Information Processing Letters 29 (1988) 171-175 North-Holland

12 November 1988

VARIETIES OF FORMAL SERIES ON TREES AND EILENBERG’S THEOREM

Symeon BOZAPALIDIS and Stavros IOULIDIS

Department of Mathematics, Aristotelian University of Thessaloniki, Thessaloniki, Greece

Communicated by L. Boasson Received April 1988 Revised June 1988

We establish Eilenberg’s theorem in the context of formal series on trees. More precisely, we prove that there is a bijection between varieties of tree series and (E, M)-varieties of tree algebras, where E and M are appropriate classes of epimorphisms

and monomorphisms, respectively, in a suitable category. An application is also given.

Keywords: Syntactic ideal, syntactic algebra, (E, M)-variety, tree-series variety

1. Preliminaries

Trees For a given ranked alphabet 2, TX (respec-

tively T=(x,,..., x,)) denotes the set of all trees (respectively of all trees indexed by xi,. . . , xn) over 2. Let P, be the subset of T=(x) consisting of all trees with just one occurrence of the variable x; Pz is a free monoid generated by the elements

its multiplication is the substitution at x. Finally, a homomorphism cp : TX + T, is said

to be very fine if, for any u E Z,,, ‘p(a(x,. , . x,))

= y(x, . . . x,), 0 Q K < n and y E r,.

F-E-algebras Let F be a field and 2 a ranked alphabet. An

F-E-algebra _&‘= (A, a) is an F-vector space A together with a family of multilinear functions a,:A”+A, UEE”, n>O.

A morphism from &= (A, a) into .97= (B, b) is a linear function h : A + B compatible with 2-operations:

h(a,(q,,...,q,)) =b,(k,...,k),

UE&, qiEA.

The vector space F(T,), freely generated by T,, is converted (in a natural way) into an F-E-algebra. Further, for any F-x-algebra .&= (A, a) we can define a function H&: TX + A by

H,(a(t,...t,))=a,(H,t,,...,H,t,);

its linear extension

H,: F(T,) + A, H, Ikit, = Ck,Hd(t,) ( 1 i i

is the unique existing F-2-morphism from F(T,) into .&. From now on, we deal with algebras .& such that Hd is surjective. The monoid Px acts on &‘= (A, a) via

q.u(t,... tr_-lXt;+l...tn)

= a,(H_&,..., H&t;_1, q, HjBt,+lr..., H&t,).

Every subspace % c A closed under this action (qE% and REP= 3 q7 E 3) is called an ideal of .&. The quotient space A/% then admits an F-Z algebra structure making the projection A --) A/% an F-2-epimorphism [ 11.

Tree series A realisation of a tree series S : TX + F is a

couple (z2, cp), consisting of an F-z-algebra zY=

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(A, a) and a linear form cp: A + F such that _ S = cp 0 Ed (S being the linear extension of S).

The syntactic ideal of S: T, + F is

zI,= {QEF(T~)@, QT)=ov(7~~p}, the quotient SB, = F(T,)/%, is the syntactic F-z-algebra of S. Since %, c Ker(S), S induces a linear form (ps : .ds --, F with S = ‘ps 0 Eds. More- over, we have the following.

Proposition 0 ([l]). The above realisation (SB,, cps) is minimal in the sense that for any other realisation (.&, ‘p) of S there is a unique F-Zepimorphism h : d-+ ds verifying ‘p = ‘ps 0 h.

Call S : TX + F recognizable if it can be realised by a couple (&, cp), with .G? finite-dimensional; hence, S is recognizable iff dim( ds) < cc.

2. (E, &Q-varieties

In a category A with finite products, we choose a class E of epimorphism and a class M of monomorphisms such that:

(i) both contain the isomorphisms of A, (ii) both are closed under composition:

e,, e,EE a e, oe2EE

and

m,, m,EM * m, 0 m,EM,

and under product:

e,,e,EE a e,Xe,EE

and

ml, m,EM 3 m,Xm,EM,

(iii) for every diagram in A,

B

I e

A-C

thereexist e’: D-tAEEandm’: D-+BEMso that the following square commutes:

D*B

e’ i 1

e

AGC

172

An (E, M)-variety is a class ^Y of objects of A with the following three properties:

(vi) A,, A, E Y- j A, x A, E V-,

(v2) AqBEMand BEV 3 AEV,

(v3) AABEE and AEV * BEV.

We denote by (V) the (E, M)-variety gener- ated by the class V c Ob( A); (V) is the smallest such variety containing V.

F’ropositionl.AE(V)iffthereareA,,...,A,EV so that

A$A’qnA;, eEE, mEM.

In the squel we shall describe the framework, where Eilenberg’s theorem will be established.

Take a field F; the category F-Alga,, has as objects all couples (2, &‘), where _X is a ranked alphabet and JX? is a finite-dimensional F-Zalge- bra such that E_, is surjective; a morphism from (2, .G’) into (r, 3?) is a couple (cp, h), where ‘p : T, + T, is a very fine homomorphism and h : d+ 93 is a linear function making the diagram below commute:

Proposition 2. The category F-Alg,, has finite products.

Proof. Let J%‘= (A, a) be an F-Z-algebra and 93 = (B, b) an F-T-algebra, both of finite dimen- sion. We construct the ranked alphabet max(,Y, r)

by

max(E, r), = { (0, Y) I (J E z,, Y E r,,

max(K, X) =n},

Volume 29, Number 4 INFORMATION PROCESSING LETTERS 12 November 1988

and we convert the product space &X .?i? into a max(2, r)-algebra by putting

i+O,y,((% PI)Y-..>(%Y PJ)

with max(K, X) = n, (Y~ E A, and pi E B. Obviously, Ha I is surjective. There are two

very fine homomorphisms

V2 : =rnax(xr) -+ =z, ‘Pr : =max(zr) + =r

respectively defined by

‘p&J, Y>(o,...w,))=a(cp~w*...~~w,),

‘p&J> Y)(~,---~,))=Y(cp,~*...~,~,)

with max(~, A) = n and wi E TmaxCP,rj. Denoting by P& and Pa the projections from

AX .%? into JX? and 99 respectively, the diagram

is the product of (2, ~2) and (r, .?+Y) inside F-Alg rin. 0

Now, let us define the E-M system. The class M consists of all morphisms (cp, h) : (E, .53?) -+ (r, ~27) with cp, h both injective and ~(2,) _C r, (Vn >, 0), while the class E consists of all mor- phisms of the form (Id, h): (2, _&) + (2, .%?), where Id is the identity function on T, and h is an F-E-epimorphism (with 2 an arbitrary ranked alphabet). Conditions (i)-(m) cited at the begin- ning of this section are now easily verified.

Proposition 3. Every (E, M)-variety Y in F-Alg,;, is generated by all the syntactic algebras that it contains.

Proof. Let (2, J%‘) E V and ‘pi,. . ., ‘p, be a dual basis of the F-z-algebra &= (A, a); further, let &, be the syntactic algebra of the tree series

- F(T,)%.&‘(P’F, i=1,2 >.*-, n.

By Proposition 0 we have an F-2-epimorphism h, : d+di, and therefore _pP, E V, for every i. Simple calculations show that the induced mor- phism

(cp, h) : (2, d> -, (mdz,. . . , z>,

d, x . . * XdJ,

cp(4=(%.4), h(q) = (h,(d.-, h,(d)

belongs to M. This proves our assertion. 0

3. Varieties of tree series

For every ranked alphabet 2, a set Y’(2) of recognizable series Tz + F is given such that:

(si) S,, S, E Y(E) and k,, k, E F * k,S, +

kA E y(E),

(sz) S E 9’(I) and 7 E P, * ST-1 E Y(2), where the series Sr-’ : TX + F is defined by (ST-~, t) = (S, t7) VT E T,,

(sj) for every very fine homomorphism cp : T2 + T, and every S E Y’(r) we have S 0 cp E

Y(E). We then say that the family Y= (Y(E)), con-

stitutes a tree series variety.

Proposition 4. Let Y be a tree series variety, S E 9’(I) and cp a linear form on &s; then, the series

belongs to Y’(Z).

Proof. We follow the proof of [3, Lemma 111.1.4] adapted in the present context or our study. Let E be the subspace of FFcTr) generated by the series Sr-i, i.e., E = (s7-l 1 r E Pz>, where S denotes the linear extension of S; the orthogonal E” of E is then just the syntactic ideal a, of S because

This implies that E has finite codimension and, consequently, E = Em; since cp 0 E, vanishes as, we have

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We are now in a position to prove Eilenberg’s theorem for tree series. Given a tree series variety 9, let V(Y) denote the (E, M)-variety in F-Alg,, generated by the syntactic algebras of the members of 9; on the other hand, for every (E, M)-variety V, Y”(V) is the variety of series whose syntactic algebras belong to V.

Theorem. The mappings -lr,Y( V) and 9’++ V( 9’) are mutual inverses; in other words, there is a bijection between (E, M)-varieties in F-Alg,,, and varieties of tree series.

Proof. The proof is inspired on that of [3, Theo- rem 111.1.11. By Proposition 1 we have V= Y( 9’( V)). The inclusion 9~ Y( “lr( 9’)) being clear, we only have to establish 9’( V(Y)) c 9’. To do this, let S E 9’( ^Y( 9’))(Z). Then,

with e E E, m E M, and S, ESP(I;) Vi. e being an F-Zepimorphism, the following triangle com- mutes:

Notice that m E M implies the existence of an injective very fine homomorphism IJI : TZ + T ,,,=(r ,,..., I-,) such that cp(U c mNrlY.. . , CL (for all k > 0) and of a linear fur@ion h : at” - .ds, x . . . Xds making the square below com- mutative (for ejery i):

‘pi = ‘PI-, o cp h;=j’; 0 h

where Pi is the ith projection from a product. Also, the linear form

can be extended into a linear form

~:.d”,x ... X.J+-+F.

Straightforward calculations show that

with 4, the restriction of 4 on “;ps, XAO} X . . . x {0}, etc. By Proposition 4, 4, 0 H E

9’(c), for every i. This fact combined with axioms (sr) and (sg) leads to S ~9’(2). 0

Let V be an (E, M)-variety and 9’ the corre- sponding series variety.

Proposition 5. For any ranked alphabet 2 = E,, U . ’ . uE,, 9 contains the characteristic series of

Z0 iff V contains an F-E-algebra .&= (A, a) con- taining a nonzero constant q, = H&(c) and verify- ing the condition

a,(ql,...,qn) =O, (1) whenever some arguments qi coincide with q,.

Proof. Suppose that (1) holds for a certain con- stant q,. Any one-dimensional vector space V= (e) can be viewed as an F-r-algebra

(To= {0}, r,=Z, for n21)

by setting H,(c)=e, uO(vI,...,v,,)=O for every (I E F, and for every ui E V. Clearly, the linear function f : V-+ A, f(e) = q, belongs to M, so that V E V. cp : V + F being the linear form cp( e) = 1, the couple (V, cp) realises the series S : T, +

F, S(c) = 1, and S(t) = 0, for every t E T, - {c}; therefore S E 9’(F). Consider now the very fine homomorphism h : TX + T,, h(2,) = c, h(u) = u

for every u E 2, (n > 1). The composition S 0 h : TX --, F is just the characteristic series of 2, and by axiom (s3) belongs to Y(Z).

In order to prove the converse, we consider the syntactic algebra &= (A, a) of the series char(2,) : T2 + F and we let

Pi= Cx..t JI J’ Jd’4ds %F j=l

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Volume 29, Number 4 INFORMATION PROCESSING LETTERS

For every r E Pz we have

(char(&), u(P, ,.--, P,_l, c, Pj+cl,..., P,>T)

= (char(Z,), CW) = 0,

where

7T = C ‘,,.I ... x~,_l,i-lAj,+l,i+l ... ‘j”,n

i,,...J,

*u(r,, . ..rj._,xtj,+, .A,).

This implies that CT E %char(P,l (is the syntactic ideal of char(_X:,)) and therefore

O=~~(c~)=a,(qt,...,qi-l, 4,t qi+l,.*.,qn)

as desired. q

12 November 1988

References

[l] S. Bozapalidis and A. Alexandrakis, RCpresentations matricielles des series d’arbres reconnaissables, RAIRO Informotique Thtkique et Applications, to appear.

[2] S. Eilenberg, Automata, Languages and Machines, Vol. A (Academic Press, New York, 1974).

[3] C. Reutenauer, SCries formelles et algtbres syntactiques, J. Algebra 66 (1980) 448-483.

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