An Equivalence Relation on A Set of Words of Finite Length
aa r X i v : . [ m a t h . C O ] A p r AN EQUIVALENCE RELATION ONA SET OF WORDS OF FINITE LENGTH
YOTSANAN MEEMARK AND TASSAWEE THITIPAK
Abstract.
In this work, we study several equivalence relations induced from the par-titions of the sets of words of finite length. We have results on words over finite fieldsextending the work of Bacher (2002, Europ. J. Combinatorics, , 141-147). Cardinali-ties of its equivalence classes and explicit relationships between two words are determined.Moreover, we deal with words of finite length over the ring Z /N Z where N is a positiveinteger. We have arithmetic results parallel to Bacher’s. Introduction
Let k be a finite field and F k denote the set of all finite words with letters in k . F k is afree monoid with identity ε , called the empty word . Consider the special linear group ofdegree two over k , SL ( k ), consisting of 2 × k of determinant one. It hasbeen proved in [B02] Lemma 2.1 that SL ( k ) generated as a monoid by the set of matrices S = − α : α ∈ k . We can view S as k and thus every word w = α . . . α l ∈ F k is corresponding to theproduct − α . . . − α l ∈ SL ( k ) . Mathematics Subject Classification.
Primary: 20G40; Secondary: 05E15.
Key words and phrases.
Equivalence relations; SL ; Words.The research of the first author was supported in part by Grants for Development of New FacultyStaff from Chulalongkorn University, Thailand. This work grows out of the second author’s master thesisat Chulalongkorn university written under the direction of the first author to which the second authorexpresses his gratitude. This gives rise to an onto homomorphism of monoids π : F k → SL ( k ) . We define an equivalence relation ∼ on k r by st ∼ uv ⇔ st = λ uv for some λ ∈ k × . Its equivalence classes are the lines spanned by x , x ∈ k , and the line spanned by ,called the infinite line , with the origin deleted. Then we usually write these classes as x , x ∈ k , and . Thus the set of all equivalence classes, denoted by P ( k ) and calledthe projective -space . The group SL ( k ) acts on P ( k ) by left multiplication. Bacherdefined the subset A of F k by A = w ∈ F k : π ( w ) = . The sets A and C = F k r A divide F k into two disjoint pieces. This partition leads to anequivalence relation on F k .For r ∈ k , we define two disjoint subsets A r and C r of F k by A r = w ∈ F k : π ( w ) = r . and C r = F k r A r . Hence A = A . In Sections 2 and 3, we investigate arithmetic andcombinatorial properties of the equivalence relation on F k induced by the partition A r and C r .Let N be a positive integer. Another route to extend Bacher’s work is to study thespecial linear group over Z /N Z , the ring of integers modulo N . We present this topic inSection 4. Write F N for the set of all finite words with letters in Z /N Z . Consider thespecial linear group of degree two over Z /N Z , SL ( Z /N Z ), consisting of 2 × N EQUIVALENCE RELATION ON A SET OF WORDS OF FINITE LENGTH 3 over Z /N Z of determinant one. Let S ′ = − α : α ∈ Z /N Z . We show that this set generates SL ( Z /N Z ) as a monoid. Our proof is different from[B02] Lemma 2.1. We use the basic fact that every closed subset of a finite group is agroup. This result shows that every element of SL ( Z /N Z ) can be written in at least oneway as a finite word with letters in S ′ .We can also consider S ′ as Z /N Z and hence every word w = α . . . α l ∈ F N is corre-sponding to the product − α . . . − α l ∈ SL ( Z /N Z ) . This yields an onto homomorphism of monoids π : F N → SL ( Z /N Z ) . For Z /N Z , we define an equivalence relation ∼ ′ on ( Z /N Z ) r by st ∼ ′ uv ⇔ st = λ uv for some λ ∈ ( Z /N Z ) × . Here ( Z /N Z ) × denotes the unit group of the ring Z /N Z . The group SL ( Z /N Z ) acts onthe set of equivalence classes by left multiplication. Parallel to Bacher’s, we set¯ A = w ∈ F N : π ( w ) = and ¯ C = F N r ¯ A . We study this partition of F N in the last two sections.The paper is organized as follows. Arithmetic and combinatorial properties implyingthe cardinalities of A r and C r are studied in Section 2. Section 3 gives an algorithm todistinguish the partition A r and C r . Words over Z /N Z and the partition ¯ A and ¯ C arepresented in Section 4. The final section is devoted to ¯ A including unique factorization,predecessors, successors and periodic words, parallel to Bacher’s A . YOTSANAN MEEMARK AND TASSAWEE THITIPAK Cardinalities of A r and C r This section presents the preliminary properties of words in A r and results on thecardinalities of A r and C r .For w ∈ F k with π ( w ) = a bc d ∈ SL ( k ), we note that w ∈ A r ⇔ r = a bc d = bd ⇔ d = br ⇔ π ( w ) = a bar − b − br with a ∈ k, b ∈ k × . Therefore we have shown
Theorem 2.1.
For r ∈ k , A r = w ∈ F k : π ( w ) = a bar − b − br for some a ∈ k, b ∈ k × . The set A has been studied by Bacher in [B02]. Our results are for the case r = 0. For l ≥
0, we write F lk for the set of words over k of length l , A lr = F lk ∩ A r and C lr = F lk ∩ C r .Unless specify, we assume r ∈ k × throughout this section. We begin with the rightinsertion. Theorem 2.2.
Let w ∈ F k . Then w ∈ A lr if and only if wα ∈ C l +1 r for all α ∈ k .Moreover, if w ∈ C lr , then there exists a unique α ∈ k such that wα ∈ A l +1 r .Proof. Assume that w ∈ A lr and let α ∈ k . Then π ( w ) = a bar − b − br for some a ∈ k and b ∈ k × . Thus π ( wα ) = π ( w ) π ( α ) = a bar − b − br − α = − b a + αb − br ar − b − + αbr . If ar − b − + αbr = ( a + αb ) r , then − b − = 0, a contradiction. Thus wα ∈ C l +1 r . Conversely,suppose that w ∈ C lr . Then π ( w ) = a bc d ∈ SL ( k ) and d = br . Note that for α ∈ k , N EQUIVALENCE RELATION ON A SET OF WORDS OF FINITE LENGTH 5 we have π ( wα ) = π ( w ) π ( α ) = a bc d − α = − b a + αb − d c + αd . Since d = br , we can choose a unique α , namely α = ( ar − c )( d − br ) − ∈ k such that π ( wα ) = − b ( d − br ) − − d r ( d − br ) − and hence wα ∈ A l +1 r . (cid:3) For the left insertion, we obtain a slightly different property.
Theorem 2.3.
Let w ∈ F k with π ( w ) = a bc d ∈ SL ( k ) . (i) If w ∈ C lr , then d = 0 if and only if αw ∈ C l +1 r for all α ∈ k . (ii) If w ∈ A lr , then there exists a unique α ∈ k such that αw ∈ A l +1 r .Proof. We first observe that for α ∈ k , π ( αw ) = π ( α ) π ( w ) = − α a bc d = c d − a + αc − b + αd . (i) Assume that w ∈ C lr . If d = 0, then b = 0, so π ( αw ) = c − a + αc − b whichmeans αw ∈ C l +1 r . If d = 0, then there exists α = ( b + dr ) d − such that π ( αw ) = c d − d − + cr dr which implies αw ∈ A l +1 r .(ii) Assume that w ∈ A lr . Then d = br . A simple calculation yields a unique α = r + r − such that π ( αw ) = c br − a + c ( r + r − ) br which means αw ∈ A l +1 r . (cid:3) Next we present results on left and right deletions of a word w ∈ A r . Theorem 2.4.
Let α . . . α l ∈ A lr . Then α . . . α l − ∈ C l − r , and α . . . α l ∈ A l − r if andonly if α = r + r − . YOTSANAN MEEMARK AND TASSAWEE THITIPAK
Proof.
Assume that α . . . α l ∈ A lr . Then π ( α . . . α l ) = a bar − b − br for some a ∈ k and b ∈ k × . Thus π ( α . . . α l − ) = π ( α . . . α l ) π ( α l ) − = a bar − b − br α l −
11 0 = α l a + b − aα l ( ar − b − ) + br b − − ar . Since b − = 0, b − − ar = − ar and so α . . . α l − ∈ C l − r . Hence π ( α . . . α l ) = π ( α ) − π ( α . . . α l ) = α −
11 0 a bar − b − br = α a − ar + b − α b − bra b . Therefore α . . . α l ∈ A l − r ⇔ b = ( α b − br ) r ⇔ α = r + r − . (cid:3) Theorem 2.2 results in |A l +1 r | ≥ |C lr | and Theorem 2.4 (i) gives rise to |A l +1 r | ≤ |C lr | .Thus |A l +1 r | = |C lr | . Since |A lr | + |C lr | = q l , we get the recurrence relation |A l +1 r | + |A lr | = q l for l ≥ |A r | = 0 . Solving this relation, we obtain the cardinalities of A lr and C lr for all l ≥
0. It should bepointing out that Bacher had the same numbers for r = 0 in [B02] Corollary 2.3. Werecord this result in Corollary 2.5.
For a finite field k with q elements, l ≥ and r ∈ k , we have |A lr | = q l − ( − l q + 1 and |C lr | = q l +1 + ( − l q + 1 . N EQUIVALENCE RELATION ON A SET OF WORDS OF FINITE LENGTH 7 Induced Equivalence Relations
Let r ∈ k . The partition A r and C r of F k induces the equivalence relation ∼ r on F k .Its properties are studied in our next theorem. Theorem 3.1.
Let x ∈ F k and β ∈ k . We have (i) If α ∈ k and α = r , then αβx ∼ r γx where γ = r − ( α − β ) r + 1 − αβr − α . (ii) rβx ∈ A r if and only if x ∈ A .Proof. Let π ( x ) = a bc d ∈ SL ( k ).(i) Assume that α, β ∈ k and α = r . Then π ( αβx ) = − α − β a bc d = − a + βc − b + βd − αa − c + αβc − αb − d + αβd and π ( γx ) = − γ a bc d = c d − a + γc − b + γd . Thus αβx ∈ A r ⇔ ( − b + βd ) r = − αb − d + αβd ⇔ − br + βdr = − αb − d + αβd ⇔ dr − αdr = − br + αb + dr − ( α − β ) dr + (1 − αβ ) d ⇔ dr = − b + ( r − ( α − β ) r + 1 − αβ ) r − α d, so αβx ∼ r γx where γ = ( r − ( α − β ) r + 1 − αβ ) r − α .(ii) Since π ( rβx ) = " − r − β a bc d = " − a + βc − b + βd − c − ar + βrc − d − br + βrd , rβx ∈A r ⇔ ( − b + βd ) r = − d − br + βrd ⇔ d = 0 ⇔ x ∈ A . (cid:3) Remark.
This result leads to an algorithm to distinguish words in F k . It extends Bacher’swork on ∼ in [B02] Proposition 2.4 (ii) to ∼ r , r ∈ k . Note that α ∼ r ε ⇔ α = r .Combined with Theorem 3.1, we completely classify all words into the partition A r and C r of F k . YOTSANAN MEEMARK AND TASSAWEE THITIPAK
We illustrate Theorem 3.1 and the above remark by the following numerical example.
Example 3.2.
Let k = F . Consider 22102 ∈ F k . r = 0. By Theorem 3.1 (i), 22102 ∼ (2 − − )102 = 0102 . By Theorem 3.1 (ii), 0102 ∼ ∼ ε. Then we have 22102 ∈ C . r = 1. By Theorem 3.1 (i),22102 ∼ (cid:20) − (2 − − · − (cid:21)
102 = 2102 ∼ (cid:20) − (2 − − · − (cid:21)
02 = 102 . Since 2 ∈ C , 102 ∈ C by Theorem 3.1 (ii). Then we have 22102 ∈ C . r = 2. By Theorem 3.1 (ii), we first consider102 ∼ (0 − − )2 = 22 ∼ (2 − − ) ε = 0 . Then 102 ∈ A , so we have 22102 ∈ A .4. Words over Z /N Z In this section, we study arithmetic properties of the partition ¯ A and ¯ C of the set F N defined parallel to Bacher’s. Let S ′ = − α : α ∈ Z /N Z . We begin by giving the proof of the following lemma.
Lemma 4.1.
The set S ′ generates SL ( Z /N Z ) as a semigroup.Proof. Recall Theorem 2 in Chapter VII of Serre’s book [S73] that the set of matri-ces − − − , − generates SL ( Z ) as a group. Since the map SL ( Z ) → SL ( Z /N Z ) obtained by reducing the matrix entries modulo N is a surjective group ho-momorphism. Then − − − , − mod N also generates SL ( Z /N Z ) as agroup. N EQUIVALENCE RELATION ON A SET OF WORDS OF FINITE LENGTH 9
Consider h S ′ i , a semigroup generated by S ′ . Since SL ( Z /N Z ) is finite, h S ′ i is a finiteclosed subset of SL ( Z /N Z ), so it is a subgroup. Note that h S ′ i contains both generators − and − − − = − − − of SL ( Z /N Z ). Hence h S ′ i =SL ( Z /N Z ). (cid:3) This lemma shows that every element of SL ( Z /N Z ) can be written in at least one wayas a finite word with letters in S ′ .Next, we establish a way to determine if words are in ¯ A . For w ∈ F N with π ( w ) = a bc d ∈ SL ( Z /N Z ), we note that w ∈ ¯ A ⇔ = a bc d = bd ⇔ d = 0 ⇔ π ( w ) = a b − b − with a ∈ Z /N Z , b ∈ ( Z /N Z ) × . Hence we have shown
Theorem 4.2.
Let N be a positive integer. Then ¯ A = w ∈ F N : π ( w ) = a b − b − for some a ∈ Z /N Z , b ∈ ( Z /N Z ) × . For l ≥
0, we write F lN for the set of words over Z /N Z of length l , ¯ A l = F lN ∩ ¯ A and¯ C l = F lN ∩ ¯ C . We first study the insertion and deletion in ¯ A . Theorem 4.3. If w ∈ ¯ A l , then αw ∈ ¯ C l +1 and wα ∈ ¯ C l +1 for every α ∈ Z /N Z .Proof. Assume that w ∈ ¯ A l and let α ∈ Z /N Z . Then π ( w ) = a b − b − where a ∈ Z /N Z and b ∈ ( Z /N Z ) × . Thus π ( αw ) = π ( α ) π ( w ) = − α a b − b − = − b − − a − αb − − b and π ( wα ) = π ( w ) π ( α ) = a b − b − − α = − b a + αb − b − . Since b = 0, αw ∈ ¯ C l +1 and wα ∈ ¯ C l +1 . (cid:3) Theorem 4.4.
Let w ∈ ¯ C l with π ( w ) = a bc d ∈ SL ( Z /N Z ) . (i) If gcd( d, N ) = 1 , i.e., d ∈ ( Z /N Z ) × , then there exist unique α, β ∈ Z /N Z such that αw ∈ ¯ A l +1 and wβ ∈ ¯ A l +1 . (ii) If gcd( d, N ) > , then αw ∈ ¯ C l +1 and wβ ∈ ¯ C l +1 for all α, β ∈ Z /N Z .Proof. We first note that for α ∈ Z /N Z , π ( αw ) = π ( α ) π ( w ) = − α a bc d = c d − a + αc − b + αd . Then αw ∈ ¯ A l +1 ⇔ − b + αd ≡ N . This congruence equation has a solution ⇔ gcd( d, N ) | b . We claim that gcd( d, N ) | b is equivalent to gcd( d, N ) = 1 and the theoremcan easily be deduced. It is obvious that gcd( d, N ) = 1 implies gcd( d, N ) | b . If gcd( d, N ) | b ,then gcd( d, N ) is a common divisor of d and b . Since ad − bc = 1, gcd( d, N ) ≤ gcd( d, b ) =1, so gcd( d, N ) = 1. Hence we have the claim. (cid:3) Theorem 4.5. If α . . . α l ∈ ¯ A l , then α . . . α l and α . . . α l − ∈ ¯ C l − .Proof. Assume that α . . . α l ∈ ¯ A l . Then π ( α . . . α l ) = a b − b − for some a ∈ Z /N Z and b ∈ ( Z /N Z ) × . Thus π ( α . . . α l ) = π ( α ) − π ( α . . . α l ) = α −
11 0 a b − b − = α a + b − α ba b and π ( α . . . α l − ) = π ( α . . . α l ) π ( α l ) − = a b − b − α l −
11 0 = α l a + b − a − α l b − b − . Since b = 0, α . . . α l and α . . . α l − ∈ ¯ C l − . (cid:3) N EQUIVALENCE RELATION ON A SET OF WORDS OF FINITE LENGTH 11
Remark.
We used to be able to derive the cardinalities of A r and C r by knowing theproperties given in the above three theorem. However, in the finite ring Z /N Z case is notthe same as the finite field k case due to this ring contains zero divisors. This makes thewords in F N behave differently as we have seen in Theorem 4.4.Another property of words in ¯ A is given in the following theorem. This result will beused in the next section. Theorem 4.6. α α . . . α l ∈ ¯ A l if and only if α l α l − . . . α ∈ ¯ A l .Proof. Consider σ = and α . . . α l ∈ F N with π ( α . . . α l ) = a bc d ∈ SL ( Z /N Z ).Since σ w xy z σ = z yx w for all w, x, y, z ∈ Z /N Z and σ = σ − , we have d cb a = σπ ( α α . . . α l ) σ = ( σπ ( α ) σ )( σπ ( α ) σ ) . . . ( σπ ( α l ) σ )= α −
11 0 . . . α l −
11 0 = − α l . . . − α − = π ( α l . . . α ) − , so π ( α l . . . α ) = a − c − b d . Thus α α . . . α l ∈ ¯ A l ⇔ d = 0 ⇔ α l α l − . . . α ∈ ¯ A l . (cid:3) The partition ¯ A and ¯ C of F k also induces the equivalence relation ∼ on F k . We recordsome relationships between two words in the next theorem. Theorem 4.7.
Let x ∈ F N and β ∈ Z /N Z . We have (i) If α ∈ ( Z /N Z ) × , then αβx ∼ ( β − α − ) x . (ii) 0 βx ∼ x . Proof.
Let π ( x ) = a bc d ∈ SL ( Z /N Z ).(i) Assume that α ∈ ( Z /N Z ) × . Then π ( αβx ) = − α − β a bc d = − a + βc − b + βd − αa − c + αβc − αb − d + αβd and π (( β − α − ) x ) = − β − α − a bc d = c d − a + ( β − α − ) c − b + ( β − α − ) d . Thus αβx ∈ ¯ A ⇔ − αb − d + αβd = 0 ⇔ − b + ( β − α − ) d = 0 ⇔ ( β − α − ) x ∈ ¯ A , so αβx ∼ ( β − α − ) x .(ii) Since π (0 βx ) = − − β a bc d = − a + βc − b + βd − c − d , 0 βx ∈ ¯ A ⇔ d = 0 ⇔ x ∈ ¯ A , so 0 βx ∼ x . (cid:3) Remark.
The above theorem yields partial answers (again due to zero divisors in Z /N Z )for determination of words into classes ¯ A and ¯ C . However, a good mathematical soft-ware such as Maple TM can easily compute the product of 2 × N . This allows us to directly distinguish words in F N .5. More on ¯ A We concentrate more on ¯ A and record its further parallel properties to Bacher’s inthis last section. This work includes unique factorization, predecessors, successors andperiodic words.In order to prove the fact about unique factorization on ¯ A , we start with the followinglemma. N EQUIVALENCE RELATION ON A SET OF WORDS OF FINITE LENGTH 13
Lemma 5.1. (i) If w, w ′ ∈ ¯ A then ww ′ ∈ ¯ C and wαw ′ ∈ ¯ A for any α ∈ Z /N Z . (ii) If exactly one of w, w ′ is an element of ¯ A then wαw ′ ∈ ¯ C for any α ∈ Z /N Z .Proof. To prove (i), let π ( w ) = a b − b − and π ( w ′ ) = a ′ b ′ − b ′− for some a, a ′ ∈ Z /N Z , b, b ′ ∈ ( Z /N Z ) × , and let α ∈ Z /N Z . Then π ( ww ′ ) = a b − b − a ′ b ′ − b ′− = aa ′ − bb ′− ab ′ − a ′ b − − b − b ′ , and π ( wαw ′ ) = a b − b − − α a ′ b ′ − b ′− = − ba ′ − ab ′− − αbb ′− − bb ′ ( bb ′ ) − . Thus wαw ′ ∈ ¯ A for any α ∈ Z /N Z . Since b, b ′ ∈ ( Z /N Z ) × , ww ′ ∈ ¯ C .To prove (ii), suppose that w ∈ ¯ A and w ′ ∈ ¯ C and let α ∈ Z /N Z . Then π ( w ) = a b − b − for some a ∈ Z /N Z and b ∈ ( Z /N Z ) × , and π ( w ′ ) = a ′ b ′ c ′ d ′ in SL ( Z /N Z )with d ′ = 0. Thus π ( wαw ′ ) = " a b − b − − α a ′ b ′ c ′ d ′ = " − ba ′ + ac ′ + αbc ′ − bb ′ + ad ′ + αbd ′ − b − c ′ − b − d ′ . Since b ∈ ( Z /N Z ) × and d ′ = 0, wαw ′ ∈ ¯ C . For another case, let w = β . . . β m ∈ ¯ C and w ′ = β ′ . . . β ′ n ∈ ¯ A for some positive integers m and n . By Theorem 4.6, we have β ′ n . . . β ′ ∈ ¯ A and β m . . . β ∈ ¯ C . The previous proof shows that β ′ n . . . β ′ αβ m . . . β ∈ ¯ C . Thus we get wαw ′ = β . . . β m αβ ′ . . . β ′ n ∈ ¯ C by Theorem 4.6. (cid:3) Let P l = (cid:8) α α . . . α l ∈ ¯ A l : α . . . α h ∈ ¯ C h for h = 1 , . . . , l − (cid:9) and P = S P l . Theorem 5.2. [Unique Factorization in ¯ A ] Let w ∈ F N . Then w ∈ ¯ A if and only if w can be written as w = p δ p δ . . . p n δ n p n +1 for some n ≥ with p , . . . , p n +1 ∈ P and δ , . . . , δ n ∈ Z /N Z . Moreover, such a factor-ization of w ∈ ¯ A is unique. Proof.
Suppose that w can be written as in this form. By Lemma 5.1, it is easy to seethat w ∈ ¯ A . Conversely, assume that w = α α . . . α l ∈ ¯ A l . Then there is the smallestpositive integer s such that α . . . α s ∈ ¯ A . Setting p = α . . . α s ∈ P and δ = α s +1 . Thus α s +2 α s +3 . . . α l must be in ¯ A l − ( s +1) by Lemma 5.1. Repeating this process we get the sets { δ , . . . , δ n } ⊂ Z /N Z and { p , . . . , p n +1 } ⊂ P so that w = p δ p δ . . . p n δ n p n +1 for some n ≥
0. The smallest length of p i for each i implies the uniqueness of this factorization. (cid:3) Given two words w, w ′ ∈ F N of the form w = α α . . . α l − and w ′ = α α . . . α l , we call w ′ an immediate successor of w and w an immediate predecessor of w ′ . Theorem 5.3.
Each element w ∈ ¯ A l has a unique immediate successor and a uniqueimmediate predecessor in ¯ A l .Proof. Assume that α α . . . α l − ∈ ¯ A l . Then π ( α α . . . α l − ) = a b − b − for some a ∈ Z /N Z and b ∈ ( Z /N Z ) × . Thus π ( α α . . . α l − ) = π ( α ) − π ( α α . . . α l − )= α −
11 0 a b − b − = α a + b − α ba b , so π ( α α . . . α l ) = π ( α α . . . α l − ) π ( α l )= α a + b − α ba b − α l = − α b α a + b − + α α l b − b a + α l b . Since b ∈ ( Z /N Z ) × , α α . . . α l ∈ ¯ A l ⇔ α l = − ab − . Hence w has a unique immediatesuccessor in ¯ A l . Similarly, we can show that w also has a unique immediate predecessorin ¯ A l . (cid:3) For w = α α . . . α l ∈ ¯ A l , by Theorem 5.3 there exists an infinite word W = . . . α − α α α α . . . N EQUIVALENCE RELATION ON A SET OF WORDS OF FINITE LENGTH 15 such that α i +1 . . . α i + l is the immediate successor in ¯ A l of α i . . . α i + l − for all integer i .That is, all subwords formed by l consecutive letters of W are elements in ¯ A l . Since ¯ A l is finite, the infinite word W associated to w is periodic. Hence for every w ∈ ¯ A l , thereexists the smallest positive integer s such that the infinite word W associated to w is s -periodic. Example 5.4.
Some infinite periodic words over Z / Z .(1) The infinite periodic word corresponding to both 121 and 212 is a 2-periodic word . . . . . . .(2) The infinite periodic word corresponding to 234 ,
343 and 432 is a 4-periodic word . . . . . . . Theorem 5.5.
Let W = . . . α s − α α . . . α s − α α . . . be an infinite s -periodic word withletters in Z /N Z . Then there exists a smallest positive integer t such that all subwords oflength ts − in W belong to ¯ A . Moreover, all subwords of length lts − l ≥ of W belong to ¯ A .Proof. We observe that the elements π ( α α . . . α s − ) , π ( α . . . α s − α ) , . . . , π ( α s − α . . . α s − ) ∈ SL ( Z /N Z )are all conjugate. Then they have a common order t ′ , we claim that t ′ has the desiredproperty. Let w be a subword of length t ′ s in W . Thus w = w ′ w ′ . . . w ′ | {z } t ′ copies where w ′ is asubword of length s in W , so π ( w ) = π ( w ′ w ′ . . . w ′ | {z } t ′ copies ) = ( π ( w ′ )) t ′ = . Assume that w = β . . . β t ′ s . The subword of length t ′ s − w is in ¯ A as a result of π ( β . . . β t ′ s − ) = π ( w ) π ( β t ′ s ) − = β t ′ s −
11 0 . Hence t ≤ t ′ exists by thewell-ordering principle. (cid:3) Remark.
In the above proof, sometimes t < t ′ . For example, consider the infinite 1-periodic word, . . . . . . . The order of π (0) = 4 but we can choose t = 2 since 0 ∈ ¯ A .Moreover, since t ′ divides | SL ( Z /N Z ) | , we know that t ≤ | SL ( Z /N Z ) | . Example 5.6. In Z / Z , consider the 4-periodic word W = . . . . . . .(1) We have t = 1 so that all subwords of length 4(1) − W belong to ¯ A .(234 , , , l = 2, all subwords of length 4(2) − W belong to ¯ A .(2343234 , , , l = 3, all subwords of length 4(3) − W belong to ¯ A .(23432343234 , , , References [B00] R. Bacher,
An equivalence relation on { , } ∗ , Europ. J. Combinatorics, (2000), 853-864.[B02] R. Bacher, SL ( k ) and a subset of words over k , Europ. J. Combinatorics, (2002), 141-147.[S73] J.-P. Serre, A Course in Arithmetic , Springer-Verlag, New York, 1973.
Yotsanan Meemark, Department of Mathematics, Faculty of Science, ChulalongkornUniversity, Bangkok, 10330 THAILAND
E-mail address : [email protected] Tassawee Thitipak, Department of Mathematics, Faculty of Science, ChulalongkornUniversity, Bangkok, 10330 THAILAND
E-mail address ::