Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics"a+b"=max(a,b) and"ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.


Introduction
The max-Łukasiewicz semiring is defined over the interval [0, 1] equipped with the operations of addition a ⊕ b = max(a, b) and multiplication a ⊗ L b = max(0, a + b − 1). These operations are extended to matrices and vectors in the usual way: (A ⊕ B) ij = a ij ⊕ b ij and (A ⊗ L B) ik = j a ij ⊗ L b jk . We consider the max-Łukasiewicz powers of matrices A ⊗ L k := k A ⊗ L · · · ⊗ L A, and the spectral problem over max-Łukasiewicz semiring: given λ ∈ [0, 1], find x ∈ [0, 1] n with not all components 0 such that A ⊗ L x = λ ⊗ L x.
Our study of max-Łukasiewicz semiring is motivated by the recent success of tropical linear algebra, developed over the max-plus semiring R max = R ∪ {−∞} equipped with operations of "addition" a ⊕ b = max(a, b) and "multiplication" a ⊗ b = a + b. We convert the problems of max-Łukasiewicz linear algebra, i.e., the linear algebra over max-Łukasiewicz semiring, to the problems of tropical (max-plus) linear algebra. Then we can take advantage of the well-developed theory and algorithms of the latter, see [2,4,5,8,14,24,25], to mention only a few possible sources. Our main ingredients are the theory of spectral problem A ⊗ x = λ ⊗ x, Bellman equations or Z-equations x = A ⊗ x ⊕ b, one-sided systems A ⊗ x = b and, occasionally, the tropical linear programming. Recall that the basic theory of onesided systems A ⊗ x = b involves residuation theory [15] as well as set coverings [5]. Also see Rashid et al. [36] for a preliminary work on the max-Łukasiewicz linear algebra exploring details of the three-dimensional case.
A basic idea behind this paper is that each fuzzy triangular norm (t-norm) as described, for instance, in Klement, Mesiar and Pap [31], leads to an idempotent semiring, which we call a max-t semiring. There are directions of abstract fuzzy sets theory which are related to the present work. Some connections of multivalued logic and fuzzy algebra with idempotent mathematics have been developed by Di Nola et al. [18][19][20][21]. These works develop certain aspects of algebra over semirings arising from fuzzy logic (MV-algebras, Łukasiewicz transform), which currently seem most interesting and useful for the fuzzy sets theory.
However, neither general MV-algebras [19] nor even the special case of Łukasiewicz MV-algebra are considered in this paper. We are rather motivated by the basic problems of the tropical linear algebra, which we are going to consider here in the context of max-Łukasiewicz semiring. We also remark that max-Łukasiewicz semiring can be seen as a special case of incline algebras of Cao, Kim and Roush [9], see also, e.g., Han-Li [28] and Tan [40]. One of the main problems considered in that algebra is to study the periodicity of matrix powers over a larger class of semirings, using lattice theory, lattice ordered groups and residuations. Let us also recall the distributive lattices as another special case of incline algebras, although this special case does not include the max-Łukasiewicz semiring. Powers of matrices over distributive lattices are studied, e.g., by Cechlárová [11].
The approach which we develop here, does not apply to incline algebras in general (or to distributive lattices in particular), but it allows to study the linear-algebraic problems over max-Łukasiewicz algebra in much more detail.
Aggressive network. Let us also recall the basic network motivation to study tropical (max-plus) linear algebra and, more generally, linear algebra over semirings , see also ). This motivation suggests a directed graph D with nodes N = {1, . . . , n} and edges E = {(i, j): i, j ∈ N }, where each edge (i, j) is weighted by a ij . Sarah is an agent travelling in the network. She is given 1 unit of money before entering it at node i (say, 1 thousand of GBP), and a ij expresses the amount of money left on her bank account after she moves from i to j . The quantity c ij := 1 − a ij expresses the cost of moving from i to j . More generally, if Sarah is given x i units of money, with 0 ≤ x i ≤ 1, then there are two cases: when x i − c ij ≥ 0 and when x i − c ij < 0. In the first case, x i − c ij = x i + a ij − 1 will be the money left on her account after she moves from i to j . In the second case, Sarah's account will be frozen, in other words, her balance will be set to 0 forever and with no excuse. In any case, x i ⊗ L a ij expresses the amount of money left on Sarah's account if she goes from i to j . More generally, if Sarah is given 1 unit of money and follows a walk P = (i 1 , . . . , i k ) on D, then the Łukasiewicz weight of P computed as w L (P ) = a i 1 i 2 ⊗ L · · · ⊗ L a i k−1i k will show how much money will remain on her bank account. Computing matrix powers over Łukasiewicz semiring, it can be seen that the entry (A ⊗ L t ) ij shows Sarah's funds at j if she chooses an optimal walk from i to j . Computing the left orbit of a vector, (x ⊗ L A ⊗ L t ) i also shows Sarah's funds if 1) for each , x is the amount of money given to her if she enters the network in state , 2) she chooses an optimal starting node and an optimal walk from that node to i, where the Łukasiewicz weight of a walk P = (i 1 , . . . , i k ) is now computed as In the context of aggressive network, we can pose the Łukasiewicz spectral problem x ⊗ L A = λ ⊗ L x if we want to control the dynamics of Sarah's funds, for instance, to know precisely when the game will be over.
, it is also natural to take a partition (K, L) of {1, . . . , n}, that is, the subsets K, L ⊆ {1, . . . , n} such that K ∪ L = {1, . . . , n} and K ∩ L = ∅, and impose that x i ≤ 1 − λ for i ∈ L and x i ≥ 1 − λ for i ∈ K. The Łukasiewicz eigenvectors satisfying these conditions are called (K, L)-Łukasiewicz eigenvectors. As we shall see, when K and L are proper subsets of {1, . . . , n}, the existence of (K, L) eigenvectors is equivalent to (K, L) being a "secure partition" of the network where we subtract λ from each edge, see Definition 3.2. This establishes a connection between the Łukasiewicz spectral problem and the combinatorics of weighted digraphs. The network sense of (K, L)-eigenvectors is also clear if we require the strict inequalities x i < 1 − λ for i ∈ L and x i > 1 − λ for i ∈ K: in this case we know how much Sarah should be given in each state, in order that the game will be over after one step if Sarah starts in any node of L, and in order that she has a chance to live longer if she starts in a node of K and chooses an optimal trajectory. For convenience, in the paper we will consider right Łukasiewicz eigenvectors and orbits.   1 represents an example of aggressive network. The weights of edges on the left denote Sarah's debts, or payments if she is able to make them, and the weights on the right stand for her balance after moving along the corresponding edge.
The rest of the paper is organized as follows. Section 2 is occupied with necessary preliminaries on tropical convexity and tropical linear algebra. Section 3 develops the theory of (K, L)-Łukasiewicz eigenvectors and secure partitions.
Here we give an explicit description of generating sets of (K, L)-Łukasiewicz eigenspaces, establish the connection with the problem of finding secure partitions of weighted digraphs, and provide an algorithm for enumeration of all possible secure partitions. In Section 4 we examine the powers and orbits of matrices in Łukasiewicz semiring, whose theory is closely related to its well-known tropical (max-plus) counterpart.

Basics of the tropical linear algebra
The main idea of this paper is that the max-Łukasiewicz linear algebra is closely related to the tropical (max-plus) linear algebra. A key observation relating the Łukasiewicz linear algebra with the tropical linear algebra is that, given a matrix A ∈ [0, 1] m×n and a vector x ∈ [0, 1] n , we have where A (1) denotes the matrix with entries a ij − 1, and 0 is the vector with m entries all equal to 0. Note that the entries of A (1) belong to [−1, 0], while the entries of x are required to be in [0, 1] n . More generally, we denote by A (α) the matrix with entries a ij − α. Sometimes we will also use the min-plus version of the tropical linear algebra, in particular, the min-plus addition a ∧ b := min(a, b) and the min-plus matrix product (A ⊗ B) ik = j a ij + b jk . Denoting by A the matrix with entries (−a ji ), known as Cuninghame-Green inverse, one obtains the following duality law: In the sequel we usually omit the ⊗ sign for tropical matrix multiplication, unlike the ⊗ L sign for the max-Łukasiewicz multiplication. We are only interested in matrices and vectors with real entries, that is, with no −∞ entries.
We note that we only need the special case of tropical (max-plus) semiring here. See, e.g., Gondran and Minoux [26,27] for more general idempotent algebras (dioïds).

Elements of tropical convexity
A set C ∈ (R ∪ {−∞}) n is called tropically convex if together with any two points x, y ∈ C it contains the whole tropical segment Note that The tropical convex hull of a set X ⊆ (R ∪ {−∞}) n is defined as tconv(X) := μ λ μ x μ : where only a finite number of λ μ are not −∞. In this case, the tropical Carathéodory theorem states that in (4), we can restrict without loss of generality to tropical convex combinations of no more than n + 1 points x μ . As in the case of the usual convexity, there exists an internal description of tropical convex sets in terms of extremal points and recessive rays, and external description as intersection of (tropical) halfspaces. See [3,13,17,24] for some of the recent references. Here we will be mostly interested in the internal description of a compact tropical convex set C based on extreme points: if represented as a point in a tropical segment of C, such a point should coincide with one of its ends.
It follows that the linear equations over max-Łukasiewicz semiring are affine equations over max-plus semiring, with the solutions confined in the hypercube [0, 1] n . The solution sets to systems of such equations are compact and tropically convex. Moreover, they are tropical polyhedra, i.e. tropical convex hulls of a finite number of points.
A tropical analogue of a theorem of Minkowski was proved in full generality by Gaubert and Katz [24], see also Butkovič et al. [7] for a part of this result. Here we are interested in the particular case of compact tropically convex sets in R n . Theorem 2.1. Let C ⊆ R n be a compact tropical convex set. Further let u (μ) be a (possibly infinite) set of its extreme points. Then, where in (5), only a finite number of λ μ are not equal to −∞. In words, any compact tropically convex subset of R n is generated by its extreme points.

Cyclicity theorem
Starting from this subsection, all matrices have only finite entries, no −∞. We now consider the sequence of max-plus matrix powers A k = k A ⊗ · · · ⊗ A, for A ∈ R n×n . The max-algebraic cyclicity theorem states that if the maximum cycle mean of A ∈ R n×n (to be defined later) equals 0, then the sequence of max-plus matrix powers A k becomes periodic after some finite transient time T (A), and that the ultimate period of A k is equal to the cyclicity of the critical graph. Cohen et al. [12] seem to be the first to discover this, see also [2,4,5,14,30]. Generalizations to reducible case, computational complexity issues and important special cases of the cyclicity theorem have been extensively studied in [5,16,22,23,35,38].
Below we need this theorem in the form of CSR-representations as in [37,39]. To formulate it precisely we need the following concepts and notation. The notions of the associated digraph, the maximum cycle mean, the critical graph and the Kleene star are of general importance in the tropical linear algebra.
For a matrix A ∈ R n×n , define the associated weighted digraph . . , n}, E = N × N and each edge (i, j) ∈ E has weight a ij . Conversely, each weighted digraph with real entries on n nodes corresponds to an n × n real matrix.
Let ρ(A) denote the maximum cycle mean of A, i.e., The cycles (i 1 , . . . , i k ) where the maximum cycle mean is attained are called critical. Further, the critical graph, denoted by C(A), consists of all nodes and edges belonging to critical cycles. As it will be emphasized later, ρ(A) also plays the role of the (unique) tropical eigenvalue of A. The sum of formal series is called the Kleene star of A ∈ R n×n . Here I denotes the tropical identity matrix, i.e., the matrix with diagonal entries equal to 0 and the off-diagonal entries equal to −∞. Series (7) converges if and only if ρ(A) ≤ 0, in which case

The Kleene star satisfies
Defining the additive weight of a walk P on D(A) as sum of the weights of all edges contained in the walk, observe that the following optimal walk interpretation of tropical matrix powers A k and Kleene star A * : 1) for each pair i, j , the (i, j) entry of A k is equal to the greatest additive weight of a walk connecting i to j with length k, 2) for each pair i, j with i = j , the (i, j) entry of A * is equal to the greatest additive weight of a walk connecting i to j (with no length restriction). 3 We introduce the notation related to CSR-representation. Let C ∈ R n×c and R ∈ R c×n be matrices extracted from the critical columns (resp. rows) of ((A − ρ(A)) γ ) * , where γ is the cyclicity of C(A). To calculate γ by definition, one needs to take the g.c.d. of the lengths of all simple cycles in each strongly connected component of C(A), and then to take the l.c.m. of these. Without loss of generality we are assuming that C(A) occupies the first c nodes of the associated graph.
Let S be defined by By A ·i and A i· we denote the ith column, respectively the ith row of A.

Theorem 2.2. For any matrix A ∈ R n×n there exists a number T (A) such that for all t ≥ T (A)
Moreover, This CSR form of the Cyclicity Theorem [12] was obtained in [37], see also [39]. [2,4,5,14,26,30] for general references.

Theorem 2.3. Let A ∈ R n×n , then the maximum cycle mean ρ(A) is the unique tropical eigenvalue of A.
The set of all eigenvectors of A with eigenvalue ρ(A) is denoted by V (A, ρ) and called the eigencone of A. It can be described in terms of the critical graph, by the following procedure. The critical graph of A (ρ) , the matrix with entries a ij − ρ(A), consists of several strongly connected components, isolated from each other. In each of the components, one selects an arbitrary index i and picks the column (A (ρ) ) * ·i of the Kleene star. It can be shown that for any other choice of an index i in the same strongly connected component, the column will be "proportional", i.e.
In what follows, by Ñ C (A) we denote an index set containing exactly one index from each strongly connected component of C(A). The following (standard) description of V (A, ρ) is standard, see Krivulin [32,33].
The tropical spectral theory can be further applied to the equation which has been studied, e.g., in [4,10,34]. It is called Bellman equation due to its relations with dynamic optimization on graphs and in particular, the Bellman optimality principle [34]. Its nonnegative analogue is known as Z-matrix equation, see [8].
We will make use of the following basic result, formulated only recently in [8,32,33]. The proof is given for the reader's convenience. We consider only the case when A and b have real entries, since this is the only case that we will encounter. The solution set of (12) will be denoted by S (A, b).
Proof. First it can be verified that any vector like on the r.h.s. of (13) satisfies x = Ax ⊕ b, using that A * = AA * ⊕ I (8).
for all k ≥ n.
This implies x ≥ A * b. In particular, a solution of (12) exists if and only if A * converges, that is, if and only if If ρ(A) < 0, then by the cyclicity theorem, vectors of {A k x} k≥1 start to fall with the constant rate ρ(A), starting from some k. This shows that for large enough k, A k x ≤ A * b and A * b is the only solution.
If ρ(A) = 0, then the orbit {A k x} k≥1 starts to cycle from some k. But as Ax ≤ x, we have x ≥ Ax ≥ . . . ≥ A k x ≥ . . . , and it is only possible that the sequence {A k x} k≥1 stabilizes starting from some k. That is, starting from some k, We will rather need the following formulation of the above result, implied by Theorem 2.4.

Max-Łukasiewicz eigenproblem
The problem is to find λ such that there exist nonzero x solving A ⊗ L x = λ ⊗ L x. Using (1) we convert this problem to the following one: Before developing any general theory, let us look at some two-dimensional examples. Take Then Take λ < 0.5. Then the terms with λ can be cancelled, and we obtain the system of inequalities which has solution set {x: This is the Łukasiewicz eigenspace 4 associated with any λ < 0.5. If λ > 0.5, then the diagonal terms on the l.h.s. of (18) can be cancelled and we obtain the system The solutions to equations of (21) can be written as and, respectively, Intersecting these sets we obtain the Łukasiewicz eigenspace If λ = 0.5 then we solve which is equivalent to The solution set is Łukasiewicz eigenspaces (20), (24) and (27)  {x: In fact, it is the solution set for the second equation, whereas solution sets for the first equation in cases λ < 0.2, λ = 0.2 and 0.2 < λ < 0.4 are: {x: respectively. All these sets contain the solution set (29), hence the intersection of the solutions for both equations is described by (29). For λ = 0.4 we obtain the solution sets for the first and the second equation and their intersection is the solution set For λ > 0.4 we obtain the solution sets for the equations Their intersection (the solution set) for 0.4 < λ < 0.7 has the form while for λ ≥ 0.7 the solution set is simply All described eigenspaces (29), (31), (33) and (34) for λ = 0.35, λ = 0.4, λ = 0.55 and λ = 0.85 are displayed on Fig. 3. We now going to describe certain parts of the Łukasiewicz eigenspace, determined by the pattern of maxima on the r.h.s. of (16), whose generating sets (containing no more than n + 1 points) can be written explicitly and computed in polynomial time.

Background eigenvectors
These are the eigenvectors that satisfy x i ≤ 1 − λ for all i. In this case (16) becomes A (1) x ≤ 0, and using (2) we obtain that the solutions are given by If λ = 1 then there are no nonzero background eigenvectors. If λ = 0 then all Łukasiewicz eigenvectors are background. If λ < 1 then nonzero background eigenvectors exist if and only if there is j such that max i a ij < 1. In other words, background eigenvectors do not exist if and only if in each column of A there is an entry equal to 1. It can be seen that in this case ρ(A) = 1.
In particular, we obtain the following.

Pure eigenvectors
These are the vectors that satisfy x i ≥ 1 − λ for all i. In this case (16) transforms to Evidently, the set of vectors satisfying (36) is tropically convex. We obtain the following description of the pure eigenvectors.
Proof. As any pure eigenvector is a solution to (36) we can use Corollary 2.1. We obtain that x ∈ R n is a pure eigenvector if and only if it can be represented as in (37) with z(x) satisfying the second constraint in (38), and has all coordinates between 1 − λ and 1. From (37) holds for all i. Substituting (37) in x ≤ 1 and using the duality law (2) we obtain the condition max((A (λ) ) * 0) ≤ λ and the first constraint in (38). 2 Definition 3.1. For λ > 0, a weighted graph D is called λ-secure if for each node of D and each walk P issuing from that node, −λ + w(P ) ≤ 0. is λ-secure, by the walk interpretation (A (λ) ) * . Conversely, the λ-security is impossible if D(A (λ) ) has a cycle with positive weight, and the condition max((A (λ) ) * ) ≤ λ follows from the optimal walk interpretation of (A (λ) ) * . 2 We now describe the generating set of pure Łukasiewicz eigenvectors.
Proof. By Theorem 3.1, u and v (k) are pure Łukasiewicz eigenvectors associated with λ, and so is any tropically convex combination of them. Further, all z(x) satisfying (38) are tropical convex combinations of z(u) and z(v (k) ).
Using max-linearity of (37), we express any pure Łukasiewicz eigenvector as a tropical convex combination of u and v (k) . 2 This result also leads to a description of all Łukasiewicz eigenvectors when λ = 1. Note that in this case it can be shown that 0 and v (i) for i = 1, . . . , l, are the extreme points of the set of Łukasiewicz eigenvectors.
We obtain the following description of (K, L)-eigenvectors, which uses the concept of tropical Schur complement, introduced by Akian, Bapat and Gaubert [1, Definition 2.13]: Observe that Schur (1) where z K (x) is any vector with components in K satisfying and x K , resp. x L are restrictions of x to the index set K, resp. to L.

Proof. By definition, (K, L) Łukasiewicz eigenvectors satisfy the following system
We start by writing out the solution of (43). Subtracting (λ − 1) from both sides of this equation we obtain Eq. (45) has solutions if and only if ρ(A KK ) ≤ λ. Let z K be any K-vector (that is, a vector with all components in K) such that Then vector x satisfies (45) if and only if it satisfies Observe that if ρ(A KK ) < λ then λ is not the (unique) eigenvalue of A KK , and N C (A KK , λ) is empty so that z K = −∞.

Eq. (44) can be rewritten as inequality
and substituting x K from (47) we get Expression in the brackets equals Schur (1) (K, λ, A). The inequality (49) is equivalent to the following three inequalities: We also must require that In particular, this implies the existence of a nontrivial (K, L) eigenvector associated with λ < 1, provided that ρ(A (λ) KK ) ≤ 0 and A (λ) LK (A (λ) KK ) * 0 K ≤ 0 L . If x K and x L satisfy (47) and (50) then in particular we have (48), and hence x K ≤ (A (1) is satisfied automatically, and the claim of the theorem follows. 2 We now give a graph-theoretic interpretation for the existence conditions in Theorem 3.2. and λ = 0.4. This is the associated matrix of the graph on Fig. 1 (right). Consider secure partitions of the graph associated with A (λ) . One of them has L = {1, 5}, K = {2, 3, 4} and is shown on Fig. 4. In the graph associated with A (λ) , it is secure to travel from any node that belongs to L to any node belonging to K and to continue a walk in K without having any deficit of the budget. On the other hand, the partition L = {5, 4}, K = {1, 2, 3} is not secure, because there is a walk from index 4 ∈ L to index 1 ∈ K with positive weight (the walk p = (4, 2, 1) with w(p) = 0.1). As a corollary of Theorem 3.2, we can describe the set of (K, L)-eigenvectors as a tropical convex hull of no more than n + 1 explicitly defined points. Both statement and proof are analogous to those of Corollary 3.2.
Corollary 3.4. The set of (K, L) Łukasiewicz eigenvectors is the tropical convex hull of vectors u, v ( ) for ∈ L, and w (k) for k ∈ K such that k ∈ N C (A KK , λ), defined by (41) and the following settings:

Finding all secure partitions
The problem of describing all Łukasiewicz eigenvectors associated with λ can be associated with the following combinatorial problem: 1) check whether graph D(A (λ) ) is λ-secure, 2) describe all (K, L) partitions which are secure in D (A (λ) ).
Node i of a weighted digraph is called secure if the weight of every walk starting in i is nonpositive.
1. There is a node k ∈ L 2 \L 1 = K 1 \K 2 , secure in D(A (λ) Proof. 1. Consider the set M := K 1 \K 2 and assume that no node of M is secure in K 1 . Then for each node i ∈ M there exists a walk P in D(A (λ) K 1 K 1 ) starting in i, whose weight is positive, and such that the weights of all its proper subwalks starting in i are non-positive. Let be the end node of the walk, and let k be the last node of M visited by the walk. If k = then the subwalk of P starting in k and ending in has a positive weight, which contradicts with the security of (K 2 , L 2 ). We conclude that for all walks in D(A (λ) K 1 K 1 ) starting in M and having positive weight, the end node also belongs to M. However, using such walks and the finiteness of M we obtain that D(A (λ) K 1 K 1 ) has a cycle with positive weight, contradicting the security of (K 1 , L 1 ). Hence M contains a node that is secure in K 1 .
2. The "if" part: Observe that all walks starting in L 1 and continuing in K 1 − k have nonpositive weight since the partition (K 1 , L 1 ) is secure. All walks starting in k and continuing in K 1 − k have nonpositive weight since k is secure in D(A (λ) The "only if" part: If k is not secure, then there is a walk in D(A (λ) K 1 K 1 ) going out of k and having a positive weight. If this walk ends in k then we obtain a cycle with positive weight, in contradiction with the security of (K 1 , L 1 ). Otherwise, take a subwalk which contains k only as the starting node, and then its weight must be also positive. This shows that (K 1 − k, L 1 + k) is not secure. 2 Let us emphasize that the above theorem also holds in the case of K 1 = [n] when we require that D(A (λ) ) is λ-secure, since this condition (by Theorem 3.1 and Corollary 3.2) implies that ρ(A (λ) ) ≤ 0. The case when L 2 = [n], in which case (K 2 , L 2 ) is secure by definition, yields the following corollary. The equivalence of security of partition (K, L) with existence of (K, L)-Łukasiewicz eigenvector implies that there is a unique minimal secure partition. D(A (λ) ) has the least secure partition, which corresponds to the greatest Łukasiewicz eigenvector associated with λ.

Proposition 3.3. Digraph
Proof. First observe that there is the set of Łukasiewicz eigenvectors contains the greatest element: this is a tropical convex hull of a finite number of extremal points, and the greatest element is precisely the "sum" (⊕) of all extremal points.
Let z be this greatest element, and let (K, L) be the partition such that z i < 1 − λ for all i ∈ L and z i ≥ 1 − λ for all i ∈ K. This partition is secure. For any other secure partition (K , L ) there exists a (K , L )-eigenvector y (associated with λ), and we have y ≤ z and hence y i < 1 − λ for all i ∈ L implying L ⊆ L . 2 The vector x whose components are solutions of (51) is the greatest Łukasiewicz eigenvector associated with λ. The partition (K, L) such that x i < 1 − λ for all i ∈ L and x k ≥ 1 − λ for k ∈ K is secure partition with the least L. Problem (51) is an instance of the tropical linear programming problems treated by Butkovic and Aminu [5,6] and Gaubert et al. [25]. It can be solved in pseudopolynomial time [5,6], by means of a psudopolynomial number of calls to a mean-payoff game oracle [25].
The Łukasiewicz eigenproblem with λ = 0.6 amounts to solving A (1) According to the algorithm, we first need to find the greatest max-Łukasiewicz eigenvector. It can be verified that (1 0.8 0.7) is a pure Łukasiewicz eigenvector (note that all components are greater than 0.4). As there is a pure eigenvector, it follows that the greatest max-Łukasiewicz eigenvector is also pure, and the minimal partition is the least partition K = {1, 2, 3}, L = {∅}. In fact, (1 0.8 0.7) is the greatest solution of the tropical Z-matrix equation x = A (0. 6) x ⊕ (0.4 + 0) satisfying x i ≤ 1, hence this is exactly the greatest (pure) max-Łukasiewicz eigenvector. Moreover, it can be found that the pure eigenspace is the max-plus segment with extremal points (1 0.8 0.7) and (0.6 0.5 0.4).
The second part of the algorithm describes how to find all secure partitions. The algorithm starts with the secure partition with minimal L, according to (3.5), and tries to increase L by adding indices from K. In the beginning we have A

Powers and orbits
As we are going to show, any vector orbit in max-Łukasiewicz algebra is ultimately periodic. This ultimate periodicity can be described in terms of CSR representation of periodic powers (A (1) ) t . We assume the critical graph of A occupies the first c nodes. Note that the CSR terms computed for A t and (A (1) ) t are the same.
Observe that if ρ(A) = 1 then the critical graph C(A (1) ) consists of all cycles, such that all entries of these cycles have zero weight. Using Theorem 2.5 we obtain the following.
Proof. We start by showing the following identity: Indeed, iterating (1) we obtain For the final reduction we used that all entries of A (1) are nonpositive, hence (A (1) ) l ⊗ 0 ≤ 0 for all l ≥ 0. Now we substitute the result of Theorem 2.2 observing that ρ t (A)CS t Rx ≤ 0 for all large enough t if ρ(A) < 1.
If ρ(A) = 1 then for each t and i = 1, . . . , c, the ith row of (A (1) ) t has a zero entry, from which it follows that ((A (1) ) t ⊗ 0) i = 0 for all large enough t and i = 1, . . . , c implying that ((A (1) ) t ⊗ x ⊕ 0) i equals (A (1) ) t (x ⊕ 0) i and hence S t R(x ⊕ 0) i for all big enough t . 2 It also follows that where 0 is the n × n matrix consisting of all zeros. Consequently, we obtain that if ρ(A) < 1 then A ⊗ L t = 0 for sufficiently large t , and if not then For any n × n matrix A in max-Łukasiewicz algebra, we can define the "matrix of ones" A [1] [1] , and that (A ⊗ L t ) [1] = (A [1] ) ⊗ L t . Also, the Łukasiewicz product of two {0, 1} matrices coincides with their product in the Boolean algebra. This implies that if ρ(A) = 1 (i.e., the "graph of ones" has nontrivial strongly connected components), then (A ⊗ L t ) [1] is nonzero for any t, and so is A ⊗ L t . As the period of (A ⊗ L t ) [1] = (A [1] ) ⊗ L t is γ , the period of A ⊗ L t cannot be less. Eq. (57), expressing the ultimate powers of A in the CSR form, assures that it cannot be more than γ . 2 since CS t R0 ≤ 0 (observe that none of the entries of A (1) and hence any of its powers exceed zero). This shows that the last (n − c) components of A ⊗ L t ⊗ L x are tropical affine combinations of the first (critical) coordinates, and hence we only need to determine the period of S t R(x ⊕ 0), which is the ultimate period of the first c coordinates of (A (1) ) t (x ⊕ 0). The latter period can be computed in O(n 3 log n) time by means of an algorithm described in [37], see also [5,29]. 2 Butkovič [5] and Sergeev [38] studied the so-called attraction cones in tropical algebra, that is, sets of vectors x such that the orbit A t x hits an eigenvector of A at some t .
In the case of max-Łukasiewicz algebra, attraction sets can be defined similarly. Then either ρ(A) < 1 and then all orbits converge to 0, or ρ(A) = 1 and then there may be a non-trivial periodic regime, in which the non-critical components of A ⊗ L t ⊗ L x are tropical affine combinations of the critical ones. It follows that (like in max algebra) the convergence of A ⊗ L t ⊗ L x to an eigenvector with eigenvalue 1 is determined by critical components only. Since these are given by S t R(x ⊕ 0), the attraction sets in max-Łukasiewicz algebra are solution sets to This system of equations can be analysed as in [38], a task that we postpone to a future work. We see that the sequence {A ⊗ L t } t≥1 is ultimately periodic with period 2, and the periodicity starts at t = 3. The term (A (1) ) t−1 A = CS t−1 RA of (56) dominates in all the columns and rows of A ⊗ L t with critical indices (1,2). In the periodic regime (t ≥ 3), the zero term dominates in almost all entries of the non-critical A ⊗ L t MM where M = {3, 4}.