we have

$$y^{\star\star}(\kappa) \le \overline{y} \quad \Leftrightarrow \quad \kappa \le \lambda^{\mathrm{m}}\_{\mathrm{max}}.\tag{42}$$

Therefore, *y*--(*κ*) <sup>∈</sup> *<sup>S</sup>*(*C*, *<sup>λ</sup><sup>m</sup>* max ⊗*<sup>L</sup> b*, *y*, *y*). The converse implication (ii) ⇒ (i) is trivial.

In case B we have 0 < *h*(*λ*) < *m*. That is,

$$0 \le \dots \le 1 - b\_{\mathbb{H}} < \lambda \le 1 - b\_{\mathbb{H}+1} \le \dots \ge 1. \tag{43}$$

Write Λ(*h*) = - 1 − *bh*, 1 − *bh*<sup>+</sup><sup>1</sup>, for brevity. In view of (19) and (20), we have *H* = {1, 2, ... , *h*} (with 0 < *λ* + *bi* − 1 for *i* ∈ *H*) and *M* \ *H* = {*h* + 1, *h* + 2, ... , *m*} (with 0 ≥ *λ* + *bi* − 1 for *i* ∈ *M* \ *H*).

Moreover, we denote

$$N(h) = \left\{ j \in N; \left( (\overline{y}\_j \wedge \overline{y}\_j^h) - \bigwedge\_{i \in H} d\_{ij} \right) > 1 - b\_h \right\}\_{j} \tag{44}$$

$$
\lambda\_{\max}^h = \left(1 - b\_{h+1}\right) \wedge \bigwedge\_{j \in N(h)} \left(\left(\overline{y}\_j \wedge \overline{y}\_j^h\right) - \bigwedge\_{i \in H} d\_{ij}\right). \tag{45}
$$

**Theorem 4.** *Case (B). Assume C* ∈ I(*m*, *n*)*, b* ∈ I(*m*) *and h* ∈ *M. Further assume the monotonicity condition* (11) *holds. Then the following statements are equivalent.*


**Proof.** For fixed *λ* ∈ Λ*h*, (i) is equivalent (in view of Lemma 5) to the statement

$$y^\*(\lambda) \in S(\mathbb{C}, \lambda \otimes\_L b, \overline{y}, \underline{y}),\tag{46}$$

which is further equivalent (in view of Lemma 2) to

$$\mathcal{Y}^\*(\lambda) \in \mathcal{S}(\mathbb{C}\_{H\prime}\lambda \otimes\_L b\_{H\prime}\overline{\mathfrak{y}} \wedge \overline{\mathfrak{y}}^{\mathfrak{h}}, \underline{\mathfrak{y}}).\tag{47}$$

The proof will be completed by demonstrating that (47) is equivalent to (ii). We assume (47) for fixed *λ* ∈ Λ(*h*), and we describe conditions under which (47) also holds for arbitrary *κ* ∈ Λ(*h*) with *λ* ≤ *κ*.

We shall verify the restrictions *<sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(*κ*) <sup>≤</sup> *<sup>y</sup>* <sup>∧</sup> *<sup>y</sup><sup>h</sup>* and the activity of the variables *<sup>y</sup>*-(*κ*) in every row *k* ∈ *H* of the matrix *CH* with vector *bH*.

The lower restriction follows by monotonicity *yj* <sup>≤</sup> *<sup>y</sup>*- *<sup>j</sup>* (*λ*) <sup>≤</sup> *<sup>y</sup>*- *<sup>j</sup>* (*κ*), for every *j* ∈ *N*. The upper restriction

$$y\_j^\*(\kappa) = \overline{y}\_j \wedge \left(\kappa + \bigwedge\_{i \in M} d\_{ij}\right) \wedge \bigwedge\_{i \in M \backslash H} (1 - c\_{i\bar{j}}) \le \overline{y}\_{\bar{j}} \wedge \overline{y}\_{\bar{j}}^{\mathbf{h}} \tag{48}$$

follows by Lemma 2 directly from

$$\mathfrak{d}\_{\mathfrak{J}}\mathfrak{\*}(\kappa) = \overline{\mathfrak{y}}\_{\mathfrak{j}} \wedge \overline{\mathfrak{Y}}\_{\mathfrak{j}}^{\mathfrak{h}} \wedge \left(\kappa + \bigwedge\_{i \in H} d\_{i\mathfrak{j}}\right) \leq \overline{\mathfrak{y}}\_{\mathfrak{j}} \wedge \overline{\mathfrak{Y}}\_{\mathfrak{j}}^{\mathfrak{h}}.\tag{49}$$

As a consequence, the activity condition (22) is fulfilled in all rows *k* ∈ *M* \ *H* for every variable *y*- *<sup>j</sup>* (*κ*) with ∈ *N*.

To verify also the second activity condition (21) for at least one variable *j* ∈ *N* in every row *k* ∈ *H*, we denote by *μ<sup>h</sup> <sup>j</sup>* the break-point at which the (min/plus)-linear function

$$y\_j^\star(\kappa) = \left(\overline{y}\_j \wedge \overline{y}\_j^\natural\right) \wedge \left(\kappa + \bigwedge\_{i \in H} d\_{i\overline{j}}\right) \tag{50}$$

of the variable *κ* changes its direction. In other words,

$$y\_j^\*(\kappa) = \begin{cases} \kappa + \bigwedge\_{i \in H} d\_{ij} & \text{if } \kappa \le \mu\_{j'}^h \\ \overline{y}\_j \wedge \overline{y}\_j^h & \text{if } \kappa \ge \mu\_j^h. \end{cases} \tag{51}$$

By condition 1 − *bi* < *κ*, for *i* ∈ *H*, we get 0 < *κ* + *bi* − 1 ≤ *κ* + *bi* − *cij* = *κ* + *dij*. At the break-point, both parts of the function (50) have the same value. That is,

$$
\mu\_{\dot{j}}^h + \bigwedge\_{i \in H} d\_{\dot{i}\dot{j}} = \overline{y}\_{\dot{j}} \wedge \overline{y}\_{\dot{j}}^h. \tag{52}
$$

or, equivalently,

$$
\mu\_{\rangle}^{h} = \left(\overline{\boldsymbol{y}}\_{\rangle} \wedge \overline{\boldsymbol{y}}\_{\rangle}^{h}\right) - \bigwedge\_{i \in H} d\_{ij}. \tag{53}
$$

**Claim 1.** *Assume <sup>j</sup>* <sup>∈</sup> *N, <sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>(*h*)*. If <sup>y</sup>*- *<sup>j</sup>* (*λ*) *is active in <sup>k</sup>* <sup>∈</sup> *H, then* <sup>1</sup> <sup>−</sup> *bh* <sup>&</sup>lt; *<sup>λ</sup>* <sup>≤</sup> *<sup>μ</sup><sup>h</sup> <sup>j</sup> . Moreover, for every <sup>κ</sup>* <sup>∈</sup> <sup>Λ</sup>(*h*) *with <sup>κ</sup>* <sup>≤</sup> *<sup>μ</sup><sup>h</sup> <sup>j</sup> , y*- *<sup>j</sup>* (*κ*) *is active in k.*

**Proof of Claim 1.** By assumption, *y*- *<sup>j</sup>* (*λ*) = *λ* + *dkj* = *λ* + *<sup>i</sup>*∈*<sup>H</sup> dij*, in view of (21) and (23). Then, *<sup>λ</sup>* <sup>≤</sup> *<sup>μ</sup><sup>h</sup> <sup>j</sup>* , and the activity of *<sup>y</sup>*- *<sup>j</sup>* (*λ*) in *k* is described by the formula

$$d\_{kj} + \lambda + \bigwedge\_{i \in H} d\_{ij} - 1 = \lambda + b\_k - 1,\tag{54}$$

while the activity of *y*- *<sup>j</sup>* (*κ*) under assumption *<sup>κ</sup>* <sup>≤</sup> *<sup>μ</sup><sup>h</sup> <sup>j</sup>* is described by

$$c\_{k\bar{j}} + \kappa + \bigwedge\_{i \in H} d\_{i\bar{j}} - 1 = \kappa + b\_k - 1. \tag{55}$$

As (54) and (55) are equivalent, the assertion of Claim 1 follows. In view of (44), (45) and (53) we have

$$N(h) = \left\{ j \in N; \; \mu\_j^h > 1 - b\_h \right\},\tag{56}$$

$$
\lambda\_{\text{max}}^h = \left(1 - b\_{h+1}\right) \wedge \bigwedge\_{j \in N(h)} \mu\_j^h. \tag{57}
$$

**Claim 2.** *Assume y*-(*λ*) <sup>∈</sup> *<sup>S</sup>*(*CH*, *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup> bH*, *<sup>y</sup>* <sup>∧</sup> *<sup>y</sup>h*, *<sup>y</sup>*)*. If <sup>λ</sup>* <sup>≤</sup> *<sup>κ</sup>* <sup>∈</sup> <sup>Λ</sup>(*h*)*, with <sup>κ</sup>* <sup>≤</sup> *<sup>μ</sup><sup>h</sup> <sup>j</sup> for every j* ∈ *N*(*h*)*, then y*-(*κ*) <sup>∈</sup> *<sup>S</sup>*(*CH*, *<sup>λ</sup><sup>h</sup>* max <sup>⊗</sup>*<sup>L</sup> bH*, *<sup>y</sup>* <sup>∧</sup> *<sup>y</sup>h*, *<sup>y</sup>*)*.*

**Proof of Claim 2.** By assumption, for every *<sup>k</sup>* <sup>∈</sup> *<sup>H</sup>* there is a *<sup>j</sup>* <sup>∈</sup> *<sup>N</sup>* such that *<sup>y</sup>*-(*λ*) is active in *k*. Then by Claim 1, under assumption *λ* ≤ *κ* ≤ *<sup>j</sup>*∈*<sup>N</sup> <sup>μ</sup><sup>h</sup> <sup>j</sup>* , for every *<sup>k</sup>* <sup>∈</sup> *<sup>H</sup>* there is a *<sup>j</sup>* <sup>∈</sup> *<sup>N</sup>*(*h*) such that *<sup>y</sup>*-(*κ*) is active in *k*. That is, *y*-(*κ*) <sup>∈</sup> *<sup>S</sup>*(*CH*, *<sup>λ</sup><sup>h</sup>* max <sup>⊗</sup>*<sup>L</sup> bH*, *<sup>y</sup>* <sup>∧</sup> *<sup>y</sup>h*, *<sup>y</sup>*).

In case C we have *H* = ∅ and *M* \ *H* = *M*. That is, *λ* ≤ 1 − *bi* for all *i* ∈ *M*. The solvability in case C is described by the following theorem.

**Theorem 5.** *Case (C). Assume C* ∈ I(*m*, *n*)*, b* ∈ I(*m*) *and y* ≤ *y* ∈ I(*n*)*, with the monotonicity condition* (11)*. Then the following statements are equivalent*


**Proof.** Assume 0 ≤ *λ* ≤ 1 − *b*<sup>1</sup> and *y* ∈ *S*(*C*, *λ* ⊗*<sup>L</sup> b*, *y*, *y*). By Lemma 5(ii), this is equivalent to *y*-(*λ*) ∈ *S*(*C*, *λ* ⊗*<sup>L</sup> b*, *y*, *y*). For every *j* ∈ *N* we have, in view of Remark 2,

$$\underline{y}\_{j} \le y\_{j}^{\star}(\lambda) = \overline{y}\_{j} \land \bigwedge\_{i \in M} (1 - c\_{i\bar{j}}).\tag{58}$$

The equivalence (i) ⇔ (ii) follows immediately.

**Theorem 6.** *Assume C* ∈ I(*m*, *n*)*, b* ∈ I(*m*) *and y* ≤ *y* ∈ I(*n*)*, with the monotonicity condition* (11)*. The bounded parametric system* (9) *and* (10) *is solvable for some λ* ∈ I *if and only if at least one of the following statements is fulfilled*


**Proof.** For the convenience of the reader, we recall the previous definitions.

$$h(\lambda) = \max\{i \in M; \lambda > 1 - b\_i\},\tag{59}$$

$$
\Lambda(h) = \begin{pmatrix} 1 - b\_{h\prime} \mathbf{1} - b\_{h+1} \end{pmatrix} \tag{60}
$$

$$N(h) = \left\{ j \in N; \left( (\overline{y}\_j \wedge \overline{y}\_j^h) - \bigwedge\_{i \in H} d\_{i\overline{j}} \right) > 1 - b\_{\overline{h}} \right\},\tag{61}$$

$$
\lambda^{m}\_{\text{max}} = 1 \land \bigwedge\_{j \in N} \bigvee\_{i \in M} (\overline{y}\_{j} - d\_{ij}),
\tag{62}
$$

$$
\lambda\_{\text{max}}^h = \left(1 - b\_{h+1}\right) \wedge \bigwedge\_{j \in \mathcal{N}(h)} \left(\left(\mathfrak{F}\_j \wedge \mathfrak{F}\_j^h\right) - \bigwedge\_{i \in H} d\_{i\bar{j}}\right). \tag{63}
$$

Assume that the system (9) and (10) is solvable for some *λ* ∈ I. Clearly, one of these possibilities takes place: (a) *h*(*λ*) = *m*, (b) 0 < *h*(*λ*) < *m*, *λ* ∈ Λ(*h*), (c) *h*(*λ*) = 0. The assertion of the theorem then follows from Theorems 3–5.

**Theorem 7.** *Suppose that C* ∈ I(*m*, *n*)*, b* ∈ I(*m*) *and y*, *y* ∈ I(*n*)*. The problem of recognizing the solvability of the bounded parametric max-Łuk linear system*

$$\mathcal{C}\otimes\_L y = \lambda \otimes\_L b \tag{64}$$

*with bounds y* <sup>≤</sup> *<sup>y</sup>* <sup>≤</sup> *y for some value of the parameter <sup>λ</sup>* ∈ I *can be solved in O*(*mn*2) *time.*

**Proof.** In view of Theorem 6, the solvability of the bounded max-Łuk linear system (64) for some *λ* ∈ I can be verified by verifying the solvability of (64) for the values *λ*<sup>1</sup> max, *λ*<sup>2</sup> max, ... *λ<sup>m</sup>* max and verifying the condition *yj* ≤ *i*∈*M* - 1 − *cij* , for every *j* ∈ *N*.

For every *h* = 1, 2, ... , *m*, *λ<sup>h</sup>* max can be computed in *O*(*n*2) time and the computation of *y*-(*λ<sup>h</sup>* max) requires *O*(*n*) time. The verification of *y*-(*λ<sup>h</sup>* max) <sup>∈</sup> *<sup>S</sup>*(*CH*, *<sup>λ</sup><sup>h</sup>* max ⊗*<sup>L</sup> bH*, *y*, *y*) needs *O*(*n*) time, while condition (ii) in Theorem 5 can be verified in *O*(*mn*) time. Thus, the total computational complexity is *O*(*mn*2).

#### **5. Numerical Examples**

**Example 1** (A numerical illustration to Q1—solvable case)**.** *Assume that transition matrix C and required state vector b are given. Our goal is to recognize whether there is* 0 < *λ* ∈ I *and y* ∈ I(*n*) *with y* ≤ *y* ≤ *y such that C* ⊗*<sup>L</sup> y* = *λ* ⊗*<sup>L</sup> b. In other words, we ask whether the system* (9) *and* (10) *with entries* (65) *is solvable for some* 0 < *λ* ∈ I*.*

$$\mathbf{C} = \begin{pmatrix} 0.6 & 0.5 & 0.5 & 0.8 & 0.8\\ 0.5 & 0.7 & 0.6 & 0.5 & 0.9\\ 0.3 & 0.9 & 0.3 & 0.3 & 0.0\\ 0.1 & 0.7 & 0.9 & 0.2 & 0.9\\ 0.9 & 0.2 & 0.2 & 0.6 & 0.8 \end{pmatrix}, b = \begin{pmatrix} 0.7\\ 0.6\\ 0.3\\ 0.1\\ 0.1 \end{pmatrix}, \underline{y} = \begin{pmatrix} 0.1\\ 0.0\\ 0.1\\ 0.3\\ 0.1 \end{pmatrix}, \overline{y} = \begin{pmatrix} 0.8\\ 0.6\\ 0.6\\ 0.9\\ 0.5 \end{pmatrix}. \tag{65}$$

*Applying Theorem 6, we get a positive answer. Namely, the system* (9) *and* (10) *is solvable for λ* ∈ (0.3, 0.4 *with solution y*- = (0.1, 0.1, 0.1, 0.3, 0.1) *and has no solution for λ* ∈ (0, 0.3 ∪ (0.4, 1*. Therefore, the orbit of the DES considered in case study in Section 2 can reach the state λ* ⊗ *b for every* 0.3 < *λ* ≤ 0.4*, if the starting state is y*-*. On the other hand, the DES cannot reach the state b* = 1 ⊗*<sup>L</sup> b nor can reach any state λ* ⊗*<sup>L</sup> b if λ* ≤ 0.3 *or λ* > 0.4*.*

*Details of the computation are shown below. We use the method described in Theorem 6. By Definition* (59)*, we get five different values of h*(*λ*) *and distinguish the following cases: (a) h*(*λ*) = 5*, (b) h*(*λ*) = 1, 2, 3 *and (c) h*(*λ*) = 0*.*

*Case (a). We have <sup>H</sup>* <sup>=</sup> {1, 2, 3, 4, 5}*, <sup>M</sup>* \ *<sup>H</sup>* <sup>=</sup> <sup>∅</sup> *and <sup>λ</sup>* <sup>∈</sup> (0.9, 1*. By* (62)*, we get <sup>λ</sup>*<sup>5</sup> max = 1*. Using* (32) *we compute the maximal possible candidate for a solution: y*-(*λ*<sup>5</sup> max) = *y*-(1)=(0.2, 0.4, 0.2, 0.5, 0.2)*T. Clearly, <sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(1)*. It remains to see whether y*-(1) *is a solution to* (9)*.*

$$\mathcal{C} \otimes\_L y^\*(1) = \begin{pmatrix} 0.6 & 0.5 & 0.5 & 0.8 & 0.8\\ 0.5 & 0.7 & 0.6 & 0.5 & 0.9\\ 0.3 & 0.9 & 0.3 & 0.3 & 0.0\\ 0.1 & 0.7 & 0.9 & 0.2 & 0.9\\ 0.9 & 0.2 & 0.2 & 0.6 & 0.8 \end{pmatrix} \otimes \begin{pmatrix} 0.2\\ 0.4\\ 0.2\\ 0.5\\ 0.2 \end{pmatrix} = \begin{pmatrix} 0.3\\ 0.1\\ 0.3\\ 0.1\\ 0.1 \end{pmatrix}$$

$$\neq \begin{pmatrix} 0.7\\ 0.6\\ 0.3\\ 0.1\\ 0.1 \end{pmatrix} = 1 \otimes \begin{pmatrix} 0.7\\ 0.6\\ 0.3\\ 0.1\\ 0.1 \end{pmatrix} = \lambda\_{\text{max}}^5 \otimes\_L b$$

*In view of Lemma 5, <sup>C</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*-(1) <sup>=</sup> *<sup>λ</sup>*<sup>5</sup> max ⊗*<sup>L</sup> b implies that the system has no solution when* 0.9 < *λ* ≤ 1*.*

*Case (b). Three subcases are considered.*

*For h*(*λ*) = 3*, we have H* = {1, 2, 3}*, M* \ *H* = {4, 5} *and* Λ(3)=(0.7, 0.9*. Using* (53) *and* (61)*, we compute <sup>μ</sup>*<sup>3</sup> = (0.1, 0.9, 0.1, 0.5, 0.4)*T, <sup>N</sup>*(3) = {2}*. Then <sup>λ</sup>*<sup>3</sup> max = 0.9 ∧ 0.9 = 0.9 ∈ Λ(3)*, in view of* (63)*. Applying this value we get y*-(0.9)=(0.1, 0.3, 0.1, 0.4, 0.1)*T, in view of* (28)*. The candidate y*-(0.9) *fulfills <sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*- *<sup>j</sup>* (0.9)*, but it is not a solution to the system, because <sup>C</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*-(0.9) <sup>=</sup> *<sup>λ</sup>*<sup>3</sup> max ⊗*<sup>L</sup> b. Therefore, the system has no solution when* 0.7 < *λ* ≤ 0.9*.*

*For h*(*λ*) = 2 *we have H* = {1, 2}*, M* \ *H* = {3, 4, 5} *and* Λ(2)=(0.4, 0.7*. Similarly to the previous subcase, we can calculate <sup>μ</sup>*<sup>2</sup> = (0, 0.2, 0.1, 0.5, 0.4)*<sup>T</sup> and <sup>N</sup>*(2) = {4}*. Then <sup>λ</sup>*<sup>2</sup> max = 0.7 ∧ 0.5 = 0.5 ∈ Λ(2)*. The obtained result y*-(0.5)=(0.1, 0.1, 0.1, 0.4, 0.1)*<sup>T</sup> fulfills the condition <sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(0.5)*, but again, y*-(0.5) *is not a solution to the system, which means that there are no solutions when* 0.4 < *λ* ≤ 0.7*.*

*For h*(*λ*) = 1 *we have H* = {1}*, M* \ *H* = {2, 3, 4, 5} *and* Λ(1)=(0.3, 0.4*. Similarly to the previous subcases, <sup>μ</sup>*<sup>1</sup> = (0, <sup>−</sup>0.1, <sup>−</sup>0.1, 0.5, 0.2)*<sup>T</sup> and <sup>N</sup>*(1) = {4}*, and so <sup>λ</sup>*<sup>1</sup> max = 0.4 ∧ 0.5 = 0.4 ∈ Λ(1)*. For λ*<sup>1</sup> max = 0.4 *we compute y*-(0.4)=(0.1, 0.1, 0.1, 0.3, 0.1)*T. This candidate fulfills <sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*- *<sup>j</sup>* (0.4) *and also is*

*a solution to the system, because of the equality <sup>C</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*-(0.4) = 0.4 ⊗*<sup>L</sup> b. It follows that the system* (9) *and* (10) *considered in this example is solvable when λ* = 0.4*.*

*Case (c). In this case we have h*(*λ*) = 0*, i.e., H* = ∅*, M* \ *H* = *M and λ* ∈ (0, 0.3*. The maximal candidate y*-(*λ*)=(0.1, 0.1, 0.1, 0.2, 0.1)*<sup>T</sup> satisfies <sup>C</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*-(*λ*) = *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup> b, but the requirement of <sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(*λ*) *is not fulfilled. As a consequence, the considered system is not solvable when* 0 < *λ* ≤ 0.3*.*

**Example 2** (A numerical illustration to Q1—insolvable case)**.** *Similarly to Example 1, transition matrix C and required state vector b are given. Again, we wish to recognize whether there is* 0 < *λ* ∈ I *and y* ∈ I(*n*) *with y* ≤ *y* ≤ *y such that C* ⊗*<sup>L</sup> y* = *λ* ⊗*<sup>L</sup> b. In this example, the matrix C is the same, only the vectors b, y and y have different entries.*

$$\mathbf{C} = \begin{pmatrix} 0.6 & 0.5 & 0.5 & 0.8 & 0.8\\ 0.5 & 0.7 & 0.6 & 0.5 & 0.9\\ 0.3 & 0.9 & 0.3 & 0.3 & 0.0\\ 0.1 & 0.7 & 0.9 & 0.2 & 0.9\\ 0.9 & 0.2 & 0.2 & 0.6 & 0.8 \end{pmatrix}, b = \begin{pmatrix} 0.8\\ 0.8\\ 0.8\\ 0.5\\ 0.5 \end{pmatrix}, \underline{\mathbf{y}} = \begin{pmatrix} 0.1\\ 0.2\\ 0.3\\ 0.8\\ 0.5 \end{pmatrix}, \underline{\mathbf{y}} = \begin{pmatrix} 0.7\\ 0.4\\ 0.7\\ 0.9\\ 0.8 \end{pmatrix}. \tag{66}$$

*Applying the method described in Theorem 6, we get a negative result: the system has no solution for any λ* ∈ I*. As a consequence, neither b, nor any of its multiples λ* ⊗*<sup>L</sup> b can be reached by the orbit of the DES.*

*The details of the computation are shown below. By Definition* (59)*, we get three different values of h*(*λ*) *and distinguish the following cases: (a) h*(*λ*) = 5*, (b) h*(*λ*) = 3 *and (c) h*(*λ*) = 0*.*

*Case (a). We have <sup>H</sup>* <sup>=</sup> {1, 2, 3, 4, 5}*, <sup>M</sup>* \ *<sup>H</sup>* <sup>=</sup> <sup>∅</sup> *and <sup>λ</sup>* <sup>∈</sup> (0.5, 1*. By* (62)*, we get <sup>λ</sup>*<sup>5</sup> max = 0.6*. From entries* (66) *we compute the maximal possible candidate for solution, y*-(*λ*<sup>5</sup> max) = *y*-(0.6)=(0.2, 0.4, 0.2, 0.5, 0.2)*T. Clearly, y* <sup>≤</sup> *<sup>y</sup>*-(0.6)*. It remains to verify whether y*-(0.6) *is a solution to* (9)*.*

*In view of Lemma 5, <sup>C</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*-(6) <sup>=</sup> *<sup>λ</sup>*<sup>5</sup> max ⊗*<sup>L</sup> b implies that the system has no solution when* 0.5 < *λ* ≤ 1*.*

*Case (b). In this case, only one subcase has to be considered.*

*For h*(*λ*) = 3 *we have H* = {1, 2, 3}*, M* \ *H* = {4, 5} *and* Λ(3)=(0.2, 0.5*. Using* (53) *and* (61)*, we compute <sup>μ</sup>*<sup>3</sup> = (−0.1, 0.4, <sup>−</sup>0.1, 0.4, 0.2)*T, <sup>N</sup>*(3) = {2, 4}*. Then <sup>λ</sup>*<sup>3</sup> max = 0.5 ∧ 0.4 = 0.4 ∈ Λ(3)*, in view of* (63)*. Applying this value we get y*-(0.4)=(0.1, 0.3, 0.1, 0.4, 0.1)*T, in view of* (28)*. The candidate y*-(0.4) *does not fulfill <sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*- *<sup>j</sup>* (0.4) *and at the same time is not a solution to the system, because <sup>C</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*-(0.4) = *λ*3 max ⊗*<sup>L</sup> b. Therefore, the system has no solution when* 0.2 < *λ* ≤ 0.5*.*

*Case (c). In this case we have h*(*λ*) = 0*, i.e., H* = ∅*, M* \ *H* = *M and λ* ∈ (0, 0.2*. The maximal candidate y*-(*λ*)=(0.1, 0.1, 0.1, 0.2, 0.1)*<sup>T</sup> satisfies <sup>C</sup>* <sup>⊗</sup>*<sup>L</sup> <sup>y</sup>*-(*λ*) = *<sup>λ</sup>* <sup>⊗</sup>*<sup>L</sup> b, but the requirement of <sup>y</sup>* <sup>≤</sup> *<sup>y</sup>*-(*λ*) *is not fulfilled. As a consequence, the considered system is not solvable when* 0 < *λ* ≤ 0.2*.*

#### **6. Conclusions**

In this study, existence of a bounded solution to a one-sided linear system in max-Łuk algebra has been considered in dependence on a given linear parameter factor on the fixed side of the system. Equivalent solvability conditions have been found and a polynomial-time recognition algorithm has been suggested. The correctness of the algorithm has been exactly demonstrated. The work of the algorithm has been illustrated by numerical examples.

The results are new: although the solvability of a one-sided linear system in max-Łuk algebra in the non-parametric case can easily be verified, the method of recognizing the solvability of the parameterized system has not yet been known.

The presented results can be applied in the study of the max-Łukasiewicz systems with interval coefficients. Łukasiewicz arithmetical conjunction can also be used in various types of optimization problems, for example, in the study of interactive cash-flows. Furthermore, the suggested recognition algorithm plays an important role in the investigation of interval eigenvectors.

An advantage of the presented algorithm is, that not only the existence or non-existence of the solution is recognized; the solution values are computed as well, in the solvable case. A possible generalization of the results for other *t*-norms, different from the Łukasiewicz *t*-norm and minimum (the Gödel *t*-norm), remains open for future research.

**Author Contributions:** All authors contributed equally to this work. 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. ˇ

**Acknowledgments:** The authors appreciate the valuable ideas and suggestions of J. Plavka (Technical University of Košice) expressed in personal discussions about this manuscript.

**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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