OOn level-induced suborders
Laurent Lyaudet ∗ March 31, 2020
Abstract
In this article, we characterize orders that are level-induced suborders anytimethey are induced suborders of a superorder. We also characterize orders that areconsecutive level-induced suborders anytime they are level-induced suborders ofa superorder. Thus characterizing orders that are consecutive level-induced subor-ders anytime they are induced suborders of a superorder.
Current version : 2020/03/29Keywords : orders, always level-induced orders, ali orders, naturally consecutivelevel-induced orders, nacli orders, (directed) cographs, transitive series parallel graphs,interval orders, series parallel orders, series parallel interval orders, semi-orders, unitinterval orders, series parallel unit interval orders, 1-weak orders
Apologies: We do science as a hobby, it is not our daily job and there is an impact on thequality of the bibliography. For an unpublished work we did in 2015, we started doingbibliographic search during 9 months, but all the gathered references were lost when ahacker erased all our files on our laptop. Since then, we chose to publish our ideas onarXiv and correct the bibliography afterwards. For this article, we found no prior workdefining kinds of induced suborders with constraints on their levels relatively to those ofthe superorder. Our search was in English and French scientific literature, and since thetopic of order theory is ancient and vast, we may have missed early references in otherlanguages. If you do know an early reference, please be kind enough to email/correctus. This is now the fifth version on arXiv, we are sorry for the errors that we publishedin the preceding four versions; we hope there is none in this version; at least, there issome (slow) progress.This article study Open problem 5.16 in Lyaudet (2019). “Characterize finite ordersthat are induced suborders of any well-founded order if and only if they are (consecu-tive) level-induced suborders of this well-founded order. Examples: chains, antichainsof size 1 and 2. Counter-examples: antichains of size at least 3.” ∗ https://lyaudet.eu/laurent/ , [email protected] a r X i v : . [ m a t h . G M ] M a r ection 2 contains most of the definitions and notations used in this article. Insection 3, we characterize orders that cannot be induced suborders without being level-induced suborders. Section 4 characterize orders that cannot be level-induced subor-ders without being consecutive level-induced suborders. In section 5, we give algo-rithms to find (level-)induced suborders of the previously defined classes. Throughout this article, we use the following definitions and notations. O will bereserved for asymptotic growth of functions. Thus P denotes an order (it may beeither a partial, or a total/linear order), in particular P , denotes the binary total or-der where < . We denote Domain( P ) , the domain of the order P (for exam-ple, Domain( P , ) = { , } ). We write x < y , and x > y as usual to expressthe order between two elements; we also write x ∼ y when two elements are in-comparable in the partial order considered. We denote OrderFunction( P ) , the or-der function of the order P defined from Domain( P ) to { = , ∼ , <, > } (for example, OrderFunction( P , ) = { ((0 , , =) , ((0 , , < ) , ((1 , , > ) , ((1 , , =) } ).We denote Inv( P ) , the inverse/reverse order of P ; for example, Inv( P , ) = P , is the order on 0 and 1 where < . Definition 2.1 (Maximum chain, height) . Let P be an order, a chain of P is maximum if it is maximal and no other chain of P has greater cardinality. The cardinal of amaximum chain is the height of P , denoted Height( P ) . When P is well-founded , weredefine a maximum chain to be one such that the corresponding ordinal is maximum;and we redefine its height to be the ordinal corresponding to its maximum chains. Thusin this case Height( P ) denotes an ordinal. Note that an infinite order may have no maximum chain, but it always have at leastone maximal chain. When there is no maximum chain,
Height( P ) is defined as thesupremum cardinal/ordinal of the cardinals/ordinals corresponding to maximal chains.In a well-founded order P , the level decomposition of P is the function Level P :Domain( P ) → Height( P ) ( Height( P ) is an arbitrary ordinal.) such that ∀ x ∈ Domain( P ) , Level P ( x ) = Supremum(Level P ( y )+1 such that y < x, y ∈ Domain( P )) .(Of course, this supremum is 0 if no element is below x .) We define the level-width of P as the supremum of the cardinals of the levels of P. Given two elements x, y ∈ Domain( P ) , Gap P ( x, y ) = Gap P ( y, x ) = Supremum(Level P ( x ) , Level P ( y )) − Infimum(Level P ( x ) , Level P ( y )) . The gap between two elements is clearly 0 if andonly if these two elements belong to the same level. Note that this ordinal gap may notcorrespond to an actual well-founded chain in P between the two elements, in particu-lar they may have an arbitrary large gap and be incomparable. When there is more thantwo elements, the gap of a set of elements is the supremum of the gaps of the pairs.We consider the following kinds of suborders: Some authors also say Noetherian. In both cases, it means that there is no strictly decreasing infinitesequence. An induced suborder P (cid:48) of an order P is such that Domain( P (cid:48) ) ⊆ Domain( P ) ,and ∀ x, y ∈ Domain( P (cid:48) ) , OrderFunction( P (cid:48) )( x, y ) = OrderFunction( P )( x, y ) .Let X ⊆ Domain( P ) , P [ X ] denotes the suborder of P induced by X . • A level-induced suborder P (cid:48) of a well-founded order P is such that Domain( P (cid:48) ) ⊆ Domain( P ) , ∀ x, y ∈ Domain( P (cid:48) ) , OrderFunction( P (cid:48) )( x, y ) = OrderFunction( P )( x, y ) ,and ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) = Level P (cid:48) ( y ) ⇔ Level P ( x ) = Level P ( y ) .(Note that we could also define two other kinds of level-induced suborders with ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) = Level P (cid:48) ( y ) ⇒ Level P ( x ) = Level P ( y ) ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) = Level P (cid:48) ( y ) ⇐ Level P ( x ) = Level P ( y ) .There is a simple proof by transfinite induction on the levels of P (cid:48) showing that ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) = Level P (cid:48) ( y ) ⇒ Level P ( x ) = Level P ( y ) implies ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) = Level P (cid:48) ( y ) ⇐ Level P ( x ) =Level P ( y ) . Moreover, the same proof shows that ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) < Level P (cid:48) ( y ) ⇔ Level P ( x ) < Level P ( y ) . Thus only two kinds of level-inducedsuborders exists ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) = Level P (cid:48) ( y )( ⇔ or ⇒ ) Level P ( x ) = Level P ( y ) , and ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) = Level P (cid:48) ( y ) ⇐ Level P ( x ) = Level P ( y ) . ) • A consecutive level-induced suborder P (cid:48) of a well-founded order P is such that Domain( P (cid:48) ) ⊆ Domain( P ) , ∀ x, y ∈ Domain( P (cid:48) ) , OrderFunction( P (cid:48) )( x, y ) =OrderFunction( P )( x, y ) , ∀ x, y ∈ Domain( P (cid:48) ) , Gap P (cid:48) ( x, y ) = Gap P ( x, y ) .(In the finite case, the equality of the gaps may be replaced by the followingconditions: ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) = Level P (cid:48) ( y ) ⇔ Level P ( x ) =Level P ( y ) , and ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x )+1 = Level P (cid:48) ( y ) ⇔ Level P ( x )+1 = Level P ( y ) .) In this section, we assume that a given well-founded order P (cid:48) is an induced suborderof a well-founded order P . We study necessary and sufficient conditions on P (cid:48) to havethat P (cid:48) is a level-induced suborder of P . Definition 3.1 (ali orders) . An ali order P (cid:48) is a well-founded order such that whenever P (cid:48) is isomorphic to an induced suborder of a well-founded order P , then P (cid:48) is alsoisomorphic to a level-induced suborder of P . Recall that an initial section I of an order P is a subset of his domain closed bytaking smaller elements: ∀ x ∈ Domain( P ) , ∀ y ∈ I, x < y ⇒ x ∈ I . Given asubdomain Y of P , InitialSection(
P, Y ) = { x ∈ Domain( P ) , ∃ y ∈ Y, x ≤ y } isthe initial section generated by Y . ( I = InitialSection( P, I ) and P [ I ] denotes thesuborder of P induced by I .) Lemma 3.2.
Let P (cid:48) be a well-founded order. • If we have a level-induced suborder isomorphic to P (cid:48) in the suborder induced byan initial section of some well-founded order P , then this level-induced suborderis also level-induced in P . If P (cid:48) is an ali order and it is isomorphic to an induced suborder P (cid:48)(cid:48) of the well-founded order P , it is isomorphic to a level-induced suborder in the restrictionof P to the initial section generated by P (cid:48)(cid:48) . Proof:The first assertion is trivially true because ∀ x ∈ InitialSection(
P, X ) , Level P [InitialSection( P,X )] ( x ) = Level P ( x ) .The second assertion follows from it and the fact that P (cid:48) is an ali order (“relativelyto P [InitialSection( P, Domain( P (cid:48)(cid:48) ))] ”). Corollary 3.3 (ali orders revisited) . An ali order P (cid:48) is a well-founded order such thatwhenever P (cid:48) is isomorphic to an induced suborder P (cid:48)(cid:48) of a well-founded order P , then P (cid:48) is also isomorphic to a level-induced suborder of P [InitialSection( P, Domain( P (cid:48)(cid:48) ))] . We now observe that :
Lemma 3.4.
Any well-founded order P (cid:48) is an induced suborder of a well-foundedorder P of level-width 2. Moreover, P has no level-induced suborder isomorphic to O obs = ( { a, b, c, d } , { a < b, c < d } ) ≡ Inv( O obs ) . Proof:We use a well-founded chain to lift each element of
Domain( P (cid:48) ) to a separatelevel. By Zermelo’s axiom, there is a bijection f between some ordinal α and Domain( P (cid:48) ) , such that ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) < Level P (cid:48) ( y ) ⇒ f ( x ) An ali order has level-width at most 2 and no level-induced suborderisomorphic to O obs = ( { a, b, c, d } , { a < b, c < d } ) ≡ Inv( O obs ) . Lemma 3.6. No ali order has a level of size 2 except maybe the first. Proof: 4ssume for a contradiction that P (cid:48) is an ali order with two elements x, y such that Level P (cid:48) ( x ) = Level P (cid:48) ( y ) > . Take x, y such that their level is minimum. By theprevious corollary, we must have an element z, z < x, z < y . (By transitivity, itis trivial to see that such a z exists in all previous levels, since only the first levelmay have two elements.) Thus we have a level-induced suborder isomorphic to O obs = ( { a, b, c } , { a < b, a < c } ) .We now show how to remove all such level-induced suborders for any well-founded order. Again by Zermelo’s axiom, there is a bijection f between some ordi-nal α and Domain( P (cid:48) ) , such that ∀ x, y ∈ Domain( P (cid:48) ) , Level P (cid:48) ( x ) < Level P (cid:48) ( y ) ⇒ f ( x ) < f ( y ) . This time, we add a distinct chain for each element of Domain( P (cid:48) ) .Let DisjointCopy( i ) be a chain isomorphic to the ordinal i such that its elements areassumed to be distinct from all other elements considered in the following formula: Domain( P ) = Domain( P (cid:48) ) (cid:116) ( (cid:70) i ∈ α DisjointCopy( i )) . • ∀ x, y ∈ Domain( P (cid:48) ) , OrderFunction( P )( x, y ) = OrderFunction( P (cid:48) )( x, y ) , • ∀ x, y ∈ DisjointCopy( i ) , OrderFunction( P )( x, y ) = OrderFunction(DisjointCopy( i ))( x, y ) , • ∀ x ∈ DisjointCopy( i ) , ∀ y ∈ DisjointCopy( j ) , OrderFunction( P )( x, y ) = ‘ ∼ ’, • ∀ x ∈ DisjointCopy( i ) , ∀ y ∈ Domain( P (cid:48) ) , OrderFunction( P )( x, y ) = ‘ < ’ if i = f − ( y ) or f ( i ) < y (OrderFunction( P (cid:48) )( f ( i ) , y ) ∈ { = , < } ) , ‘ ∼ ’ otherwise .Clearly, each element f ( i ) of Domain( P (cid:48) ) is now on a distinct level, since DisjointCopy( i ) is a longest chain below it. Moreover, if some element in DisjointCopy( i ) is lessthan two elements on the same level, then clearly, f ( i ) must be less than thesetwo elements, and this is impossible since f ( i ) may only be less than elements in Domain( P (cid:48) ) , that are now scattered. Theorem 3.7. An ali order is either • a well-founded total order, • an antichain of size 2, • the disjoint union of a well-founded chain and an incomparable element, wherethe well-founded chain has height 2 or is isomorphic to a regular cardinal/ordinalor its successor, • the order composition of two incomparable elements and a well-founded chain(we call this case a “(1,1)-based chain”), • or the order composition of – the disjoint union of a well-founded chain and an incomparable element,where the well-founded chain has height 2 (we call this case a “(2,1)-basedchain”) or is isomorphic to a regular cardinal/ordinal or its successor, and a well-founded chain. Proof:It is trivial to see that a well-founded total order is an ali order.A well-founded chain and an incomparable element may or may not form an aliorder. • Clearly an antichain of size 2 is an ali order. • A chain of height 2 and an incomparable element is an ali order: Indeed,consider x, y, z with x < y, x ∼ z, y ∼ z . – If Level P ( x ) = Level P ( z ) , there is nothing to do. – If Level P ( x ) < Level P ( z ) , then there is another element t such that Level P ( x ) = Level P ( t ) , and t < z (since there is an element in eachlevel below z , such that z is more than it, and x ∼ z ). Clearly, t, z, x give a level-induced suborder isomorphic to a chain of height 2 and anincomparable element. – If Level P ( x ) > Level P ( z ) , then there is another element t such that Level P ( z ) = Level P ( t ) , and t < x (since there is an element in eachlevel below x , such that x is more than it, and x ∼ z ). Clearly, t, y, z (or t, x, z ) give a level-induced suborder isomorphic to a chain of height2 and an incomparable element. Note that the same argument applies towell-founded chains of any height with a lowest element x and an incom-parable element z . • A well-founded chain corresponding to a successor of a successor ordinal α +2 more than 2 ( α > ) and an incomparable element is not an ali order. (Such achain is ended by a chain of height 2 on two consecutive levels α and α + 1 .)Indeed, consider the order P with Domain( P ) = DisjointCopy( α + 2) (cid:116) DisjointCopy( α + 1) , such that: – ∀ x, y ∈ DisjointCopy( α +2) , OrderFunction( P )( x, y ) = OrderFunction(DisjointCopy( α +2))( x, y ) , – ∀ x, y ∈ DisjointCopy( α +1) , OrderFunction( P )( x, y ) = OrderFunction(DisjointCopy( α +1))( x, y ) , – ∀ x ∈ DisjointCopy( α +1) , ∀ y ∈ DisjointCopy( α +2) , OrderFunction( P )( x, y ) = ‘ < ’ if x corresponds to an ordinal less than y and x does not correspond to α, ‘ ∼ ’ otherwise .Any chain in P isomorphic to α + 2 must have its greatest element to be thegreatest element corresponding to α +1 in DisjointCopy( α +2) . Then clearlythe only element incomparable with it is the greatest element corresponding to α in DisjointCopy( α + 1) . Thus, since α > it cannot yield a level-inducedsuborder. • A well-founded chain corresponding to (a successor of) a limit ordinal α thatis singular (not a regular cardinal/ordinal) and an incomparable element is notan ali order. (Such a chain is isomorphic to γ = α (resp. γ = α + 1 ).) Since6 is not a regular cardinal, let β + 1 < γ be a successor ordinal such that α \ β is not isomorphic to α . Now, consider the order P with Domain( P ) =DisjointCopy( γ ) (cid:116) DisjointCopy( β + 1) , such that: – ∀ x, y ∈ DisjointCopy( γ ) , OrderFunction( P )( x, y ) = OrderFunction(DisjointCopy( γ ))( x, y ) , – ∀ x, y ∈ DisjointCopy( β +1) , OrderFunction( P )( x, y ) = OrderFunction(DisjointCopy( β +1))( x, y ) , – ∀ x ∈ DisjointCopy( β +1) , ∀ y ∈ DisjointCopy( γ ) , OrderFunction( P )( x, y ) = ‘ < ’ if x corresponds to an ordinal less than y and x does not correspond to β, ‘ ∼ ’ otherwise .Since β + 1 < γ , any chain in P isomorphic to γ must have its final segmentin DisjointCopy( γ ) . Then clearly the only element incomparable with it isthe greatest element corresponding to β in DisjointCopy( β + 1) . Thus, since β > , and α \ β is not isomorphic to α , it cannot yield a level-inducedsuborder. • A well-founded chain corresponding to (a successor of) a regular limit ordi-nal/cardinal α and an incomparable element is an ali order. (Such a chain isisomorphic to γ = α (resp. γ = α + 1 ).) Let ( x i ) i ∈ γ and an incomparable ele-ment z form such an induced suborder in some partial order P . From what wenoted for the case of a chain of height 2 and an incomparable element, we justneed to consider the case where Level P ( x ) < Level P ( z ) . Clearly, there isa chain ( y j ) j ∈ Level P ( z ) below z intersecting all levels below Level P ( z ) , suchthat, by transitivity, x i (cid:54) < y j , i ∈ γ, j ∈ Level P ( z ) . – If Level P ( z ) ≥ Supremum(Level P ( x i ) , i ∈ γ ) , then ( y j ) j ∈ Level P ( z ) ,j ≥ Level P ( x ) contains a subchain isomorphic to γ , and x ∼ y j , j ∈ Level P ( z ) , j ≥ Level P ( x ) . Taking the subchain starting on the same level than x (made of the elements ( y j ) j ∈ Level P ( z ) ,j ≥ Level P ( x ) ,j ∈{ Level( x i ) ,i ∈ γ } ), to-gether with z on top of this subchain (if needed) and x , we obtain thesought level-induced suborder. – If Level P ( x ) < Level P ( z ) < Supremum(Level P ( x i ) , i ∈ γ ) , it ismore complicated. If ( x i ) i ∈ γ, Level P ( x i ) ≥ Level P ( z ) is reduced to the sin-gleton x α (hence γ = α + 1 ), then we are in a situation equivalent tothe previous case Level P ( z ) ≥ Supremum(Level P ( x i ) , i ∈ γ ) , but weshould replace it by Level P ( z ) > Level P ( x i ) , ∀ i ∈ α ; it is clear in thatcase that ( y j ) j ∈ Level P ( z ) ,j ≥ Level P ( x ) contains a subchain isomorphic to α starting on the same level than x ; together with z on top of this sub-chain and x , it yields the sought level-induced suborder.Assume that no element x i , i ∈ γ belongs to Level P ( z ) . There may be noelement in Level P ( z ) ordered with all elements x i , i ∈ γ ; nevertheless,there is a lowest element x k such that Level P ( x k ) > Level P ( z ) ; andthere is an element x (cid:48) k < x k such that Level P ( x (cid:48) k ) = Level P ( z ) . If thereis an element x i , i ∈ γ that belongs to Level P ( z ) , we also name thiselement x (cid:48) k .Clearly, { x (cid:48) k } ∪ ( x i ) i ∈ γ, Level P ( x i ) ≥ Level P ( z ) is cofinal in a chain isomor-phic to γ , and all elements of the chain are incomparable with z . More-over, even in the case γ = α + 1 , we can remove the element x α of7his chain, and still obtain a chain cofinal in a chain isomorphic to α ,since we already studied the case where ( x i ) i ∈ γ, Level P ( x i ) ≥ Level P ( z ) isreduced to the singleton x α . In both cases, by regularity of α , the sub-chain { x (cid:48) k } ∪ ( x i ) i ∈ γ, Level P ( x i ) ≥ Level P ( z ) is isomorphic to γ . Thus, to-gether with the incomparable element z , it yields the sought level-inducedsuborder.The only case left to study is then when the second element on level 0, y , isless than some element of the chain ( x i ) i ∈ γ . Let ( x i ) i ∈ α be the initial segment ofelements that are incomparable with y .Let P be an order containing the chain ( x i ) i ∈ γ and the element y with appro-priate order relationship. If ( x i ) i ∈ α and y form an ali order, then clearly by tran-sitivity all elements in P [InitialSection( P, { x i , i ∈ α } ∪ { y } )] will be less thanelements ( x i ) i ∈ γ \ α . Thus, a level-induced suborder isomorphic to ( x i ) i ∈ α and y in P [InitialSection( P, { x i , i ∈ α } ∪ { y } )] , together with the chain ( x i ) i ∈ γ \ α , willimmediately yield a level-induced suborder isomorphic to ( x i ) i ∈ γ and y .If ( x i ) i ∈ α and y does not form an ali order, consider an order P counter containing ( x i ) i ∈ α and y with appropriate order relationship such that no level-induced subor-der isomorphic to ( x i ) i ∈ α and y exists. Let P be the order composition of P counter and the chain ( x i ) i ∈ γ \ α . Clearly, any element of ( x i ) i ∈ γ \ α is comparable with allother elements of P . Thus, no (level-)induced suborder of P isomorphic to ( x i ) i ∈ α and y may contain an element of ( x i ) i ∈ γ \ α . Hence, no level-induced suborder iso-morphic to ( x i ) i ∈ α and y exists in P , and no level-induced suborder isomorphic to ( x i ) i ∈ γ and y exists in P .This ends this technical proof. Corollary 3.8. An ali order is a well-founded order without induced suborder isomor-phic to O obs = ( { a, b, c, d } , { a < b, c < d } ) ≡ Inv( O obs ) , O obs = ( { a, b, c } , { a
Lemma 4.2. For any well-founded order P (cid:48) containing an induced suborder isomor-phic to O obs = ( { a, b, c } , { a < b, a < c } ) or O obs = ( { a, b, c } , { a > b, a > c } ) ,there is a well-founded order P such that P (cid:48) is a level-induced suborder of P , but P (cid:48) is not isomorphic to any consecutive level-induced suborder of P . Proof:We use a disjoint sum of well-founded chains of same height to lift each level of P (cid:48) so that any two levels of P (cid:48) are now γ levels apart, where γ ≥ ω β +1 , andthe cardinal of Domain( P (cid:48) ) is at most ℵ β . Again by Zermelo’s axiom, there isa bijection f between some ordinal α and Domain( P (cid:48) ) . We add a distinct well-founded chain for each element of Domain( P (cid:48) ) . Let DisjointCopy( i ) be a chainisomorphic to the ordinal i such that its elements are assumed to be distinct from allother elements considered in the following formula: Domain( P ) = Domain( P (cid:48) ) (cid:116) ( (cid:70) i ∈ α DisjointCopy( γ × Level P (cid:48) ( f ( i )))) . • ∀ x, y ∈ Domain( P (cid:48) ) , OrderFunction( P )( x, y ) = OrderFunction( P (cid:48) )( x, y ) , • ∀ x, y ∈ DisjointCopy( γ × Level P (cid:48) ( f ( i ))) , OrderFunction( P )( x, y ) = OrderFunction(DisjointCopy( γ × Level P (cid:48) ( f ( i ))))( x, y ) , • ∀ x ∈ DisjointCopy( γ × Level P (cid:48) ( f ( i ))) , ∀ y ∈ DisjointCopy( γ × Level P (cid:48) ( f ( j ))) , OrderFunction( P )( x, y ) = ‘ ∼ ’, 11 ∀ x ∈ DisjointCopy( γ × Level P (cid:48) ( f ( i ))) , ∀ y ∈ Domain( P (cid:48) ) , OrderFunction( P )( x, y ) = ‘ < ’ if i = f − ( y ) or f ( i ) < y (OrderFunction( P (cid:48) )( f ( i ) , y ) ∈ { = , < } ) , ‘ ∼ ’ otherwise .Clearly, P (cid:48) is a level-induced suborder of P , and any two levels of P (cid:48) are now γ levels apart, since DisjointCopy( γ × Level P (cid:48) ( f ( i ))) is a longest chain belowelement f ( i ) .Let ( x, y, z ) be a triple of elements of Domain( P ) , such that x < y, x < z, y ∼ z ( O obs ), or x > y, x > z, y ∼ z ( O obs ). It naturally defines one ordinal Gap P ( x, y, z ) = Supremum(Gap P ( x, y ) , Gap P ( x, z )) .Observe that no element of DisjointCopy( γ × Level P (cid:48) ( f ( i ))) is more than anelement, unless that element is also in DisjointCopy( γ × Level P (cid:48) ( f ( i ))) . Hence,it cannot be more than two incomparable elements.Clearly, if it is less than two incomparable elements like x , then these two ele-ments are in Domain( P (cid:48) ) , and x ∈ { f ( i ) } (cid:116) DisjointCopy( γ × Level P (cid:48) ( f ( i ))) implies that f ( i ) is also less than these two elements. Moreover, Gap P ( x, y, z ) ≥ Gap P ( f ( i ) , y, z ) ≥ γ , in that case.Thus, if there is an induced suborder isomorphic to O obs = ( { a, b, c } , { a b, a > c } ) in P (cid:48) . Consider such an induced suborder ( x, y, z ) in P . We alreadynoted that x must be in P (cid:48) ; if both y, z are in DisjointCopy( γ × Level P (cid:48) ( f ( k ))) ,then they are comparable, a contradiction. Hence, without loss of generality, y ∈{ f ( j ) }(cid:116) DisjointCopy( γ × Level P (cid:48) ( f ( j ))) , for some f ( j ) (cid:54) = x, f ( j ) ∈ Domain( P (cid:48) ) .It is now trivial to see that ( { x, f ( j ) , z } , { x > f ( j ) , x > z } ) is also an inducedsuborder isomorphic to O obs = ( { a, b, c } , { a > b, a > c } ) with Gap P ( x, y, z ) ≥ Gap P ( x, f ( j ) , z ) . But since Gap P ( x, f ( j ) , z ) ≥ γ , again we have that no consec-utive level-induced suborder isomorphic to P (cid:48) exists in P . Corollary 4.3. A nacli order is the disjoint union of well-founded chains. Lemma 4.4. No nacli order has more than one level of size at least 2. Proof:Again, we create a gap between consecutive levels of P (cid:48) . We use a unique well-founded chain of height γ × Height( P (cid:48) ) to lift all levels of P (cid:48) so that any two levelsof P (cid:48) are now γ levels apart, where γ ≥ ω β +1 , and the cardinal of Domain( P (cid:48) ) isat most ℵ β . Domain( P ) = Domain( P (cid:48) ) (cid:116) DisjointCopy( γ × Height( P (cid:48) )) . Sinceadded levels have size 1 and original levels are too far apart, at most one level canhave size more than one in a consecutive level-induced suborder.12 heorem 4.5. A nacli order is a well-founded chain, an antichain, or the disjoint unionof a well-founded chain and an antichain. Equivalently, a nacli order is a well-founded order without induced suborder isomorphic to O obs = ( { a, b, c, d } , { a < b, c Acknowledgements We thank God: Father, Son, and Holy Spirit. We thank Maria. They help us throughour difficulties in life. References C. Crespelle and C. Paul. Fully dynamic recognition algorithm and certificate for di-rected cographs. Discrete Applied Mathematics , 154(12):1722–1741, 2006. doi: 10.1016/j.dam.2006.03.005. URL https://doi.org/10.1016/j.dam.2006.03.005 .M. J. Fischer, R. A. DeMillo, N. A. Lynch, W. A. Burkhard, and A. V. Aho, editors. 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