Idempotent and Compact Matrices on Linear Lattices: a Survey of Some Lattice Results and Related Solutions of Finite Relational Equations

After a survey of some known lattice results, we determine the greatest idempotent (resp. compact) solution, when it exists, of a finite square rational equation assigned over a linear lattice. Similar considerations are presented for composite relational equations.


IDEMPOTENT AND COMPACT MATRICES ON LINEAR LATTICES: A SURVEY OF SOME LATTICE RESULTS AND RELATED SOLUTIONS
OF FINITE RELATIONAL EQUATIONS determines, if (resp.S) is nonempty, the greatest (with respect to the partial ordering pointwise induced naturally in (resp.S) from the total ordering of L) element of (resp.S) defining the matrix S AaB (resp.S* Q-laT, where Q-1 is the transpose of Q): x x XL as resp.Sij (AaB)ij AiaBj (QlaTik)-l(QijaTik), Sk (Q-laT)Jk i= i= for any i,j, k E I n.The concept of max-min transitivity is the most widely used one for investigating properties of finite square Boolean matrices [10].For square matrices over L, we say that R is max-rain transitive if R2_< R (i.e., Rib A Rhj <_ Rij for any i,h,j In) and this notion is also widely dealt in several areas of research like clustering technology [5], information retrieval [20], preference relations [7,8].Moreover, if L is linear, EQ. (1.1) can be seen as a description of a finite-state system in which R represents a transition relationship between the assigned input A and the given output B. Following Kolodziejczyk  [11] (cfr. also [3]), applying k times the transition, let B (k) be the resulting output defined by B (k) R k o A (assume, of course, B (1) B).The problem is to determine a solution/ , with an assigned type of transitivity, which guarantees the necessary speed of convergence (/ is convergent if R k+l R k for some integer k).Similar considerations can be made on Eq. (1.2).The authors of [3] and [11] have proved that in (resp.S) exist elements with different type of transitivity, mainly in (resp.S) were entirely characterized the max-min transitive elements (in particular, those having Schein rank equal to 1), determining the greatest and the minimal ones.
Here we prove the existence in , when nonempty, of compact (in particular, idempotent) elements For each of them, a specific rule of convergence of their powers holds, so giving further information on the speed of convergence of the entire system.Related but different considerations are presented for the analogous elements of S. 2. A SURVEY OF SOME LATTICE RESULTS.
We refer to Birkhoff [1] for terminology of lattice theory.Now we recall some well known facts.
If P,Q,R X x XL are assigned matrices, P < R means Pij <-Rij for, any i,j e I n in L, P < R means P<_R and PR, (PAR), (PVR): XX are matrices pointwise defined as (P A R)i j Pij A Rij (PV R)i j Pij V Rij for any i,j e I n, similarly it is defined the infimum and the supremum of any finite set of matrices.It is well known that the max-rain composition (1.2) is associative and the following properties hold: since L is linear and hence distributive.
REMARK 2.1.Since x is finite, the results of Sanchez [15] hold in the more weak hypothesis that L is a Brouwerian lattice, but it is easy to see that, under this last hypothesis, " continues to be a bounded residuated 1-monoid.
Following Shmuely [18], let e {R E: R < R2} be the set of all non-negative (or compact) elements of "$ and, as denoted usually, ^be the set of all non-positive (or max-min transitive or subidempotent [1, p. 328]) elements of ".
However, as it is known, ^becomes a lattice defining the sup operation "kJ" as R IR2 R v R2, where R1,R 2 R A and "_" stands for the max-min transitive closure of any matrix R defined as REMARK 2.3.If is complete, then , being a complete residuated lattice, satisfies properties (J1) and (J2) of Shmuely [18] (cfr.also Prop. 2.1 of [5]).Thus, by Corollary of [18], the smallest element e ^including R e ' is given by R v R 2 v..-v R n v However, it is proved, like in Kaufmann [9, p.95], that R v R2v.v Rn.This result holds for Boolean matrices ([10, The study of the powers of a square Boolean matrix is useful in automata theory, information theory, etc., (e.g., [10]).The sequence R, R2,R3, depends on two parameters p (the period of R) and k (the index of convergence or R), p and k being the smallest positive integers such that R k+p tlk.For Boolean matrices, these indices were widely studied (e.g., see Shao and Li [17] and references therein).These parameters can be defined also for a matrix R E "$ which either converges (i.e., p 1) to an idempotent matrix or oscillates with finite period [21].If C e C, then C n =C n-1 [21] and R n+l R n if R e ^[6], thus we use these simple facts to prove the following: THEOREM 2.4.The set f n ^{R e :R R2} of all the idempotent elements of ' is a lattice under the partial ordering induced by in f and the operations: where I, J e .
PROOF.For any I,J _ , from above, we have that (I ^j)n and (I v j)n-1 are elements of f.
Concerning Eq. (1.1), it was proved in [3] that the set "J'=Y(A,B)=n^of all max-rain transitive solutions is nonempty iff O, w e '/being the matrix, defined as Wij Bj if Bi > Bj and Wij Sij if B <_ Bj for any i,j In, the greatest element of 'J', i.e., W _> R for any R E '.Let 3 n t be the set of all idempotent solutions of Eq. (1.1).Of course, 3 g "J" and let B " be defined as Bij Bj for any i,j I n.Then B 3 and hence B B n <_ W n <_ W, i.e., the matrix I W n belongs to by Remark 2.6.Since I wn= W n+l [6], we deduce that I 3 and further, R R n <_ wn= I for any R 3. Thus, we have proved that THEOREM 3.1.iff I 3. Further, 1 > R for any R 3. REMARK 3.2.It is easily seen that an alternative definition of W is w S ^(BaB), where (BaB) is defined pointwise as (BaB)i j Blabj for any i,j I n. 4. ON COMPACT SOLUTIONS OF EQ. (1.1).
Assume here that #-O too.Let e 0 =en be the set of all compact solutions of Eq. (1.1) and now we need to define the matrix C t as C S ^S2 ^... ^Sn in order to prove that O iff C e: 0.
We give some preliminary propositions and lemmas.
PROPOSITION 4.2.If Si2j < Sij for some i,j In, then we have that S2hj < Shj for any h I n.PROOF.Let S2kj > Ski for some k I n.Since S2kj=SktAStj for some I n, we should have that S2j=S}t^Sj>Skj.Then t#j (otherwise Skj>_Sj>Skj, a contradiction) and Stj > Sj > Ski Bj, which should imply that Stj 1.If Skt= Bt, then B Skt > Sj > Ski Bj.If Skt= 1, then Bj < A k < B and thus B > Bj in any case.On the other hand, Sij > Si > Bj, hence Sij, i.e., Si Bj by Prop. 4.1.But this contradicts the fact that Si > Sit A Stj Sit A Sit > B > Bj. then we have that S2mj < Sr"j (resp.Sirn > r" PROOF.We prove the thesis (a) since the thesis (b) can be proved similarly.The thesis (a) is certainly true for k 2 (it suffices to choose i).Hence assume k > 2 and let r" I n such that Sij Sr"j > Sij Sire A Srnj.If skin 2 A Smj Sir " Hence Sir " ^Sr"j Srnj and therefore Sr"j > Sij > Sij > Sr"j.
If Si2j 1, the set Lij {t In:Sit Stj 1} is certainly nonempty.Then the following results hold: PROPOSITION 4.4.Let Sij Si2j and Stt < for any Lij.k (a) If Sih < (resp.(b) If Shi < 1) for some h In, then we have that Sih < (resp.Sj < 1) for any integer k >_ 1.   PROOF.Let h I n be such that Sih < and assume that Skih for some integer k _> 1.We could certainly suppose, without loss of generality, k( _> 2) to be the smallest integer such that 2 Si > Sihk-1.By Prop. 4.3 (b), then Sim > Sire Br" for some r" I n.Let I n such that 2 Sir " Sit A Str . 1,thus we should have that A < B m < A < Bt, i.e., Stt 1.On the other hand, we know that Brn < A < Bj since Sij 1, thus (since Str" 1) A < Br" < Bj which should imply that Stj 1, i.e., Lij and therefore the contradiction =Stt < 1.Thus the thesis (a) is true and similarly one proves the thesis (b).
k k LEMMA 4.5.If Sij > Bj for any integer k > 1, then we have that Sij for any such k.
PROOF.Let S. Sih 1AShh2 A'"AShn_Ij for some hl,h2,...,hn_ ln(h 0 i,h n j).Then we should have that Sih and sn-1 hi j Shlh2 A...ASh,,_j 1A---A 1, i.e., Sh j by Prop. 4.4(b), thus h Lij.Now Sh Sih 1/ Shlh A and Sj 2 1, hence $ih=Shj= by Prop. 4.4 (a), (b) and then we should deduce that h2 E L too.Now otherwise Bh < Ah <_ Bh2 Bh a contradiction.So continuing, we should get that h Lij for any In_ and hs h for any s,t In_ since Bh < Bh2 < < Bhn_l.This should imply that card Lij > n-1, a contraAiction to the hypothesis that crd Lij < n-2 (since i,j f Lij ).Now we are able to show that THEOREM 4.7.The matrix C ^= 1Sk belongs to C.
PROOF.We must prove that C2j _> Cij for y i,j e I n.In order to avoid trivial situations, assume that j.Since CiJ =tV=l(Cit^C tj > Cij ^Cjj >_ Bj, k the thesis is clearly true if Cij Bj.Let Cij > Bj, i.e., Sij > Bj for any t E I n.By Lernma 4.5, we k have that $ij for any k I n (in particular, Sij Si Sj 1) and hence Cii. 1.If Lii ,. then k we deduce that Cii because Sii for any k I, and hence Ci2; Cil.A Ci; A 1. Similarly one gets Ci if j Lij.Now assume that i, j Lij.By Lemma 4.6, there exists m Lij such that k--1 Thus Sire Sire ^Smm A and Sire Srnj S,n, 1, hence Smm for any integer k > 2 k k-1 Smm ^Smj ^for any k I n.Then we have that 6' Cim A Crnj ^1, hence we get (Tij (7i2j in any case.
If i # , let I w, by Thm.3.1, be the greatest element of i for y In, then we deduce the following rult: THEOREM 5.1.* # iff t* *, where 1" a = lli) n.Further, I* (w*) n R for y RE*.
PROOF.Let 3", then i# d hence i for y il n.By Thin. 3.1, we c consider I i for y I n d since I for y In, we have that I* longs to by [6].Let R3"i, thus Rifor y iIn, i.e., RS =llid then R=R nS(=lli)n=l*Sl=li.
Ts mes that I* i for y I n y Retook 2.6, i.e., I* $ d hence I* li in *.We note n lot y In, so that [6] explicitly that (w*) n S W } nn X li)n=i,.
We c have that *# (hence # ), then * but S# d * do not imply nesi]y that * it is shown in the following: Note that, as easy examples prove, we can have that S # O and '*= 13, hence I*= 13.For sake of completeness and in accordance to Remark 3.2, we point out the following result: THEOREM 5.3.If "J'* 13, then w* (T-1aT)^S*, where (T-laT) is defined pointwise (T-laT)j k =i (TijTik) for any j,k .I n.
V Ahb _< Bj < Bhb, i.e., S k -2 h b h b Shbhb Sih and hence A Bj for some R C 0. Let k I n be such that A k ^RkjBj.