On the length of shortest 2-collapsing words

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1 Introduction and preliminaries.
In this paper by an automaton A = Q, Σ, δ we mean a finite deterministic automaton with state set Q, input alphabet Σ, and transition function δ : Q × Σ → Q.The action of Σ on Q given by δ will be denoted by concatenation: qa = δ(q, a).This action extends naturally, by composition, to the action of the words of Σ * on Q.Given a word w ∈ Σ * , we will be interested in the difference of the cardinalities |Q| − |Qw|.This difference is called the deficiency of the word w with respect to A and will be denoted df A (w).
For a fixed k ≥ 1, a word w ∈ Σ * is called k-compressing for A if df A (w) ≥ k.An automaton A is k-compressible if there exists a k-compressing word for A. A word w ∈ Σ * is k-collapsing (over Σ) if it is k-compressing for every k-compressible automaton with the input alphabet Σ.A word v is called k-synchronizing if it is k-compressing for all k-compressible automata with k + 1 states.Of course a k-collapsing word is also k-synchronizing.
The concept of a k-collapsing word is a natural automata-theoretic interpretation of the notion of words with the property ∆ k introduced for algebraic motivations in [12].Actually k-collapsing words can be seen as universal testers for checking whether an automaton is k-compressible.They are a blackbox versions of words considered in the so called generalized Černý conjecture that extends the celebrated Černý conjecture [5].Recall that the former conjecture deals with the length of the shortest k-compressing word for a given k-compressible automaton.In [12] it was proved that k-collapsing words always exist, for each Σ and each k ≥ 1, by means of a recursive construction which gives a word whose length is doubly exponential as function of k.Then better lower and upper bounds for the length c(k, t) of the shortest k-collapsing words on an alphabet of cardinality t were given in [9] and the lower bound for the case k = 2 was slightly improved in [10].So far for the case k = 2 the best known bounds are the following: 2t 2 ≤ c(2, t) ≤ t 3 + 3t 2 + 2t.
In [1,2], two different algorithms for deciding whether a word is 2-collapsing are given and another more combinatorial approach to the same problem is presented in [8].Here we get a better upper bound for c(2, t), namely c(2, t) ≤ t 3 +6t 2 +5t 2 , by means of the latter approach.Moreover we prove that this bound can be improved at least for small values of t: we build, again by using this approach, shorter 2-collapsing words on alphabets of cardinality 4 and 5.The reader is referred to [9] and to [4] for some further references and connections to some topics in Theoretical Computer Science, Language Theory and Combinatorics.Here for sake of completeness we shortly recall notation and main properties.
We view an automaton A = Q, Σ, δ as a set of transformations on Q labelled by letters of Σ rather than as a standard triple.By transformations of A we mean those transformations on Q that are induced via δ by letters of Σ.In order to define an automaton it is enough just to assign to every letter of Σ a transformation on Q.Now, for a ∈ Σ, df A (a) = 0 if and only if the corresponding transformation is a permutation on Q.If df A (a) = 1, then there is a state z ∈ Q which does not belong to the image Qa and two different states x, y ∈ Q satisfying xa = ya; in such a case the corresponding transformation will be referred to as a transformation of type {x, y}\z (x, y have the same image under the transformation, z is missed).The semigroup generated by the transformations of A consists precisely of the transformations corresponding to words in Σ * .It contains a group generated by those transformations that are permutations; this group is called the group of permutations of A.
It is well known ( [9]) that each 2-collapsing word over a fixed alphabet Σ is 2-full, i.e. it contains any word of length 2 on Σ among its factors, hence to characterize 2-collapsing words it is enough to consider 2-full words compressing each 2-compressible automata such that no word of length 2 is 2-compressing for them (proper 2-compressible automata, for short).Proper 2-compressible automata must have both non-permutation and permutation transformations and they are characterized by the following: 3. there are two states x, y such that each non-permutation transformation is of type {x, y}\x or {x, y}\y, both types occur, and the group of permutations does not fix the set {x, y}.
Automata of the first two types are called MONO-automata, of the latter one are called STEREOautomata.
In [8] it is proved that a word w ∈ Σ * is 2-collapsing if and only if certain systems of conditions on permutations have only trivial solutions.We recall some notations from [6].In the sequel, as no confusion can raise, we view the set Q of the states of A as a set of natural numbers: Q = {1, 2, ..., n}, so that permutations on Q can be viewed as elements of the symmetric group Sym n .The two states x, y occurring in Proposition 1 will be then denoted by 1 and 2. By a role assignment we mean an arbitrary partition of the alphabet Σ with a distinguished nonempty block Π ⊂ Σ. Roughly speaking letters in Π play the role of permutation letters of Σ and the remaining letters play the role of non-permutation letters.Let Υ = Σ \ Π.Since non-permutation letters in proper 2-compressible automata can be of different types, a role assignment induces on Υ a further partition in non empty blocks.We denote this partition either as {D 2 , . . ., D h }, h ≥ 2, where letters in D i will play the role of the non-permutation letters of the form {1, i}\1 or as {E 1 , E 2 } where letters in E 1 , E 2 will play the role of non-permutation letters of the form {1, 2}\1 and {1, 2}\2 respectively.In the first case the partition of Σ will be called a MONO-role assignment (DB-partition in [8]) and in the latter a STEREO-role assignment (3DB-partition in [8]).
Let (Π, Υ) be an arbitrary role assignment, then each word w ∈ Σ + can be uniquely represented in the following form: where p 1 , . . ., p m ∈ Π + , u 1 , . . ., u m−1 ∈ Υ + , u 0 , u m ∈ Υ * and m is a positive integer.We say that a factor p i of the decomposition ( 1) is an inner factor if both u i−1 and u i are non-empty.Then for a MONO-role assignment (Π, Υ) for each a ∈ Υ we denote by S a the set of the inner factors p i of w such that a is the first letter in u i .Otherwise if (Π, Υ) is a STEREO-role assignment (Π, {E 1 , E 2 }) we denote by S k , k = 1, 2, the sets of the inner factors p i of w such that the last letter of u i−1 belongs to E k .We assign to the word w ∈ Σ + , for each role assignment and for each inner factor p i , a permutation condition of the form: -1p i ∈ {1, j} if the role assignment is MONO and p i ∈ S a with a ∈ D j , -kp i ∈ {1, 2}, k = 1, 2, if the role assignment is STEREO and p i ∈ S k .Then each role assignment associates to w a system of permutation conditions formed by all the permutation conditions corresponding to the inner factors of w.In case of a MONO-role assignment (Π, Υ) such system will be denoted Γ w (Π, Υ) otherwise it will be denoted Γ ′ w (Π, Υ).Since letters of Π are regarded as permutations acting on a finite set of positive integers {1, 2, . . ., n} (representing Q), conditions of the form jv ∈ A with j positive integer, v ∈ Π + , A ⊆ {1, 2, . . ., n} mean that the image of j under the product v of permutations belongs to the set A. We say that a system of permutation conditions has a solution if there exists an assignment of permutations on a finite set {1, 2, . . ., n} to letters in Π such that all the conditions in the system are satisfied.Obviously the systems associated to w by each role assignment have always the solution where all permutations are the identity.A solution of the system Γ w (Π, Υ), associated to w by a MONO-role assignment, is called trivial if all permutations fix 1.Also, in the special case when Υ consists of the unique block D 2 (and as a consequence, all j's on the right hand side of the conditions are equal to 2), a solution with all permutations fixing the set {1, 2} is considered trivial.The remaining solutions are nontrivial.Similarly a solution of the system Γ ′ w (Π, Υ), associated to w by a STEREO-role assignment (Π, Υ) is called trivial if all the permutations fix {1, 2}.
In [8] 2-collapsing words were characterized in terms of permutation conditions in the following way.

Theorem 2 ([8]
) A word w ∈ Σ + is 2-collapsing if and only if it is 2-full and the following conditions hold: 1.For each MONO-role assignment (Π, Υ) of Σ all the solutions of the system Γ w (Π, Υ) are trivial.
By the above Theorem 2 one can readily derive the following characterization of 2-synchronizing words.
Corollary 3 A word w ∈ Σ + is 2-synchronizing if and only if it is 2-full and for each role assignment (Π, Υ) of Σ all the solutions of the systems Γ w (Π, Υ) (in case of MONO-role assignments) and Γ ′ w (Π, Υ) (in case of STEREO-role assignments) in the semigroup T 3 of all transformations on 3 letters are trivial.

Main result.
We introduce some useful notations.Definition 1 .A permutation p on {1, 2, ..., n} is special with respect to the ordered pair (i, j), i, j ∈ {1, 2, ..., n}, whether it fixes either {i} or {i, j}.Two permutations p, z are special of the same type with respect to the ordered pair (i, j) if both p and z fix {i} or both of them fix {i, j}.
Note that the property of being special of the same type with respect to (i, j) is not transitive; e.g. both the permutations p = (1)(2...n) and z = (12)(3...n) are special of the same type as y = (1)(2)(3...n) with respect to (1, 2), but they are not of the same type.Note however that two special permutations p and z can be of the same type of y with respect to (i, j) without being of the same type only if one of them fixes {i} and does not fix j, the other has a cycle (ij) while y fixes both {i} and {j}.Trivial solutions of systems introduced before Theorem 2 are special of the same type with respect to (1, 2), so we are interested in sets of equations whose occurrence in a system guarantees that solutions are special of the same type with respect to (1, 2).Lemma 4 Let p, z be two permutations on {1, 2, ..., n} and let i, j ∈ {1, 2, ..., n}.The permutations p, z fulfil the conditions: if and only if they are special of the same type with respect to (i, j).

Proof:
The "if part" is trivial.Conversely from condition 1. we immediately get that either p fixes i or it is of the form p = (ij...)... and condition 2. gives that in such case p = (ij).... Similarly conditions 3. and 4. give that z either fixes i or has the form z = (ij).... Now if z = (ij)... and p fixes i then p fixes also j otherwise condition 5. is not fulfilled.If p = (ij)... then jz = ipz ∈ {i, j} by condition 5. whence z fixes {i, j}.✷ Similarly, the following can be proved: Lemma 5 Let p, z be two permutations on {1, 2, ..., n} and let i, j ∈ {1, 2, ..., n}.The permutations p, z fulfil the conditions: if and only if they are special of the same type with respect to (i, j).

Theorem 6
The minimal 2-collapsing word on t letters has length less than or equal to Proof: To show our bound we want to exhibit a 2-collapsing word, similar to the one introduced in [9].Consider an alphabet Σ = {a 1 , a 2 , ..., a t } on t letters in the lexicographic order and let u = a 1 a 2 ...a t and We want to prove that the word w is 2-collapsing.Let us consider different role assignments (Π, Υ) of Σ.
Assume that (Π, Υ) is a STEREO-role assignment with Υ = {E 1 , E 2 }; let h, k be respectively the least and the greatest index of an element in Υ and let a j be an element in Υ belonging to a different block than a k (eventually h = j).Each a i , i > k (if any) fixes {1, 2}, since the factors a k a i a k , a j a i a j occur in w whence both the equations 1a i ∈ {1, 2} and 2a i ∈ {1, 2} occur in Γ ′ w (Π, Υ).More the product a 1 a 2 ...a h−1 fixes {1, 2}.In fact a j (a 1 a 2 ...a h−1 )a h and a k (a 1 a 2 ...a h−1 )a h are factors of w and all a 1 , ..., a h−1 are in Π hence both the equations 1(a 1 a 2 ...a h−1 ) ∈ {1, 2} and 2(a 1 a 2 ...a h−1 ) ∈ {1, 2} occur in Γ ′ w (Π, Υ).This yields that also any a i , i < h (if any) fixes {1, 2}, since also a i a j a i (a 1 a 2 ...a h−1 )a h and a i a k a i (a 1 a 2 ...a h−1 )a h are factors of w whence both the equations 1a i (a 1 a 2 ...a h−1 ) ∈ {1, 2} and 2a i (a 1 a 2 ...a h−1 ) ∈ {1, 2} occur in Γ ′ w (Π, Υ) with a 1 a 2 ...a h−1 fixing {1, 2}.Similarly, at last, each In fact either a h and a k are in different blocks or there is an a j ∈ Υ belonging to a different block with respect to a h and a k , so for each i, h < i < k there are two indices s, r, h ≤ s < i < r ≤ k such that a s and a r are in different blocks of Υ, hence a i a r a i (a 1 a 2 ...a h−1 )a h and a s a i a s are factors of w.
Let k be the greatest index such that a k ∈ Υ then a k ∈ D k ′ for some k ′ .Let D j ′ be a block of Υ different from D k ′ and let a j ∈ D j ′ .We will prove that every a i ∈ Π fixes {1}.In fact if i > k then a k a i a k , a j a i a j are factors of w hence both the equations 1a i ∈ {1, k ′ }, 1a i ∈ {1, j ′ } occur in the system Γ w (Π, Υ), whence a i fixes {1} for all a i ∈ Π with i > k, if any.Now let i < k.If j < i < k then a j a i a j , a k a k+1 ...a t a i a k a i are factors of w, if i < j then a k a k+1 ...a t a i a j a i , a k a k+1 ...a t a i a k a i are factors of w and all elements a k+1 , ..., a t are in Π, hence they fix {1}.Then in the system Γ w (Π, Υ) there are either the equations 1a i ∈ {1, j ′ } and 1a k+1 ...a t a i ∈ {1, k ′ } or 1a k+1 ...a t a i ∈ {1, j ′ } and 1a k+1 ...a t a i ∈ {1, k ′ } from which in both cases we get that 1a i ∈ {1, k ′ } and 1a i ∈ {1, j ′ }, thus all a i ∈ Π fix {1}.
At last suppose that Υ consists of only one block, i.e. g = 2, and let h, k respectively be the least and the greatest index of an element in Υ. Denote v 1 = a 1 ...a h−1 and v 2 = a k+1 ...a t , then a h v 1 a h , a k v 2 a k are factors of w whence 1v 1 , 1v 2 ∈ {1, 2} are equations occurring in Γ w (Π, Υ).
In this case, suppose first that 1v 2 = 1.Since for all for every a i ∈ Σ.If all the letters a i fix 1, then we are done.Otherwise suppose there is a letter a i such that 1a i = 2. Then we get 2v 1 ∈ {1, 2} and since 1v 1 ∈ {1, 2} we obtain that v 1 fixes {1, 2}.Thus from 1a i 2 v 1 ∈ {1, 2} we get 2a i = 1 and a i = (12).... To prove that in this case all permutations are of the same type, note that whenever a factor a k v 2 a i a j a i v 1 a h appears in w we have 1a i a j a i ∈ {1, 2} and then apply Lemma 4.
Suppose now 1v 2 = 2. Then 2a i v 1 , 2a 2 i v 1 ∈ {1, 2} for every a i ∈ Π.Let us prove that v 1 must fix {1, 2}.This is trivial if h = 1 and v 1 is empty, else let {1, 2} = {1v 1 , zv 1 } for some z and prove that z = 2.For all i : 1 ≤ i < h a k v 2 a i a h a i and a i a h a i v 1 a h are factors of w, hence 1v 2 a i ∈ {1, 2} and 1a i v 1 ∈ {1, 2} are equations in Γ w (Π, Υ), so that 2a i ∈ {1, 2} and a i v 1 fixes {1, 2}.If for some i : 1 ≤ i < h: 2a i = 2a 2 i then a i = (2)... and 2v 1 ∈ {1, 2} whence z = 2; else {2a i , 2a 2 i } = {1, z} which immediately yields a i = (2 1 z...)... since 2 = 2a i ∈ {1, 2}.If there exist two such elements a i , a j , 1 ≤ i < j < h then the factor a k v 2 a i a j a i v 1 a h occurs in w so that 1v 2 a i a j a i v 1 = (2a i a j )a i v 1 ∈ {1, 2} is an equation in Γ w (Π, Υ) whence 2a i a j = z ∈ {1, 2} and z = 2; but if only one element a i of this type exists then z = 1a i = 1v 1 ∈ {1, 2}.Thus v 1 fixes {1, 2}, which immediately yields 2a i , 1a i ∈ {1, 2} for every a i ∈ Π, i < k by the occurrence of the factor a k v 2 a i a k a i v 1 a h , and for every a i ∈ Π, i > k by the occurrence of the factors a k a i a k and a k v 2 a i v 1 a h ; hence all elements in Π fix {1, 2} so that they are special of the same type.
Thus w is 2-collapsing because the systems associated to w for each role assignment (Π, Υ) have only trivial solutions and the length of the word w is: ✷ 3 Short 2-collapsing words.
The bound found in Theorem 6 is lower than the one given in [9], but still for small values of n it reveals to be not efficient: indeed for n = 3, 4, 5 it respectively gives |w| ≤ 48, 90, 150.It was proved in [3] that the shortest 2-collapsing words on 3 letters have length 21 and the shortest 2-synchronizing words on 3 letters have length 20, while a 2-collapsing word on 4 letters is known whose length equals 58 ([4]).But we can use our lemmas to get shorter 2-collapsing and 2-synchronizing words on 4 and 5 letters.
We will make a systematic use of the following Lemma 7 Let w be a word on an alphabet Σ with |Σ| ≥ 2, let Σ ′ ⊆ Σ and w ′ ∈ Σ ′+ be a factor of w, which is a 2-collapsing word on Σ ′ .For a role assignment (Π, Υ) of Then for each role assignment (Π, Υ) such that ∅ = Π ′ and ∅ = Υ ′ the following properties hold: w (Π, Υ) all the elements in Π ′ fix the set {1, 2}.In all the above cases, if Π ′ = Π then the system associated to w for (Π, Υ) has only trivial solutions.
, hence all solutions of the latter system are also solutions of the former.Hence if Υ and consequently Υ ′ has only one block, then all elements in Π ′ are special permutations relative to (1, 2) which all fix {1} or all fix {1, 2}, because w ′ is 2-collapsing.Similarly if Υ ′ is formed by at least two blocks then all elements in Π ′ fix {1}.If (Π, Υ) is a STEREO-role assignment of Σ, and Υ ′ has two non trivial blocks, then (Π ′ , Υ ′ ) is a STEREO-role assignment of Σ ′ hence the equations of Γ ′ w ′ (Π ′ , Υ ′ ) occur in Γ ′ w (Π, Υ) and so all elements in Π ′ are special permutations relative to (1, 2) which all fix {1, 2}.The last statement is trivial.✷ It is important to remark, using the same notation and the same arguments of the above Lemma 7, that if (Π, Υ) is a MONO-role assignment with more than one block different from Π, while Υ ′ is formed by a unique block D ′ i , then in each solution of Γ w (Π, Υ) all the elements in Π ′ are special of the same type with respect to {1, i}.Similarly if (Π, Υ) is a STEREO-role assignment of Σ and E ′ 1 = ∅ then in each solution of Γ ′ w (Π, Υ) all the elements in Π ′ are special of the same type with respect to (2, 1) while if E ′ 2 = ∅ then all the elements in Π ′ are special of the same type with respect to (1, 2).Proposition 8 c(2, 4) ≤ 56 Proof: Let Σ = {a, b, c, d} and let Consider the word w = r(bab)su.Of course |w| = 56; we want to prove that w is 2-collapsing.In order to exhibit w we made use of two of the minimal 2-collapsing words on 3 letters.Namely u 1 = r(bab) is the 55-th word in the list in [3] on the alphabet {c, a, b}, while u 2 = (bab)s(ad) is the 11-th one in that list on the alphabet {b, a, d}.Both the words u 1 and u 2 are 2-collapsing, as shown in [3].Put Σ 1 = {a, b, c} and Σ 2 = {a, b, d}.
Let us consider different role assignments (Π, Υ) of Σ and put Let |Π| = 1 and Π ⊂ {a, b} then Π 1 = Π 2 = Π.Moreover if (Π, Υ) induces on Υ a partition in more than one block then there is at least an i = 1, 2 such that Υ i contains two non empty blocks, so the system associated to w for (P, Υ) has only trivial solutions by Lemma 7. Then let Π = {c}.If (Π, Υ) is a MONO-role assignment and Υ consists of only one block, then all the solutions of Γ w (Π, Υ) are trivial by Lemma 7. If Υ consists of more than one block then one has to consider the factors acb, bca of w if the letters a and b belong to different blocks, else consider the factors acd and dca.In both cases, if (Π, Υ) is a MONO-role assignment then the equations 1c ∈ {1, 2}, 1c ∈ {1, 3} occur in Γ w (Π, Υ) hence c fixes {1}.If (Π, Υ) is a STEREO-role assignment then the equations 1c ∈ {1, 2}, 2c ∈ {1, 2} occur in Γ ′ w (Π, Υ), hence c fixes the set {1, 2}.The same argument applies when Π = {d}, considering the factors adb, bda or bdc, cdb.
Let |Π| = 2. Assume Υ = {a, b}.Then for all i = 1, 2, ∅ = Π i = Π and Υ = Υ i .Then if Υ is formed by two blocks then all the solutions of the system associated to w for (Π, Υ) are trivial: by Lemma 7 if (Π, Υ) is a MONO-role assignment then both c, d fix {1}, if (Π, Υ) is a STEREO-role assignment then both c, d fix {1, 2}.Otherwise if Υ is formed by a unique block we only get that c and d are special permutations relative to (1,2).But since w contains the factors acdb, bdca, the equations 1cd ∈ {1, 2}, 1dc ∈ {1, 2} occur in Γ w (Π, {a, b}) hence c, d are special permutations of the same type with respect to (1, 2), i.e. the system Γ w (Π, {a, b}) has only trivial solutions.Then let Υ = {a, b}, whence Π = Π i for some i = 1, 2. Hence if Υ is formed by a unique block then Γ w (Π, Υ) has only trivial solutions by Lemma 7. Otherwise we only get that the solutions of the systems associated to w for (Π, Υ) are either special permutations relative to (1, 2) or fix {2}.Denote by β 1 , β 2 the elements in Π and by α 1 , α 2 the elements in Υ; for every choice of (Π, Υ) where Υ = {a, b} is formed by two blocks, there is an i = 1, 2 such that α h β i α h and either α h β ǫ i β 3−i α h or α h β ǫ i β 3−i α 3−h (where ǫ ∈ {0, 1} and β 0 i is the empty word) are factors of w for all h = 1, 2. Then if (Π, Υ) is a STEREO-role assignment then the equations , and this yields that the solutions of the systems are pairs of special permutations which both fix the set {1, 2}, i.e. the system has only trivial solutions.Otherwise if (Π, Υ) is a MONO-role assignment with three blocks the equations 1β i ∈ {1, 2}, 1β i ∈ {1, 3}, 1β ǫ i β 3−i ∈ {1, 2}, 1β ǫ i β 3−i ∈ {1, 3} occur in Γ w (Π, Υ), and this yields that the solutions of the systems are pair of trivial permutations which both fix {1}, hence again the system has only trivial solutions.
If Υ = {c} then a, b are special permutations of the same type with respect to (1, 2) since they are solutions of Γ u1 ({a, b}, {c}), whence 1a, 1a Consider the words w 1 = r(bab)su ∈ {a, b, c, d} + and w 2 = bs ′ u ′ r(bab) ∈ {a, b, c, e} + .Like in the proof of Proposition 8 the word w 1 is 2-collapsing on the letters {a, b, c, d} and symmetrically the word w 2 is 2-collapsing on the letters {a, b, c, e}: we only changed the order of some parts, but still we involved as factors some of the minimal 2-collapsing words on three letters.The factors in u and u ′ (ca) strictly correspond to the one we considered in the proof of Proposition 8.
More, let v = cedbeacdebdedbedade 2 d 2 ecedce and finally w = s ′ u ′ r(bab)suv.(Note that w contains all the factors of w 2 ).Of course |w| = 119; we want to prove that w is 2-collapsing.Let us consider different role assignments (Π, Υ) of Σ and put If |Π| = 1 arguments like the ones applied in the proof of Proposition 8 can be used to prove that the system associated to w for (Π, Υ) has only trivial solutions.
Let |Π| = 2, 3. Assume that, for some i = 1, 2, Π ⊆ Σ i .If Υ is formed by a unique block, then the system associated to w for (Π, Υ) has only trivial solutions by Lemma 7; the same happens when both Υ i and Υ are decomposed in more than one block.So, for all i such that Π ⊆ Σ i , Υ i is formed by a unique block while Υ is formed by more than one block.It is easy to check that if Π ⊂ {a, b, c} then either Υ 1 or Υ 2 are formed by two blocks.Then assume Π = {a, b, c}, whence Υ = ({d}, {e}).Hence all the solutions of the system associated to w for (Π, Υ) fix {1, 2} by the remark after Lemma 7. Assume that Π is not a subset of {a, b, c} and Π ⊆ Σ 1 , then Π 1 = Π and Υ 2 = Υ = (Υ 1 , {e}).In all the solutions of the system associated to w for (Π, Υ) by Lemma 7 the elements in Π 2 fix {1, 2} if (Π, Υ) is a STEREO-role assignment and fix {1} if it is a MONO-role assignment.Moreover each solution of the system associated with w for (Π, Υ) is formed by permutations which either fix {1, 2} or fix {1} or fix {2} (by remark after Proof: Let Σ = {a, b, c, d} and let The word w = (cd 2 cbcd)r(ab)s(cdadc 2 d) is 2-synchronizing and |w| = 52.In order to exhibit w we used two of the minimal 2-synchronizing words on 3 letters listed in [3].Namely u 1 = r(ab) is the 11-th word in the list in [3], on the alphabet Σ 1 = {c, a, b}, while u 2 = (ab)s is the 15-st one in that list, on the alphabet Σ 2 = {a, b, d}.Both the words u 1 and u 2 are 2-collapsing ( [3]).
Let |Π| = 3.We are looking for solutions of our systems of equations on 3 permutations t, y, z in the symmetric group Sym 3 .If a system has only special solutions with respect to {1, 2} and t, y are of different types while both of them are of the same of z, then without loss of generality t = ( 12 The word w 1 (resp.w 2 ) is 2-synchronizing on the alphabet {a, b, c, d} (resp.{a, b, d, e}), by similar arguments as in the proof of Proposition 10.In fact it is built from two words in the list in [3] for suitable alphabets: r(ab) is the 6-th and (ab)s is the 16-th (resp.(ab)s ′ is the 16-th while (dad)r ′ is the 1-st) and u (resp.u ′ ) gives us the other necessary factors.