Profile and hereditary classes of ordered relational structures
PPROFILE AND HEREDITARY CLASSES OF ORDERED RELATIONALSTRUCTURES
DJAMILA OUDRAR AND MAURICE POUZETA
BSTRACT . Let C be a class of finite combinatorial structures. The profile of C isthe function ϕ C which counts, for every integer n , the number ϕ C ( n ) of membersof C defined on n elements, isomorphic structures been identified. The generatingfunction of C is H C ( x ) := (cid:80) n (cid:61) ϕ C ( n ) x n . Many results about the behavior ofthe function ϕ C have been obtained. Albert and Atkinson have shown that thegenerating series of several classes of permutations are algebraic. In this paper,we show how their results extend to classes of ordered binary relational structures;putting emphasis on the notion of hereditary well quasi order, we discuss some oftheir questions and answer one. AMS Subject Classification:
Keywords : profile, well quasi-ordering, indecomposability, permutations.1. I
NTRODUCTION
The context of this paper is the enumeration of finite relational structures. Arelational structure R is embeddable in a relational structure R (cid:48) , in notation R ≤R (cid:48) , if R is isomorphic to an induced substructure of R (cid:48) . The embeddability relationis a quasi order. Several significant properties of relational structures or classes ofrelational structures can be uniquely expressed in term of this quasi order. This istypically the case of hereditary classes: a class C of structures is hereditary if itcontains every relational structure which can be embedded in some member of C .Interesting hereditary classes abound. In the late forties, Fra¨ıss´e, following the workof Cantor, Hausdorff and Sierpinski, pointed out the role of the quasi-ordering ofembeddability and hereditary classes in the theory of relations (see his book [13]for an illustration). Recent years have seen a renewed interest for the study ofthese classes, particularly those made of finite structures. Many results have beenobtained. Some are about obstructions allowing to define these classes, others on thebehavior of the function ϕ C , the profile of C which counts, for every integer n , thenumber ϕ C ( n ) of members of C defined on n elements, isomorphic structures beingidentified. General counting results have been obtained, as well as precise results,for graphs, tournaments and ordered graphs (see the survey [20]). Enumerationresults on permutations, motivated by the Stanley-Wilf conjecture , solved by Marcus
Date : November 9, 2018.2000
Mathematics Subject Classification.
Key words and phrases. ordered set, well quasi-ordering, relational structures, profile, indecom-posability, graphs, tournaments, permutations.*The author was supported by CMEP-Tassili grant. a r X i v : . [ m a t h . C O ] S e p D.OUDRAR AND M.POUZET and Tard¨os (2004), fall also under this frame, an important fact due to Cameron [10].Indeed, to each permutation σ of [ n ] := { , . . . , n } we may associate the relationalstructure C σ := ([ n ] , ≤ , ≤ σ ) , that we call bichain , made of two linear orders on [ n ] ( ≤ being the natural order on [ n ] and ≤ σ the linear order defined by i ≤ σ j ifand only if σ ( i ) ≤ σ ( j ) ). As it turns out, the order defined on permutations andthe embeddability between bichains coincide (see Subsection 3.2 for details andexamples).In this paper, we show how some results obtained by Albert and Atkinson [1] forclasses of permutations extend to classes of ordered binary relational structures. Weprove notably Theorem 5.7. For this purpose, we recall in Section 2 some basic def-initions of the theory of relations, we survey in Section 3 some results concerningclasses of permutations and show how permutations are related to relational struc-tures. Then, we illustrate the role of indecomposable structures (see Section 4) andof well quasi order (see Section 5) in enumeration results. Finally, in Section 6, wepresent a conjecture and a partial solution, a special case answering a question ofAlbert and Atkinson [1].Our results have been presented at the international conference on Discrete Math-ematics and Computer Science (Dimacos’11) held in Mohammedia, Morocco, May-5-8, 2011, and at the International Symposium on Operational Research (Isor’11),held in Algiers, Algeria , May 30-June 2, 2011 [25]. We are pleased to thanks theorganizers of these meetings for their help.2. B ASIC NOTIONS , EMBEDDABILITY , HEREDITARY CLASSES AND PROFILE
Our terminology agree with [13]. Let n be a positive integer. A n -ary relationwith domain E is a subset ρ of the n -th power E n of E ; for n = 1 and n = 2 we usethe words unary relation and binary relation , in this later case we rather set xρy in-stead of ( x, y ) ∈ ρ . A relational structure with domain E is a pair R := ( E, ( ρ i ) i ∈ I ) made of a set E and a family ( ρ i ) i ∈ I of n i -ary relations ρ i on E , each ρ i being asubset of E n i . The family µ := ( n i ) i ∈ I is the signature of R . We will denote by V ( R ) the domain of R . We denote by Ω µ the class of these structures and by Ω µ the subclass of the finite ones. A relational structure R is ordered if it can be ex-pressed as R := ( E, ≤ , ( ρ j ) j ∈ J ) where ” ≤ ” is a linear order on E and the ρ j ’s are n j -ary relations; the (truncated) signature in this case is µ = ( n j ) j ∈ J . A relationalstructure R := ( E, ( ρ i ) i ∈ I ) is a binary relational structure , binary structure forshort, if each ρ i is a binary relation; the class of those finite binary structures willbe denoted by Ω I instead of Ω µ . Basic examples of ordered binary structures arechains ( J = ∅ ) , bichains ( J = { } and ρ is a linear order) and multichains ( ρ j is alinear order for all j ∈ J ). We denote by Θ d the collection of finite ordered binarystructures made of a linear order and d binary relations. Let R := ( E, ( ρ i ) i ∈ I ) bea relational structure; the substructure induced by R on a subset A of E , simplycalled the restriction of R to A , is the relational structure R (cid:22) A := ( A, ( ρ i (cid:22) A ) i ∈ I ) , where ρ i (cid:22) A := ρ i ∩ A n i . Let R := ( E, ( ρ i ) i ∈ I ) and R (cid:48) := ( E (cid:48) , ( ρ (cid:48) i ) i ∈ I ) be tworelational structures of the same signature µ := ( n i ) i ∈ I . A map f : E → E (cid:48) is an isomorphism from R onto R (cid:48) if f is bijective and ( x , . . . , x n i ) ∈ ρ i if and only LASSES OF ORDERED STRUCTURES 3 if ( f ( x ) , . . . , f ( x n i )) ∈ ρ (cid:48) i for every ( x , . . . , x n i ) ∈ E n i , i ∈ I. The relationalstructure R is isomorphic to R (cid:48) if there is some isomorphism from R onto R (cid:48) , itis embeddable into R (cid:48) , and we set R ≤ R (cid:48) , if R is isomorphic to some restrictionof R (cid:48) . The embeddability relation (called “abritement” by
Fra¨ıss´e in french) is aquasi-order. A class C of relational structures is hereditary if R ∈ C and S ≤ R imply
S ∈ C ; relational structures which are not in C are obstructions to C . The age of a relational structure R is the class Age ( R ) of finite S which are embeddableinto R (equivalently, this is the set of finite restrictions of R augmented of theirisomorphic copies). An age is non-empty, hereditary and up-directed (that is forevery S , S (cid:48) ∈ Age ( R ) there is some T ∈
Age ( R ) which embeds S and S (cid:48) ). In theterminology of posets, this is an ideal of Ω µ . If the signature is finite, every idealof Ω µ is the age of some relational structure ( Fra¨ıss´e B is a subset of Ω µ then F orb ( B ) denotes the subclass of members of Ω µ which embed no member of B . Clearly, F orb ( B ) is an hereditary class. Moreover, every hereditary subclass C of Ω µ has this form. This fact, due to Fra¨ıss´e , is based on the notion of bound: a bound of an hereditary subclass C of Ω µ is every finite R not in C such that every R (cid:48) which strictly embeds into R belongs to C . Clearly, every finite obstruction to C contains a bound. Hence, if B ( C ) denotes the collection of bounds of C consideredup to isomorphism then C = F orb ( B ( C )) .The profile of an hereditary class C is the function ϕ C : N −→ N which counts, forevery n , the number of members of C defined on n elements, isomorphic structuresbeen identified. The generating function for C is H C ( x ) := (cid:80) n (cid:61) ϕ C ( n ) x n . Thesetwo notions are the specialization to hereditary classes of basic notions in enumer-ation. Many results on the enumeration of classes of permutations are about theenumeration of relational structures. Indeed, as mentioned in the introduction, per-mutations can be considered as special cases of binary structures, and more specifi-cally ordered binary structures, in fact bichains. We introduce these notions belowand point out the relationship between permutations and bichains in the next section.3. P ERMUTATIONS , BICHAINS AND THEIR PROFILE
Permutations.
Let n be a non negative integer. Let S n be the set of per-mutations on [ n ] := { , . . . , n } and S := (cid:83) n ∈ N S n . An order relation on S isdefined as follows: the permutation π of [ n ] contains the permutation σ of [ k ] andwe write σ ≤ π if some subsequence of π of length k is order isomorphic to σ .More precisely, σ ≤ π if there exist integers ≤ x < · · · < x k ≤ n such that for ≤ i, j ≤ k, σ ( i ) < σ ( j ) if and only if π ( x i ) < π ( x j ) . For example, π := 391867452 contains σ := 51342 , as it can be seen by consideringthe subsequence ( = π (2) , π (3) , π (5) , π (6) , π (9) ).A subset C of S is hereditary if σ < π ∈ C implies σ ∈ C . Its counting function,that we call the profile of C , is ϕ C ( n ) := | C ∩ S n | . How much does ϕ C ( n ) drop D.OUDRAR AND M.POUZET from ϕ S ( n ) = n ! if C (cid:54) = S ? The
Stanley-Wilf conjecture asserted that it drops toexponential growth. The conjecture was proved in by Marcus and Tard¨os [23]:
Theorem 3.1. If C is a proper hereditary set of permutations, then, for some con-stant c, ϕ C ( n ) < c n for every n .Kaiser and Klazar [19] proved that if C is hereditary, then, either ϕ C is boundedby a polynomial and in this case is a polynomial, or is bounded below by an expo-nential, in fact the generalized Fibonacci function F n,k .We recall that the generalized Fibonacci number is given by the recurrence F n,k =0 for n < , F ,k = 1 and F n,k = F n − ,k + F n − ,k + . . . + F n − k,k for n > . F n,k is the coefficient of x n in the power series expansion of the expression − x − x − . . . − x k . The Kaiser and Klazar theorem reads as follow:
Theorem 3.2. If C is an hereditary set of permutations, then exactly one of the fourcases occurs.(1) For large n , ϕ C ( n ) is eventually constant.(2) There are integers a , . . . , a k , k ≥ and a k > , such that ϕ C ( n ) = a (cid:18) n (cid:19) + . . . + a k (cid:18) nk (cid:19) for large n . Moreover, ϕ C ( n ) ≥ n for every n. (3) There are constants c, k in N , k ≥ , such that F n,k ≤ ϕ C ( n ) ≤ n c F n,k forevery n .(4) One has ϕ C ( n ) ≥ n − for every n . In the cases (1) to (3) the generating function is rational.
Albert and Atkinson gavein 2005 examples of hereditary classes whose generating function is algebraic [1].In order to state their result, we recall first that a power series F ( x ) := (cid:80) n ≥ a n .x n with a n in C is algebraic if there exists a nonzero polynomial Q ( x, y ) in C [ x, y ] such that Q ( x, F ( x )) = 0 . The series F is rational if Q has degree 1 in y , that is, F ( x ) = R ( x ) /S ( x ) for two polynomials in C [ x ] where S (cid:54) = 0 . Recall next that apermutation π = a a . . . a n of [ n ] is simple if no proper interval of [ n ] ( (cid:54) = [ n ] , ∅ or { x } ) is transformed into an interval. In other words, { a i , a i +1 , . . . , a j } is not aninterval in [ n ] for every ≤ i < j ≤ n , and either i (cid:54) = 1 or j (cid:54) = n . If n ≤ all permutations are simple and called trivial. Albert and Atkinson’s theorem is thefollowing: Theorem 3.3. If C is an hereditary class of permutations containing only finitelymany simple permutations, then the generating series of C , namely (cid:80) n (cid:61) ϕ C ( n ) x n is algebraic. As an illustration of this result, let us mention that the class of permutationsnot above and contain no non trivial simple permutation (these permu-tations are called separable permutations ). The generating series of this class is − x − √ − x + x (see [3]).The simple permutations of small degree are , , , , . Let S n bethe number of simple permutations of [ n ] . The values of S n for n = 1 to are: LASSES OF ORDERED STRUCTURES 5 , , , , , , [32]. Asymptotically, S n goes to n ! e , a result obtained indepen-dently in [24, 2].3.2. Permutations and bichains.
Let σ be a permutation of [ n ] . To σ we associatethe bichain C σ := ([ n ] , ≤ , ≤ σ ) where ≤ is the natural order on [ n ] and ≤ σ the linearorder defined by i ≤ σ j if and only if σ ( i ) ≤ σ ( j ) .For example, let σ be the permutation of given by the sequence of its values: . The sequence of elements of ordered according to ≤ σ is: < σ < σ < σ < σ < σ < σ < σ < σ < σ . Hence, this is the sequence ofvalues of σ − , the inverse of σ . Let us represent σ by its graph in the product n × n ,that is the set G ( σ ) := { ( i, σ ( i )) : i ∈ n } and order this set componentwise, thatis set ( i, σ ( i )) ≤ ( j, σ ( j )) if i ≤ j and σ ( i ) ≤ σ ( j ) . Since σ is bijective, the poset G ( f ) is the intersection of two linear orders, given respectively by the natural orderon the first and on the second coordinate. If we identify each i to ( i, σ ( i )) , the orderinduced on n is the intersection of ≤ and ≤ σ . See Figure 1. F IGURE
1. Representation of a permutation of ten elements.
Lemma 3.4. (1) If B := ( E, L , L ) is a finite bichain then B is isomorphic toa bichain C σ for a unique permutation σ on [ | E | ] . (2) If σ and π are two permutations then σ ≤ π if and only if C σ ≤ C π . The correspondence between permutations and bichains was noted by Cameron[10] (who rather associated to σ the pair ( ≤ , ≤ σ − )) . It allows to study classesof permutations by means of the theory of relations. In particular, via this corre-spondence, hereditary classes of permutations correspond to hereditary classes of D.OUDRAR AND M.POUZET bichains and, as we will see below, simple permutations correspond to indecompos-able bichains.4. I
NDECOMPOSABILITY AND LEXICOGRAPHIC SUM
Let R := ( E, ( ρ i ) i ∈ I ) be a binary structure. A subset A of E is an interval of R if for each i ∈ I : ( xρ i a ⇔ xρ i a (cid:48) ) and ( aρ i x ⇔ a (cid:48) ρ i x ) for all a, a (cid:48) ∈ A and x / ∈ A. The empty set, the singletons and the whole set E are intervals and said trivial . If R has no non trivial interval it is indecomposable . For example, if R := ( E, ≤ ) isa chain, its intervals are the ordinary intervals. If R := ( E, ≤ , ≤ (cid:48) ) is a bichain then A is an interval of R if and only if A is an interval of ( E, ≤ ) and ( E, ≤ (cid:48) ) . Hence: Fact 1.
A permutation σ is simple if and only if the bichain C σ is indecomposable. The notion of indecomposability is rather old. The notion of interval goes back toFra¨ıss´e [14], see also [15]. A fundamental decomposition result of a binary struc-tures into intervals was obtained by Gallai [16] (see [12] for further extensions).Hence, it is not surprising that several results on simple permutations were alreadyknown (for example their asymptotic evaluation). Albert and Atkinson result re-casted in terms of relational structures asserts that if C is an hereditary class offinite bichains containing only finitely many indecomposable bichains then the gen-erating series of C is algebraic . The paper [21] contains several examples of infinitebichains B whose infinitely many members of Age ( B ) are indecomposable.We will establish an extension to ordered binary structures in Theorem 5.7.In the sequel, we recall the facts we need on lexicographic sums and the links withthe indecomposability notion. Some are old (the notion of lexicographic sum goesback to Cantor).Let R := ( E, ( ρ i ) i ∈ I ) be a binary structure and F := ( S x ) x ∈ E be a family of binarystructures S x := ( E x , ( ρ ix ) i ∈ I ) , indexed by the elements of E . We suppose that E and the E x are non-empty. The lexicographic sum of F over R , denoted by ⊕ x ∈R S x ,is the binary structure T obtained by replacing each element x ∈ E by the structure S x . More precisely, T = ( Z, ( τ i ) i ∈ I ) where Z := { ( x, y ) : x ∈ E, y ∈ E x } and foreach i ∈ I , ( x, y ) τ i ( x (cid:48) , y (cid:48) ) if either x (cid:54) = x (cid:48) and xρ i x (cid:48) or x = x (cid:48) and yρ ix y (cid:48) .Trivially, if we replace each S x by an isomorphic binary structure S (cid:48) x , then ⊕ x ∈R S (cid:48) x is isomorphic to ⊕ x ∈R S x . Hence, we may suppose that the domains of the S x ’s arepairwise disjoint. In this case we may slightly modify the definition above, setting Z := ∪ x ∈ E E x and for two elements z ∈ E x and z (cid:48) ∈ E x (cid:48) , zτ i z (cid:48) if either x (cid:54) = x (cid:48) and xρ i x (cid:48) or x = x (cid:48) and zρ ix z (cid:48) . With this definition, each set E x is an interval of the sum T . Furthermore, let Z/ ≡ be the quotient of Z made of blocks of this partition into intervals, let p : Z → Z/ ≡ be the natural projection and let R (cid:48) be the image of T (that is R (cid:48) = ( Z/ ≡ , ( ρ (cid:48) i ) i ∈ I ) LASSES OF ORDERED STRUCTURES 7 where ρ (cid:48) i = { ( p ( x ) , p ( x )) : ( x , x ) ∈ ρ i } . If we identify each block E x to theelement x, then R and R (cid:48) coincide on pairs of distinct elements. They coincide ifwe consider only reflexive relations. Conversely, if T := ( Z, ( τ i ) i ∈ I ) is a binarystructure and ( E x ) x ∈ E is a partition of Z into non empty intervals of T , then T isthe lexicographic sum of ( T (cid:22) E x ) x ∈ E over the quotient Z/ ≡ . In simpler words:
Fact 2.
The decompositions of a binary structure into lexicographic sums are incorrespondence with the partitions of its domain into intervals.
An important property of these decompositions is the following:
Fact 3.
The set of partitions of E into intervals of R , once ordered by refinement,is a sublattice of the set of partitions of E . Let us illustrate. Let us say that a lexicographic sum ⊕ x ∈R S x is trivial if | E | = 1 or | E x | = 1 for all x ∈ E , otherwise it is non trivial ; also a binary structure is sum-indecomposable if it can not be isomorphic to a non trivial lexicographic sum.We have immediately: Fact 4.
A binary structure is sum-indecomposable if and only if it is indecompos-able.
Proposition 4.1.
Let R be a finite binary structure with at least two elements. Then R is isomorphic to a lexicographic sum ⊕ x ∈S R x where S is indecomposable with atleast two elements. Moreover, when S has at least three elements, the partition of R into intervals is unique. If the set S in Proposition 4.1 has two elements, then the decomposition is notnecessary unique, a fact which leads to the notion of strong interval. We recall thatan interval A of a binary structure R is strong if it is non-empty and overlaps noother interval, meaning that if B is an interval such that A ∩ B (cid:54) = ∅ then either B ⊆ A or A ⊆ B . We say that A is maximal if it is maximal for inclusion amongstrong intervals distinct from the domain E of R . The maximal strong intervalform a partition of E , provided that some maximal exists; in this case E is non-limit [18] or, equivalently, robust [11]. Evidently, this partition exists whenever E is finite. The reader will easily check that when this partition exists and thequotient is indecomposable then every other non-trivial partition into intervals isfiner. Hence, in Proposition 4.1 above, if S has at least three elements, the intervalsin the decomposition are strong and thus the decomposition is unique.Let us say that R := ( E, ( ρ i ) i ∈ I ) is chainable if there is a linear order, ≤ , on E suchthat, for each i , xρ i y ⇔ x (cid:48) ρ i y (cid:48) for every x, y, x (cid:48) , y (cid:48) such that x ≤ y ⇔ x (cid:48) ≤ y (cid:48) . If R is reflexive, this amounts to the fact that each ρ i is either the equality relation (cid:52) E ,the complete relation E × E or a linear order; moreover if ρ i and ρ j are two linearorders, they coincide or are opposite. Note that if furthermore R is ordered then the ρ i ’ which are linearly ordered are equal or opposite to the given order.We arrive to the fundamental decomposition theorem of Gallai [16] (see for ex-ample [18], [12], [11] for extensions to infinite structures) D.OUDRAR AND M.POUZET
Theorem 4.2.
Let R be a finite binary structure with at least two elements, then R is a lexicographic sum ⊕ x ∈S R x where S is either indecomposable with at leastthree elements or a chainable binary structure with at least two elements and the V ( R x ) ’s are strong maximal intervals of R . For our purpose, we need to introduce the following notion.Let τ be an ordered structure with two elements. An ordered structure S is said τ - indecomposable if it cannot be decomposed into a lexicographic sum indexed by τ. If A is a class of structures, we denote by A ( τ ) the set of members of A whichare τ -indecomposable. Lemma 4.3.
Let S := ( { , } , ≤ , ( ρ i ) i ∈ J ) , with < , be an ordered structure withtwo elements. If R is a lexicographic sum ⊕ x ∈S R x and R is S -indecomposable,then the partition R , R is unique. We extend the notion of lexicographic sum to collections of non-empty binary struc-tures. Given a non-empty binary structure R and classes A x of non-empty binarystructures for each x ∈ R , let us denote by ⊕ x ∈R A x the class of all binary structuresof the form ⊕ x ∈R S x with S x ∈ A x . If A x = A for every x ∈ V ( R ) we denote thisclass by ⊕ R A . If R := ( { , } , ≤ , ( ρ i ) i ∈ J ) , with < , A := A and A := B , weset ⊕ x ∈R A x = A ⊕ R B . Also if A and B are two classes of binary structures, we set ⊕ A B := { ⊕ x ∈R S x : R ∈ A , S x ∈ B for each x ∈ V ( R ) } .We say that a collection C of binary structures is sum-closed if ⊕ C C ⊆ C . The sum-closure cl ( C ) of C is the smallest sum-closed set that contains C . If we define C := C and C n +1 := ⊕ C C n , then cl ( C ) = ∞ ∪ n =1 C n . If C is a class of bichains, cl ( C ) isalso a class of bichains and the class of corresponding permutations is said wreath-closed [1]. If C is made of reflexive structures and contains a one element structure,say , then cl ( C ) = ⊕ cl ( C ) cl ( C ) . If in addition C contains the empty structure then cl ( C ) is hereditary.We denote by Ind (Ω I ) the collection of finite indecomposable members of Ω I . If R is a binary structure, we denote by Ind ( R ) the collection of its finite inducedsubstructures which are indecomposable. For example, if R is a cograph or a serie-parallel poset then the members of Ind ( R ) have at most two elements (a graph(undirected) is a cograph if no induced subgraph is isomorphic to P , the path on vertices, and a poset is serie-parallel if its comparability graph is a cograph).Let D be a hereditary class of Ind (Ω I ) . Set (cid:80) D := {R ∈ Ω I : Ind ( R ) ⊆ D } . Theorem 4.4.
If all members of D are reflexive, then (cid:80) D = cl ( D ) . Proof.
Inclusion (cid:80) D ⊇ cl ( D ) holds under assumption that all members of D arereflexive. Conversely, if R / ∈ cl ( D ) then either R is indecomposable in which case R / ∈ D or R can not be expressed as a lexicographic sum of structures of D hence R / ∈ (cid:80) D . LASSES OF ORDERED STRUCTURES 9
In the sequel, we consider only ordered structures made of reflexive binary rela-tions. Let Γ d be the subclass of reflexive members of Θ d .Let A be a subclass of Γ d ; for i ∈ N let A ( i ) , resp. A ( ≥ i ) , be the subclass made ofits members which have i elements, resp. at least i elements. Lemma 4.5.
Let D be a class made of non-empty indecomposable members of Γ d such that D (1) is reduced to the one-element structure . Let A be the sum-closure of D and for each S ∈ D (2) , let A ( S ) be the subclass of S -indecomposable membersof A . Set A S := ⊕ S A if S ∈ D ( ≥ and otherwise set A S := A ( S ) ⊕ S A if S ∈ D (2) and S := ( { , } , ≤ , ( ρ i ) i ∈ J ) with < . Then: (4.1) A = { } ∪ (cid:91) S∈ D ( ≥ ) A S and (4.2) A ( S ) = A \ A S for every S ∈ D (2) .Furthermore, all sets in equation (4.1 ) are pairwise disjoint. Proof.
Let’s denote by (1) (respectively by (2) ) the left-hand side (respectivelythe right-hand side) of Equation 4.1. Inclusion (2) ⊆ (1) is obvious because A issum-closed according to Theorem 4.4. To prove inclusion (1) ⊆ (2) , let R be in (1) , if R has one element then it is in (2) , otherwise, according to Theorem 4.2, R is a lexicographic sum ⊕ x ∈S R x where S is either indecomposable with at least threeelements or a chainable binary structure with at least two elements and each R x ∈ A is a strong interval of R for each x ∈ S . In the first case, R is in (cid:83) S∈ D ( ≥ ) ⊕ x ∈S A ,hence in (2) . In the second case, S is chainable with n elements, n ≥ , and wemay set S := ( { , , · · · , n − } , ≤ , ( ρ i ) i ∈ I ) with < < · · · < n − . Set S (cid:48) := S (cid:22) { , } , S (cid:48)(cid:48) := S (cid:22) { , ··· ,n − } , R (cid:48) := R and R (cid:48) := ⊕ x ∈S (cid:48)(cid:48) R x . We have obviously R = R (cid:48) ⊕ S (cid:48) R (cid:48) . Since R is a strong interval of R , R (cid:48) is S (cid:48) -indecomposable, hence R belongs to (cid:83) S (cid:48) ∈ D (2) ( A ( S (cid:48) ) ⊕ S (cid:48) A ) which is a subset of (2) . The fact that these setsare pairwise disjoint follows from Proposition 4.1 and Lemma 4.3.Equality 4.2 is obvious: the S -indecomposable members of A are those whichcannot be writen as S -sums.In the sequel, we count. Our structures being ordered we may choose a uniquerepresentative of an n -element structure on the set { , . . . , n − } , the ordering beingthe natural order. Let H and K be the generating series of A and D ( ≥ and let K ( H ) be the series obtained by substituting the indeterminate x by H . Let H A ( S ) and A S be the generating series of A ( S ) and A S for S ∈ D ( ≥ . And let p be the cardinalityof D (2) . Lemma 4.6. (4.3) ( p − H + ( x − K ( H )) H + x + K ( H ) = 0 . (4.4) H A ( S ) = H H for every S ∈ D (2) . Proof.
Let us prove that Equation 4.4 holds. Let
S ∈ D (2) . Since by definition in Lemma4.5, A S = A ( S ) ⊕ S A , we have H A S = H A ( S ) . H . From Equation (4.2), we deduce H A S = H − H A S . H . Since the coefficients of H are non-negative, the series H is invertible, hence H A S = H H as claimed in Equation (4.4).Let us prove that Equation 4.3 holds. Let S ∈ D ( n ) , with n ≥ . Since by definitionin Lemma 4.5, A S = ⊕ S A , we have H A S = H n . From this, we deduce that thegenerating series of (cid:83) S∈ D ( ≥ A S is equal to K ( H ) . From Equation (4.4), we deducethat the generating series of H A S is H H . Hence the generating series of (cid:83) S∈ D (2) A S isequal to p H H .Substituting these values in Equation (4.1), we obtain(4.5) H = x + p H H + K ( H ) . A straightforward computation yields Equation (4.3).Let us say that a class of finite structures is algebraic if its generating series isalgebraic.
Corollary 4.7.
Let D be a class made of non-empty indecomposable members of Γ d such that D (1) is reduced to the one-element structure . If D is algebraic thenits sum-closure and the subclass A S consisting of the S -indecomposable membersof the sum-closure A are algebraic for each S ∈ D (2) , .
5. W
ELL - QUASI - ORDERED HEREDITARY CLASSES
Let C be a subclass of Ω µ and A be a poset. Set C . A := { ( R , f ) : R ∈ C , f : V ( R ) → A} and ( R , f ) ≤ ( R (cid:48) , f (cid:48) ) if there is an embedding h : R → R (cid:48) such that f ( x ) ≤ f (cid:48) ( h ( x )) for all x ∈ V ( R ) .We recall that A is well-quasi ordered (wqo) if A contains no infinite antichain andno infinite descending chain. We say that C is hereditary wqo if C . A is wqo forevery wqo A . It is clear that every class which is hereditary wqo is wqo. If C isreduced to a single structure R , it is hereditary wqo provided that R is finite (thisfollows from the fact that if A is wqo then its power A n ordered coordinatewise iswqo for each integer n ). If R is infinite, this does not hold. Also, a finite unionof hereditary wqo classes is hereditary wqo; hence every finite subclass C of Ω µ ishereditary wqo. LASSES OF ORDERED STRUCTURES 11
A longstanding open question ask whether C is hereditary wqo whenever the class C . of the elements of C labelled by , the -element antichain, is wqo.If Ch is the class of finite chains, Ch. A identifies to the set A ∗ of finite wordsover the alphabet A equipped with the Higman ordering. The fact that Ch is hered-itary wqo is a famous result due to Higman [17]. We also note the following fact: Fact 5.
If a subclass C of Ω µ (with I finite) is hereditary wqo, then ↓ C , the leasthereditary subclass of Ω µ containing C , is hereditary wqo. We recall the following result of [26].
Theorem 5.1.
If the signature is finite, a subclass of Ω µ which is hereditary andhereditary wqo has finitely many bounds. Behavior of the profile of special hereditary classes, the ages of Fra¨ıss´e, and the linkwith wqo classes were considered by the second author in the early seventies (see[27] and [28] for a survey). The case of graphs, tournaments and other combinato-rial structures was elucidated more recently (see the survey of [20]).
Proposition 5.2.
If a hereditary class D of Ind (Ω I ) is hereditary wqo then (cid:80) D ishereditary wqo and (cid:80) D has finitely many bounds. Proof.
The second part of the proposition follows from Theorem 5.1 above.In our case of binary structures, we may note that the proof is straightforward.The first part uses properties of wqo posets, and follows from Higman’ theoremon algebras preordered by divisibility (1952) [17]. Instead of recalling the resultwe give a direct proof. Let A a poset which is wqo and consider ( (cid:80) D ) . A . If ( (cid:80) D ) . A is not wqo, then according to one of preliminary result of Higman, itcontains some non finitely generated final segment ( F is a final segment if x ∈ F and x ≤ y imply y ∈ F ). According to Zorn lemma, there is a maximal one,say F , with respect to inclusion among final segments having this property. Let I := ( (cid:80) D ) . A \ F be the complement of F in ( (cid:80) D ) . A . The set I is then wqo.Let R := ( R , f ) , · · · , ( R n , f n ) , · · · be an infinite antichain of minimal elementsof F . As D . A is wqo because D is hereditary wqo, we can suppose that no ele-ment of this antichain is in D . A . Then, according to Proposition 4.1 and Theorem4.2, for every i ≥ there exists an indecomposable structure S i and non-emptystructures ( R ix ) x ∈ V ( S i ) such that R i = ⊕ x ∈S i R ix . Since ( R ix , f i (cid:22) V ( R ix ) ) strictlyembeds into ( R i , f i ) we have ( R ix , f i (cid:22) V ( R ix ) ) ∈ I for every i ≥ and x ∈ S i .Since I is wqo, and D is hereditary wqo, D . I is wqo, thus the infinite sequence ( S , g ) , · · · , ( S i , g i ) , · · · of D . I , where g i ( x ) := ( R ix , f i (cid:22) V ( R ix ) ) , contains anincreasing pair ( S p , g p ) ≤ ( S q , g q ) for some p < q . Which means that there isan embedding h : S p → S q such that g p ( x ) ≤ g q ( h ( x )) for all x ∈ V ( S p ) ,that is ( R px , f p (cid:22) V ( R px ) ) ≤ ( R qh ( x ) , f q (cid:22) V ( R qh ( x ) ) ) for all x ∈ S p . It follows that ( R p , f p ) = ⊕ x ∈S p ( R px , f p (cid:22) V ( R px ) ) ≤ ⊕ x ∈S q ( R qx , f q (cid:22) V ( R qx ) ) = ( R q , f q ) which con-tradicts that R is an antichain. Thus ( (cid:80) D ) . A is wqo and hence ( (cid:80) D ) is hereditarywqo. Proposition 5.2 particulary holds if D is finite. If D is the class Ind k (Ω I ) of inde-composable structures of size at most k then according to a result of Schmerl andTrotter, 1993 ([31]), the bounds of (cid:80) Ind k (Ω I ) have size at most k + 2 . When D is made of bichains, Proposition 5.2 was obtained by Albert and Atkinson [1].An immediate corollary is: Corollary 5.3.
If a hereditary class of Ω I contains only finitely many indecompos-able members then it is wqo and has finitely many bounds. We say that a class C of relational structures is hereditary rational , resp. hereditaryalgebraic if the generating function of every hereditary subclass of C is rational,resp. algebraic. Albert, Atkinson and Vatter [3] proved that hereditary rationalclasses of permutations are wqo. This fact can be extended to hereditary algebraicclasses. Lemma 5.4.
A hereditary class C which is hereditary algebraic is wqo. Proof. If C contains an infinite antichain, there are uncountably many heredi-tary subclasses of C and in fact an uncountable chain of subclasses; these classesprovides uncountably many generating series. Some of these series cannot be al-gebraic. Indeed, according to C. Retenauer [29], a generating series with rationalcoefficients which is algebraic over C is algebraic over Q . Since the generatingseries we consider have integer coefficients, there are algebraic over Q , hence thereare only countably many such series.If C and D are two hereditary classes, then the generating series satisfy the iden-tity H C ∪ D = H C + H D − H C ∩ D . From this simple equality we have: Lemma 5.5.
The union of two hereditary rational (resp. algebraic) classes is hered-itary rational (resp. algebraic).
Corollary 5.6.
A minimal non-hereditary rational or a minimal non-hereditary al-gebraic class C is the age of some relational structure. Proof.
According to Lemma 5.5, C cannot be the union of two proper hereditarysubclasses, hence this is an ideal, thus an age. Theorem 5.7.
Let d be an integer. If an hereditary class C of Γ d contains onlyfinitely many indecomposable members then it is algebraic. We follows essentially the lines of Albert-Atkinson proof. We do an inductive proofover the hereditary subclasses of C . But for that, we need to prove more, namelythat C and each C ( S ) for S ∈
Ind ( C ) (2) , are algebraic (this is the only differencewith Albert-Atkinson proof). To avoid unessential complications, we take out theempty relational structure of Γ d , that is we suppose that C is made of non-emptystructures. Let A := (cid:80) Ind ( C ) . If C = A then by Corollary 4.7, C and each C ( S ) for S ∈
Ind ( C ) (2) , are algebraic. Thus the result is proved. If C (cid:54) = A , we maysuppose that for each proper hereditary subclass C (cid:48) of C , both C (cid:48) and C (cid:48) ( S ) for each S ∈
Ind ( C (cid:48) ) (2) , are algebraic. Indeed, otherwise, since by Corollary 5.3, C is wqo,it contains a minimal hereditary subclass not satisfying this property and we mayreplace C by this subclass. Let S ∈
Ind ( C ) (2) . Let C ( S ) be the subclass of C made LASSES OF ORDERED STRUCTURES 13 of S -indecomposable members of C . Let and , with < , be the two elementsof V ( S ) , we set C S := ( C ( S ) ⊕ S C ) ∩ C . Let S ∈
Ind ( C ) ( ≥ we set C S := ( ⊕ S C ) ∩ C .As in Lemma 4.5 we have(5.1) C = { } ∪ (cid:91) S∈ Ind ( C ) ( ≥ ) C S and(5.2) C ( S ) = C \ C S for every S ∈
Ind ( C ) (2) . Let H and K be the generating series of C and Ind ( C ) ( ≥ respectively.Let H C (2) and H C ( ≥ be the generating series of C (2) := (cid:83) S∈ Ind ( C ) (2) C S and of C ( ≥ := (cid:83) S∈ Ind ( C ) ( ≥ C S .We have:(5.3) H C = x + H C (2) + H C ( ≥ and(5.4) H C ( S ) = H C − H C S for every S ∈ D (2) . We deduce that H C and H C ( S ) are algebraic for every S ∈
Ind ( C ) (2) , from thefollowing claims that we will prove afterwards Claim 1.
The generating series of H C ( ≥ is a polynomial in the generating series H C whose coefficients are algebraic series. Claim 2.
For each
S ∈
Ind ( C ) (2) , the generating series H C ( S ) of C ( S ) is either alinear polynomial in the generating series H C of the form (5.5) H C ( S ) = (1 − α ) H C − δ β ; whose coefficients are algebraic series or is a rational fraction of the form (5.6) H C ( S ) = H C H C . Substituting in formula 5.3 the values of H C ( ≥ and H C (2) given by Claim 1 andClaim 2 we obtain a polynomial in H C whose coefficients are algebraic series. Thispolynomial is not identical to zero. Indeed, it is the sum of a polynomial A = a + a H C + a H C and B = b + b H C + · · · + a k H k C whose coefficients arealgebraic series (in fact, B = H C ( ≥ (1 + H C ) ). The valuation of A and B as seriesin x are distinct. Indeed, the valuation of A is (notice that a = x + δ where δ iseither zero or an algebraic series of valuation at least ). Hence, if B (cid:54) = 0 (when Ind ( C ) ( ≥ is non empty) its valuation is at least . Since A and B don’t have thesame valuation, then A + B is not identical to zero. Being a solution of a non zeropolynomial, H C is algebraic. With this result and claim 2, H C ( S ) is algebraic. Withthis, the proof of Theorem 5.7 is complete. In order to prove our claims, we need the following lemmas (respectively Lemma15 and Lemma 18 in [1]).
Lemma 5.8.
Let S be an indecomposable ordered structure and A := ( A x ) x ∈S , B := ( B x ) x ∈S be two sequences of subclasses of ordered binary structures indexedby the elements of S . If S has at least three elements then ( ⊕ x ∈S A x ) ∩ ( ⊕ x ∈S B x ) = ⊕ x ∈S ( A x ∩ B x ) . If S := ( { , } , ≤ , ( ρ i ) i ∈ J ) with < then ( A ( S ) ⊕ S A ) ∩ ( B ( S ) ⊕ S B ) = ( A ( S ) ∩ B ( S )) ⊕ S ( A ∩ B ) . Proof.
The first equality follows from Proposition 4.1 and the second one followsfrom Lemma 4.3.Let C be a class of finite structures and B := B , B , ..., B l be a sequence of finitestructures, we will set C < B > := C < B , B , ..., B l > := F orb ( {B , B , ..., B l } ) ∩ C . If C is hereditary, a proper hereditary subclass C (cid:48) of C is strong if every bound of C (cid:48) in C is embeddable in some bound of C . Note that the intersection of strongsubclasses is strong.Let A := ( A x ) x ∈S , where S is indecomposable with at least three elements. A decomposition of a binary structure B over A is a map h : B → S such that B = ⊕ x ∈S (cid:22) range ( h ) B (cid:22) h − ( x ) and B (cid:22) h − ( x ) ∈ A x for all x ∈ range ( h ) . Hence, each B (cid:22) h − ( x ) is an interval of B . Let H B be the set of all such decompositions of B . Lemma 5.9.
Let S be an indecomposable ordered structure, A := ( A x ) x ∈S be asequences of subclasses of ordered binary structures indexed by the elements of S , B := B , B , ..., B l be a sequence of finite structures, and C := ( ⊕ x ∈S A x ) < B > . If S has at least three elements then C is a union of sets of the form ⊕ x ∈S D x where each D x is either A x < B > or one of its strong subclasses. Proof.
We prove the result for l = 1 and we set B := B . For that we prove that:(5.7) C = (cid:92) h ∈ H B (cid:91) x ∈ range ( h ) ⊕ y ∈S A ( x ) y where A ( x ) x := A x < B (cid:22) h − ( x ) > and A ( x ) y := A y for y (cid:54) = x .Let’s call by (1) (respectively by (2) ) the left-hand side (respectively the right-hand side) of Equation 5.7. Inclusion (1) ⊆ (2) holds without any assumption.Indeed, let T in (1) . We prove that T is in (2) . If h is a decomposition of B ,we want to find x ∈ range ( h ) such that T ∈ ⊕ y ∈S A ( x ) y . Since T is in (1) , it has adecomposition over S . Let h ∈ H B , since B (cid:2) S , there exist x ∈ S (cid:22) range ( h ) suchthat B (cid:22) h − ( x ) (cid:2) T , hence T ∈ ⊕ y ∈S A ( x ) y . LASSES OF ORDERED STRUCTURES 15
Inclusion (2) ⊆ (1) holds under the assumption that a structure in (2) has a uniquedecomposition over S and that it is ordered, (what means that S is rigid, that is S hasno automorphism distinct from the identity). Let T in (2) , then for every h ∈ H B there exist x h ∈ range ( h ) such that T ∈ ⊕ y ∈S A ( x h ) y , thus, T ∈ (cid:84) h ∈ H B ⊕ y ∈S A ( x h ) y . Wehave
T ∈ ⊕ y ∈S A y because, A ( x h ) y ⊆ A y for every h. Hence, T = ⊕ y ∈S T y . We claimthat B (cid:2) T . Suppose B ≤ T let f be an embedding of B into T and h := pof ,where p is the projection map from T into S , we must have B (cid:22) h − ( x ) ≤ T x for x ∈ rang ( h ) which is a contradiction with the fact that T ∈ (cid:84) h ∈ H ⊕ y ∈S A ( x h ) y . Using distributivity of intersection over union, we may write (2) as a union of terms,each of which is an intersection of terms like ⊕ y ∈S A y < B x > , where B x is an intervalof B such that, there exist a decomposition h of B and B x = B (cid:22) h − ( x ) . Theseintersections, by lemma 5.8 and the fact that among all decompositions of B are allones which send B into a single element x of S have the form ⊕ x ∈S D x where each D x is of the form A x < B , · · · > where the structures occurring after B (if any) areintervals of B . Hence, D x is either A x < B > or one of its strong subclasses. Thecase l > follows by induction. Proof of Claim 1.
Since A is wqo and C is a proper hereditary subclass, we have C = A < B > for some finite family B := B , B , ..., B l of elements of A . Let S ∈
Ind ( C ) ( ≥ , Lemma 5.9 asserts that C S is an union of classes, not necessarilydisjoint, of the form ⊕ x ∈S C x where each C x is either C or one of its strong subclasses.The generating series of ⊕ x ∈S C x is a monomial in the generating series H C of C whosecoefficient is a product of generating series of proper strong subclasses of C . Fromthe induction hypothesis, the generating series of of these strong subclasses arealgebraic series, hence this coefficient is an algebraic series. Using the principle ofinclusion-exclusion, we get that the generating series H C S of C S is a polynomial inthe generating series H C whose coefficients are algebraic series. Since the C S ’s arepairwise disjoint, the generating series H C ( ≥ is also a polynomial in the generatingseries H C whose coefficients are algebraic series. (cid:50) Lemma 5.10. If S has two elements and , S := ( { , } , ≤ , ( ρ i ) i ∈ J ) with < ,then ( A ( S ) ⊕ S A ) < B > is an union of classes of the form ( A (cid:48) ( S ) < B > ) ⊕ S ( A (cid:48)(cid:48) < B > ) , where A (cid:48) < B > and A (cid:48)(cid:48) < B > are either equal to A < B > or to somestrong subclasses of A < B > . Proof.
As above we suppose first l = 1 . Equation 5.7 yields(5.8) (cid:0) A ( S ) ⊕ S A (cid:1) < B > = (cid:92) h ∈ H B (cid:20)(cid:0) A ( S ) < B (cid:22) h − (0) > ⊕ S A ) (cid:91) ( A ( S ) ⊕ S ( A < B (cid:22) h − (1) > )) (cid:21) An induction take care of the case l > . Proof of Claim 2.
Let
S ∈
Ind ( C ) ( ≥ . Lemma 5.10 asserts that C S is an union ofclasses, not necessarily disjoint, of the form C (cid:48) ( S ) ⊕ S C (cid:48)(cid:48) , where C (cid:48) and C (cid:48)(cid:48) are eitherequal to C or to some strong subclasses of C . The generating series of these classesare of the form H C ( S ) H C or α H C or β H C ( S ) , where α and β are algebraic series.Using the principle of inclusion-exclusion, we get that the generating series H C S iseither of the form H C ( S ) H C or of the form α H C + β H C ( S ) + δ , where α, β and δ are algebraic series. In particular H C S is of the form α S H C ( S ) + β S where α S and β S are polynomials in H C of degree at most with algebraic series as coefficients.Using Equation 5.2 we obtain(5.9) H C ( S ) = H C H C ; when all bounds B i of C in A are S -indecomposable or(5.10) H C ( S ) = (1 − α ) H C − δ β ; if at least one bound B i is not S -indecomposable. (cid:50) The conclusion of Theorem 5.7 above does not hold with structures which arenot necessarily ordered.
Example 5.11.
Let K ∞ , ∞ be the direct sum of infinitely many copies of the completegraph on an infinite set. As it is easy to see the generating function of Age ( K ∞ , ∞ ) is the integer partition function. This generating series is not algebraic. However, Age ( K ∞ , ∞ ) contains no indecomposable member with more than two elements.More generally, note that the class F orb ( P ) of finite cographs contains no inde-composable cograph with more than two vertices and that this class is not hered-itary algebraic. Finite cographs are comparability graphs of serie-parallel posetswhich in turn are intersection orders of separable bichains. By Albert-Atkinson’stheorem, the class of these bichains is hereditary algebraic. This tells us that alge-braicity is not necessarily preserved by the transformation of a class into an othervia a process as above ( processes of this type are the free-operators of Fra¨ıss´e [13] ).
6. A
CONJECTURE AND SOME QUESTIONS
In their paper [1], Albert and Atkinson indicate that there are infinite sets of sim-ple permutations whose sum closure is algebraic but, as it turns out, some hereditarysubclasses are not necessarily algebraic. An example is the collection of decreasingoscillations (see the end of the section). In order to extend their proof to some otherclasses, they ask whether there exists an infinite set of simple permutations whosesum-closure is well quasi ordered . As we indicate in Proposition 6.1 below, the setof exceptional permutations has this property. In fact, it is hereditary wqo. We guessthat this notion of hereditary wqo is the right concept for extending Albert-Atkinsontheorem.
LASSES OF ORDERED STRUCTURES 17
Exceptional permutations correspond to bichains which are critical in the senseof Schmerl and Trotter. Let us recall that a binary structure R with domain E is critical if R is indecomposable but R (cid:22) E \{ x } is not indecomposable for every x ∈ E . Schmerl and Trotter [31] gave a description of critical posets. They fallinto two infinite classes: P := {P n : n ∈ N } and P (cid:48) := {P (cid:48) n : n ∈ N } where P n := ( V n , ≤ n ) , V n := { , . . . , n − } × { , } , ( x, i ) < n ( y, j ) if i < j and x ≤ y ; P (cid:48) n := ( V n , ≤ (cid:48) n ) and ( x, i ) < (cid:48) n ( y, j ) if j ≤ i and x < y .These posets are two-dimensional. That is, they are intersection of two linear orderswhich are respectively L n, := (0 , < (0 , < · · · < ( i, < ( i, · · · < ( n − , < ( n − , and L n, := ( n − , < · · · < ( n − i, < · · · < (0 , < ( n − , < · · · < ( n − i, · · · < (0 , for P n and L (cid:48) n, := L n, and L (cid:48) n, := ( L n, ) ∗ for P (cid:48) n . As it is well known, an indecomposable two-dimensional poset P := ( V, L ) hasa unique realizer (that is there is a unique pair { L , L } of linear orders whoseintersection is the order L of P ). Hence, there are at most two bichains, namely ( V, L , L ) and ( V, L , L ) such that L ∩ L = L . The critical posets describedabove yield four kind of bichains, namely ( V n , L n, , L n, ) , ( V n , L n, , L n, ) , ( V n , L n, , ( L n, ) ∗ ) and ( V n , ( L n, ) ∗ , L n, ) . These bichains are critical. Indeed, a bichain is indecompos-able if and only if the intersection order is indecomposable ([30] for finite bichainsand [33] for infinite bichains). The isomorphic types of these bichains are describedin Albert and Atkinson’s paper in terms of permutations of , . . . , m for m ≥ : ( i ) 2 . . .... m. . . .... m − . ( ii ) 2 m − . m − .... . m. m − .... . ( iii ) m + 1 . .m + 2 . .... m.m. ( iv ) m. m.m − . m − .... .m + 1 . For example, the type of the bichain ( V m , L m, , L m, ) is the permutation given in ( iv ) , whereas the type of ( V m , L m, , L m, ) is its inverse, given in ( ii ) (enumerate theelements of V m into the sequence , . . . , m , this according to the order L m, , thenreorder this sequence according to the order L m, ; this yields the sequence σ − := σ − (1) , . . . , σ − (2 m ) ; according to our definition the type of ( V m , L m, , L m, ) is thepermutation σ , this is the one given in ( iv ) ). For m = 2 , the permutations given in ( i ) and ( iv ) coincide with whereas those given in ( ii ) and ( iii ) coincide with ; for larger values of m , they are all different.The four classes of indecomposable bichains are obtained from B := { ( V n , L n, , L n, ) : n ∈ N } by exchanging the two orders in each bichain or by reversing the order ofthe first one, or by reversing the second one. Hence the order structure w.r.t. em-bedabbility of these classes is the same, and it remains the same if we label theelements of these bichains. Proposition 6.1.
The class of critical bichains is hereditary wqo.
Proof.
This class is the union of four classes hence, in order to prove that it ishereditary wqo, it suffices to prove that each one of these classes is hereditary wqo.According to the observation above, it suffices to prove that one, for example B , is hereditary wqo. Let A be a wqo poset. We have to prove that B . A is wqo. For that,set B := A , where A := { e : { , } → A} , and order B componentwise. Let B ∗ be the set of all words over the ordered alphabet B . We define an order preservingmap F from B ∗ onto B . A . This will suffice. Indeed, B is wqo as a product of twowqo sets; hence, according to Higman theorem on words over ordered alphabets[17], B ∗ is wqo. Since B . A is the image of a wqo by an order preserving map, itis wqo. We define the map F as follows. Let w := w (0) w (1) · · · w ( n − ∈ B ∗ .Set F ( w ) := ( R , f w ) ∈ B . A where R := ( V n , L n, , L n, ) and f w ( i, j ) := w ( i )( j ) for j ∈ { , } . We observe first that w ≤ w (cid:48) in B ∗ implies F ( w ) ≤ F ( w (cid:48) ) in B . A .Indeed, if w ≤ w (cid:48) there is an embedding h of the chain < · · · < n − into thechain < · · · < n (cid:48) − such that w ( i ) ≤ w (cid:48) ( h ( i )) for all i < n . Let h : { , . . . , n − } × { , } → { , . . . , n (cid:48) − } × { , } defined by setting h ( i, j ) := ( h ( i ) , j ) . Asit is easy to check, h is an embedding of F ( w ) into F ( w (cid:48) ) . Next, we note that F is surjective. Indeed, if ( R , f ) ∈ B . A with R := ( V n , L n, , L n, ) , then the word w := w (0) w (1) · · · w ( n − with w ( i )( j ) := f ( i, j ) yields F ( w ) = ( R , f ) .With Proposition 5.2, we have: Corollary 6.2.
The sum-closure of the class of critical bichains is wqo.
In [1] it is mentioned that this class has finitely many bounds. The generating seriesof the class of critical bichains is rational (the class is covered by four chains).According to Corollary 13 of [1] their sum-closure is algebraic.
Question 1.
Is the sum-closure of the class of critical bichains hereditary alge-braic?
We conjecture that the answer is positive. This will be a consequence of a con-jecture for hereditary classes of ordered binary structures that we formulate below.
Conjecture 6.3. If D is a hereditary class of indecomposable ordered binary struc-tures which is hereditary wqo and hereditary algebraic, then its sum-closure ishereditary algebraic. The requirement that D is wqo will not suffice in Conjecture 6.3.Indeed, let P Z be the doubly infinite path whose vertex set is Z and edge set E := { ( n, m ) ∈ Z × Z : | n − m | = 1 } . The edge set E has two transitive orientations, e.g. P := { ( n, m ) ∈ Z × Z : | n − m | = 1 and n is even } and its dual P ∗ . As an order, P is the intersection of the linear orders L := · · · < n < n − < n + 1) < n + 1 < · · · and L := · · · < n + 1) < n + 3 < n < n + 1 < · · · . Let C := ( Z , L , L ) and D := Ind ( C ) . Lemma 6.4. D is wqo but not hereditary wqo. Proof.
Members of D of size n are obtained by restricting C to intervals of size n , n (cid:54) = 3 , of the chain ( Z , ≤ ) (observe first that the graph P Z is indecomposableas all its restrictions to intervals of size different from of the chain ( Z , ≤ ) andfurthermore there are no others indecomposable restrictions; next, use the fact thatthe indecomposability of a comparability graph amounts to the indecomposabilityof its orientations [18], and that the indecomposability of a two-dimensional poset LASSES OF ORDERED STRUCTURES 19 amounts to the indecomposability of the bichains associated with the order [33]).Up to isomorphy, there are two indecomposable bichains of size n , n (cid:54) = 3 , namely C n := C (cid:22) { ,...,n − } and C ∗ n := C ∗ (cid:22) { ,...,n − } where C ∗ := ( Z , L ∗ , L ∗ ) . These twobichains embed all members of D having size less than n . Being covered by twochains, D is wqo. To see that D is not hereditary wqo, we may associate to eachindecomposable member of D the comparability graph of the intersection of the twoorders and observe that this association preserves the embeddability relation, eventhough label are added. The class of graphs obtained from this association consistsof paths of size distinct from . It is not hereditary wqo. In fact, as it is immediateto see, if a class G of graphs contains infinitely many paths of distinct sizes, then G . is not wqo. Indeed, if we label the end vertices of each path by and label theother vertices by , we obtain an infinite antichain. Thus D . is not wqo.The generating series of D is rational (its generating function is x + x − x ). In fact, D is hereditary algebraic (every hereditary subclass of D is finite). By Corollary 13of [1], the sum-closure (cid:80) D is algebraic. (in fact, if D is the generating function of (cid:80) D , then D + 2 D − D + (2 − x ) D − D + x = 0 . ). But (cid:80) D is not hereditaryalgebraic. For that, it suffices to observe that it is not wqo and to apply Lemma 5.4.The fact that (cid:80) D is not wqo is because we may embed the poset D . into (cid:80) D viaan order preserving map. A simpler argument consist to observe first that the family ( G n ) n ∈ N , where G n is the graph obtained from the n -vertex path P n by replacing itsend-vertices by a two-vertex independent set, is an antichain, next that these graphsare comparability graphs associated to members of D .The permutations corresponding to the members of D are called decreasing os-cillations . They have been the object of several studies:The downward closure ↓ D is Age ( C ) , the age of C ; this age has four obstructions,it is rational: the generating series is − x − x − x , the generating function being thesequence A05298 of [32], starting by 1, 1, 2, 5, 11, 24. For all of this see [9].6.1. Questions.
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