Geometry of antimatroidal point sets

Geometry of an timatroidal p oin t sets Y ulia Kempner 1 1 Holon Institute of T ec hnology , ISRAEL yulia k @hit.ac.il V adim E. Levit 1 , 2 2 Ariel Univ ersi t y Center of Sa maria, ISRAEL levitv @ariel.ac. il Abstract The notion of ”ant imatroid with rep etition” w as conceiv ed b y Bjorner, Lo v asz and Shor in 1991 as a multiset extension of the n o - tion of an timatroid [2]. When the und e r l yin g set consists of only t wo elemen ts, suc h t wo-dimensional anti matroids corresp ond to p oin t sets in the plane. I n this researc h w e concen trate on geometrica l prop er- ties of ant imatroidal p o int sets in th e plane and prov e that these sets are exactly parallelogram p oly omino es. Our results imply that t wo- dimensional an timatroids ha ve co nv ex d imension 2 . The second part of the researc h is dev oted to geometrica l pr op erties of three-dimensional an timatroids closed und er intersectio n . Keyw ords: an timatroid, con v ex dimension, l attice animal, p oly- omino. 1 Preliminaries An antimatroid is an accessible set system closed under union [3 ]. An al- gorithmic c hara cterization of an timat roids based on the language definition w as introduced in [6]. Another algo rithmic c haracterization of antimatroids that depicted them as set systems w as dev elop ed in [12]. While classical examples of a n timatroids connect them with p osets, c hordal graphs, con v ex 1 geometries, etc., game theory give s a framew ork in whic h an timatroids are in terpreted as p ermission structures for coalitions [1]. There are also ric h connections b et w een an timatroids and cluster analysis [13]. In mathematical psyc hology , an timat r o ids are used to desc rib e feasible states of kno wledge of a human learner [10]. In this p a p er we in ve stigate the corresp ondence betw een antimatroids and p oly omino es. In the digital pla ne Z 2 , a p olyomino [11 ] is a finite connected union of unit squares without cut p oints. If we replace eac h unit square of a p oly omino b y a v ertex at its cen ter, w e obtain an equiv alen t ob ject named a lattic e animal [5]. F urther, w e use t he name p oly omino for the tw o equiv alen t ob jects. A p o lyomino is called column-conv ex (resp. row-con v ex) if all its columns (resp. rows ) are connected (in other words, eac h column/ro w has no holes). A con v ex p oly omino is b oth row - and column-con vex . The parallelogram p oly omino es [7, 8, 9], sometimes known as staircase p olygons [4, 17], are a particular case of this family . They are defin ed b y a pair of monotone paths made only with north and east steps, such that the paths are disjoin t, except at their common ending p oin ts. In this pap er w e prov e that a n timatroida l p oin t sets in the plane and parallelogram polyominoes are equiv alen t. Let E b e a finite set. A set system o ver E is a pair ( E , F ), where F is a family of sets ov er E , called fe asible sets. W e will use X ∪ x for X ∪ { x } , and X − x for X − { x } . Definition 1.1 [15]A finite non-empty set system ( E , F ) is an antimatr oid if ( A 1) for e ach non-empty X ∈ F , ther e exists x ∈ X such that X − x ∈ F ( A 2) for al l X , Y ∈ F , and X * Y , ther e e xists x ∈ X − Y such that Y ∪ x ∈ F . An y set system satisfying ( A 1) is called a c c e ssible. In addition, w e use the following c hara cterization of antimatroids. Prop osition 1.2 [15] F or an ac c essible set system ( E , F ) the fol lowing state- ments ar e e quivalent: ( i ) ( E , F ) is an antimatr oid ( ii ) F is close d under union ( X , Y ∈ F ⇒ X ∪ Y ∈ F ) Consider another prop ert y of antimatroids. 2 Definition 1.3 A set system ( E , F ) satisfies the chain pr op erty if for al l X , Y ∈ F , and X ⊂ Y , ther e exists a chain X = X 0 ⊂ X 1 ⊂ ... ⊂ X k = Y such that X i = X i − 1 ∪ x i and X i ∈ F for 0 ≤ i ≤ k . It is easy to see that the chain prop erty follow s from ( A 2), but these prop erties are not equiv alen t. A p oly-an tima t r oid [16] is a generalization of the notion of the antimatroid to m ultisets. A p oly-antimatr oid is a finite non-empty multiset system ( E , S ) that satisfies the an timatroid prop erties ( A 1) and ( A 2). Let E = { x, y } . In this case eac h p o in t A = ( x A , y A ) in the digital plane Z 2 ma y b e considered as a multiset A o v er E , where x A is a n um b er of rep etitions of an elemen t x , and y A is a num ber of rep etitions of an elemen t y in m ultiset A . Consider a set o f p oints in the digital plane Z 2 that satisfies the prop erties of an an timatroid. That is a t w o-dimensional p oly-antimatroid. Definition 1.4 A set of p oi n ts S in the digital pla n e Z 2 is an antimatr oidal p oint set if ( A 1) for every p oint ( x A , y A ) ∈ S , such that ( x A , y A ) 6 = (0 , 0) , either ( x A − 1 , y A ) ∈ S or ( x A , y A − 1) ∈ S ( A 2) for al l A * B ∈ S , if x A ≥ x B and y A ≥ y B then either ( x B + 1 , y B ) ∈ S or ( x B , y B + 1) ∈ S if x A ≤ x B and y A ≥ y B then ( x B , y B + 1 ) ∈ S if x A ≥ x B and y A ≤ y B then ( x B + 1 , y B ) ∈ S Accessib ility implies that ∅ ∈ S . F or example, see an an timat r o idal po in t set in Figure 1 . ✲ ✻ s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s x y 0 1 2 1 A B Figure 1: An antimatroidal point set. 3 A three-dimensional p oin t set is defined similarly to a t wo-dimens io nal set. 2 Tw o-dimens ional an t i matroidal p oint se t s and p oly o mino es In this section w e consider a geometric c haracterization o f tw o-dimensional an timatroida l p oin t sets. The follo wing notation [14] is used. If A = ( x, y ) is a p oin t in a digital plane, the 4-neighb orho o d N 4 ( x, y ) is the set of p oints N 4 ( x, y ) = { ( x − 1 , y ) , ( x, y − 1) , ( x + 1 , y ) , ( x, y + 1) } and 8-ne i g hb orho o d N 8 ( x, y ) is the set of p oints N 8 ( x, y ) = { ( x − 1 , y ) , ( x, y − 1) , ( x + 1 , y ) , ( x, y + 1) , ( x − 1 , y − 1) , ( x − 1 , y + 1) , ( x + 1 , y − 1 ) , ( x + 1 , y + 1) } . Let m b e any of the n um b ers 4 or 8. A sequenc e A 0 , A 1 , ..., A n is called an N m - p ath if A i ∈ N m ( A i − 1 ) for eac h i = 1 , 2 , ...n . Any tw o p oin ts A, B ∈ S are said to be N m -connected in S if t here exists an N m -path A = A 0 , A 1 , ..., A n = B from A to B suc h t ha t A i ∈ S for eac h i = 1 , 2 , ...n . A digital set S is an N m - c onne cte d set if a ny t wo po ints P , Q from S are N m -connected in S . An N m - c onne cte d c omp onent o f a set S is a maximal subset of S , whic h is N m -connected. An N m -path A = A 0 , A 1 , ..., A n = B f r om A to B is called a monotone incr e asing N m -path if A i ⊂ A i +1 for a ll 0 ≤ i < n , i.e., ( x A i < x A i +1 ) ∧ ( y A i ≤ y A i +1 ) or ( x A i ≤ x A i +1 ) ∧ ( y A i < y A i +1 ) . The c ha in prop ert y and the fact that the family of feasible sets of a n an timatroid is closed under union mean that for eac h t w o p oin ts A, B : if B ⊂ A , then there is a monotone increasing N 4 -path from B to A , and if A is non-comparable with B , then there is a monotone increasing N 4 -path from b oth A and B to A ∪ B = (max( x A , x B ) , max( y A , y B )). In particular, for each A ∈ S there is a monotone decreasing N 4 -path from A to 0. So , w e can conclude that an an timatro idal p oin t set is an N 4 -connected comp onen t in the digital plane Z 2 . 4 Definition 2.1 A p oint set S ⊆ Z 2 is define d to b e ortho gonal ly c o nvex if, for every line L that is p ar al lel to the x-axi s ( y = y ∗ ) or to the y-ax i s ( x = x ∗ ), the interse ction of S with L is empty, a p oint, or a single interval ( [( x 1 , y ∗ ) , ( x 2 , y ∗ )] = { ( x 1 , y ∗ ) , ( x 1 + 1 , y ∗ ) , ..., ( x 2 , y ∗ ) } ). It follo ws immediately from the c hain prop ert y that an y an timatroidal p oint set S is a n orthogona lly con vex conne cted comp onent . Th us, an timatroidal p oint sets are conv ex polyominoes. In the follo wing w e pro v e that antimatroidal p oint sets in the plane closed not only under union, but under in tersection as w ell. Lemma 2.2 An antimatr oidal p oint s e t in the p l a ne is close d under interse c- tion, i.e., if two p oints A = ( x A , y A ) and B = ( x B , y B ) b elong to an antima- tr oidal p oi nt set S , then the p oint A ∩ B = ( min ( x A , x B ) , min ( y A , y B )) ∈ S . Pro of. The prop osition is eviden t for tw o comparable po ints. Consider tw o non-comparable p oints A and B , and a ssume without loss of generality that x A < x B and y A > y B . Then there is a monotone decreasing N 4 -path from A to 0, and so there is a po int C = ( x C , y B ) ∈ S on this path with x C ≤ x A . Hence, the p oin t A ∩ B b elongs to S , since it is lo cated on the monotone increasing N 4 -path from C to B . Lemma 2.2 implies that ev ery an timatroida l p oin t set is a union of rect- angles built on eac h pair o f non-comparable p oints. The following theorem sho ws that antimatroidal p oin t sets in the plane and parallelogram p oly omi- no es are equiv a lent. Theorem 2.3 A set of p oints S in the digital plane Z 2 is a n antimatr oidal p oint set if and o n ly if it is an ortho gonal ly c onvex N 4 -c onne cte d set that is b ounde d by two monotone incr e asing N 4 -p aths b etwe en (0 , 0) and ( x max , y max ) . T o prov e the ”if” part of Theorem 2.3 it remains to demonstrate that ev ery an timatroidal p oin t set S is b ounded b y t wo monotone increasing N 4 - paths b etw een (0 , 0) and ( x max , y max ). T o give a definition of the b oundary w e begin with the following notions. A p o in t A in set S is called an interior p oint in S if N 8 ( A ) ∈ S . A p oint in S which is not an in terior p oint is called a b oundary p oint . All b oundary p oints o f S constitute the b oundary of S . W e can see a n antimatroidal p oint set with its boundary in Figure 2. 5 ✲ ✻ s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s x y B 0 B 1 B 4 B 8 B 18 = ( x max , y max ) B 10 Figure 2: A b oundary of an an timatroidal p oin t set. Since an timatroidal sets in the plane are closed under union a nd under in tersection, there ar e six ty p es of b oundary p oints that w e divide into t w o sets – lo we r and upp er boundary: B low er = { ( x, y ) ∈ S : ( x + 1 , y ) / ∈ S ∨ ( x, y − 1) / ∈ S ∨ ( x + 1 , y − 1) / ∈ S } B upper = { ( x, y ) ∈ S : ( x − 1 , y ) / ∈ S ∨ ( x, y + 1) / ∈ S ∨ ( x − 1 , y + 1) / ∈ S } It is p ossible that B low er ∩ B upper 6 = ∅ . F or example, t he p oin t B 10 in Figure 2 b elongs to b oth t he low er and upp er boundar ies. Lemma 2.4 The lower b oundary is a monotone incr e as i n g p a th b etwe en (0 , 0) and ( x max , y max ) . Pro of. Let B 0 = (0 , 0) , B 1 , ..., B k = ( x max , y max ) b e a lexicographical order of B low er . W e will pro v e t ha t it is a monot one increasing path. Supp ose the opp osite. Then there is a pa ir of non-comparable p oin ts B i and B j , suc h that i < j , x B i < x B j and y B i > y B j . Then Lemma 2.2 implies that there is a rectangle built on the pair of non-comparable p oints. Hence, the po int B i do es not b elong to the lo wer b oundary , whic h is a con tradiction. Lemma 2.5 F or e ach B k = ( x, y ) ∈ B low er holds: (i) if A = ( x, z ) ∈ S and A / ∈ B low er then z > y ; (ii) if A = ( z , y ) ∈ S and A / ∈ B low er then z < x . Pro of. Supp o se the opp osite, i.e., there is A = ( x, z ) ∈ S suc h that A / ∈ B low er and z < y . Then t he p oint ( x + 1 , z ) ∈ S , so there is the la st right p oint ( x ∗ , z ) ∈ S , suc h that x ∗ > x . Hence, the p oint ( x ∗ , z ) b elongs to B low er , con tradicting Lemma 2 .4. The Prop ert y ( ii ) is prov ed similarly . 6 Lemma 2.6 The lower b oundary is an N 4 -p ath b etwe en (0 , 0) and ( x max , y max ) . Pro of. Prov e tha t | B i +1 − B i | = 1 for every 1 ≤ i < k . If x B i = x B i +1 . then from the c hain pro p ert y and Lemma 2.5 it immediately fo llo ws that y B i +1 = y B i + 1 . If x B i < x B i +1 , then the c hain prop erty implies t ha t either ( x B i + 1 , y B i ) ∈ S or ( x B i , y B i + 1) ∈ S . Since the p oin t ( x B i , y B i + 1) / ∈ B low er , w e can conclude that ( x B i + 1 , y B i ) ∈ S in a n y case. No w Lemma 2.4 implies that this p oin t b elongs to B low er , i.e., B i +1 = ( x B i + 1 , y B i ) . The upper b oundary case is v alidat ed in the same wa y . Th us an an timatroidal point set is an orthogonally con v ex N 4 -connected set b ounded b y t w o monotone increas ing N 4 -paths. The follo wing lemma is the ”only-if ” part o f Theorem 2.3. Lemma 2.7 An ortho gonal ly c onvex N 4 -c onne cte d set S that is b ounde d by two mo notone incr e asing N 4 -p aths b etwe en (0 , 0) and ( x max , y max ) is an an- timatr oidal p oint set. Pro of. By Definition 1.4 we ha v e t o c hec k the tw o prop erties (A1) and (A2): (A1) Let A = ( x, y ) ∈ S . If A is an in terior p oint in S then ( x − 1 , y ) ∈ S and ( x, y − 1 ) ∈ S . If A is a b oundar y p oin t, then the previous p oint on the b oundary (( x, y − 1) or ( x − 1 , y )) b elongs t o S . (A2) Let A * B ∈ S . Consider t w o cases: (i) x a ≥ x b and y a ≥ y b . If B is an in terior p oin t in S then ( x b + 1 , y b ) ∈ S and ( x b , y b + 1) ∈ S . If B is a b oundary p oint, then the next p o in t on the b oundary (( x b , y b + 1 ) or ( x b + 1 , y b )) b elongs to S . (ii) x a ≤ x b and y a ≥ y b . W e hav e to pro v e that ( x b , y b + 1) ∈ S . Supp ose the opp osite. Then the p oin t B is an upp er b oundary p oint. Since y a ≥ y b there exists an upp er b oundary p oint ( x a , y ) with y ≥ y b that con tradicts the monotonicit y of the b o undar y . Corollary 2.8 Any antimatr oidal p oint set S may b e r epr esente d by its b oundary in the fol lo w ing form: S = B low er ∨ B upper = { X ∪ Y : X ∈ B low er , Y ∈ B upper } This result sho ws that the c onvex dimen s ion of a tw o-dimensional p oly- an timatroid is t w o. 7 Definition 2.9 Conv e x dim ension [15 ] c dim( S ) of an y antimatr o i d S is the minimum numb er of maxima l chains ∅ = X 0 ⊂ X 1 ⊂ ... ⊂ X k = X max with X i = X i − 1 ∪ x i whose union gives the a n timatr oid S . The b oundary of an y p oly-antimatroid ma y b e found b y using the fol- lo wing algorithm. Sta rting with the maxim um p o in t ( x max , y max ) it follo ws the upp er b oundary do wn to p oin t (0 , 0) in t he fir st pass, and it follows the lo w er b oundary in the second pass. Algorithm 2.10 Upp er b oundary tr acing algorithm 1. i := 0 ; x := x max ; y := y max ; 2. B i := ( x, y ) 3. do 3.1 if ( x − 1 , y ) ∈ S then x := x − 1 , else y := y − 1 ; 3.2 i := i + 1 ; 3.3 B i := ( x, y ) ; until B i = (0 , 0) 4. R eturn the se quenc e B = B i , B i − 1 , ..., B 0 It is easy to c hec k tha t Algorithm 2.10 returns a monotone increas- ing N 4 -path that only passes o v er the upp er b oundary p oints from B upper (for eac h i p oin t B i = ( x i , y i ) has the maxim um y − co ordinate, i.e., y i = arg max y { ( x i , y ) ∈ S } ). Hence, the Upp er b o undary tracing algorithm r e- turns t he upp er b oundary of p oly-a n timatroids. The Lo we r b oundary tracing algorit hm differs from Algorithm 2.10 on searc h order only: ( y , x ) instead of ( x, y ). Step 3.1 o f Algorithm 2.10 will b e as follo ws: 3 . 1 if ( x, y − 1) ∈ S then y := y − 1 ,else x := x − 1; Corresp ondingly , the L o we r b oundary tracing algorithm returns the low er b oundary of p oly-an timatr o ids. It turned out that a tw o-dimensional p oly-antimatroid kno wn in this sec- tion as an antimatroidal p oin t set is equiv a lent to sp ecial cases of p olyominoes or staircase p olygons and is defined b y a pair of monotone paths made only with north and east steps. In the next section w e researc h the path structure of three-dimensional antimatroidal p oin t sets. 8 3 Three-dimen s ional an timatroidal p oint set In this section w e consider a particular case of three-dimensional a n tima- troidal point sets. Definition 3.1 A finite non-empty set C of p oints in d igital 3 D sp ac e is a digital cub oid if C = { ( x, y , z ) ∈ Z 3 : x min ≤ x ≤ x max ∧ y min ≤ y ≤ y max ∧ z min ≤ z ≤ z max } and | C | > 1 A digital cub oid is sp ecified b y the co ordina t es of opp o site corners. Consider the following construction of n cub oids: Definition 3.2 The se quenc e of n cub oids C 1 , C 2 , ..., C n is c al le d r e gular i f (a) x 0 min = y 0 min = z 0 min = 0 (b) x i min ≤ x i +1 min ∧ y i min ≤ y i +1 min ∧ z i min ≤ z i +1 min and at le ast one of the in e quality is str ong for e ach 1 ≤ i ≤ n − 1 (c) x i +1 min ≤ x i max ∧ y i +1 min ≤ y i max ∧ z i +1 min ≤ z i max for e ach 1 ≤ i ≤ n − 1 (d) x i max ≤ x i +1 max ∧ y i max ≤ y i +1 max ∧ z i max ≤ z i +1 max and at le ast one of the in e quality is str ong for e ach 1 ≤ i ≤ n − 1 Definition 3.3 The union of elements of r e gular se quenc e C = C 1 ∪ C 2 ∪ ... ∪ C n is c al le d an n -step stair c ase. An ex ample of a 2-step staircase is depicted in Fig ure 3. ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈                   Figure 3: 2-step staircase C . Eac h p oin t A = ( x A , y A , z A ) in the digital space Z 3 ma y b e considered as a m ultiset A o v er { x, y , z } , where x A is t he n um b er of rep etitions of an 9 elemen t x , and y A is the n um b er of rep etitions of an elemen t y , and z A is the n um b er of rep etitio ns of an elemen t z in m ultiset A . Then a 1-step staircase (cub oid) is a p o ly-an timatro id, since the family of p oints is access ible and closed under union (( x 1 , y 1 , z 1 ) , ( x 2 , y 2 , z 2 ) ∈ C ⇒ ( x 1 , y 1 , z 1 ) ∪ ( x 2 , y 2 , z 2 ) = (max( x 1 , x 2 ) , max( y 1 , y 2 ) , max( z 1 , z 2 )) ∈ C ). Lemma 3.4 An n -step stair c ase C is a p oly-antimatr oi d. Pro of. Consider some p oin t A 6 = (0 , 0 , 0) ∈ C . Then there exists i such that A ∈ C i . If A 6 = ( x i min , y i min , z i min ), then, without lost of g eneralit y , x i min < x A , and so ( x A − 1 , y A , z A ) ∈ C i ⊆ C . If A = ( x i min , y i min , z i min ), then, from Definition 3.2 (b,c) it f ollo ws that A ∈ C i − 1 and A 6 = ( x i − 1 min , y i − 1 min , z i − 1 min ). Hence, ( x A − 1 , y A , z A ) ∈ C i − 1 ⊆ C or ( x A , y A − 1 , z A ) ∈ C i − 1 ⊆ C or ( x A , y A , z A − 1) ∈ C i − 1 ⊆ C . Thu s, an n -step staircase C is acces sible. T o pro v e that C is closed under union consider tw o p oin ts A, B ∈ C . Let A ∈ C i , B ∈ C j , and i ≤ j . F rom Definition 3.2 (b) it fo llo ws that x j min ≤ max( x A , x B ). F r o m Definition 3.2 (d), max( x A , x B ) ≤ x j max . So, A ∪ B = (max( x A , x B ) , max( y A , y B )) ∈ C j , i.e., A ∪ B ∈ C . Our goal is to find t he minimum n umber o f maximal c hains whic h describe an n -step staircase. First, consider a cub oid C giv en b y tw o p oints ( x min , y min , z min ) and ( x max , y max , z max ). W e will denote b y P X the maximal chain connecting the p oints ( x min , y min , z min ) and ( x max , y min , z min ), i.e., P X = ( x min , y min , z min ) , ( x min + 1 , y min , z min ) , ..., ( x max , y min , z min ). P Y and P Z are defined corresp ondingly . It is easy to see that C = P X ∨ P Y ∨ P Z = { A 1 ∪ A 2 ∪ A 3 : A 1 ∈ P X , A 2 ∈ P Y , A 3 ∈ P Z } So, if w e ha ve three maximal c hains B 1 , B 2 , B 3 from the cub oid C , whic h pass o v er all the p oin ts o f P X , P Y and P Z , then C ⊆ B 1 ∨ B 2 ∨ B 3 . On the other hand, since C is closed under union, w e hav e B 1 ∨ B 2 ∨ B 3 ⊆ C . Th us, B 1 ∨ B 2 ∨ B 3 = C . No w, consider the three-dimensional ve rsion of Algorithm 2 .10 for searc h order ( x, y , z ). Let ( x max , y max , z max ) b e t he maxim um p oint of n -step stair- case C . The alg o rithm builds the c hain connecting the maxim um p oin t and the p oint (0 , 0 , 0). 10 Algorithm 3.5 XYZ-Boundary tr a cing algo rithm 1. i := 0 ; x := x max ; y := y max ; z := z max ; 2. B i := ( x, y , z ) 3. do 3.1 if ( x − 1 , y , z ) ∈ S then x := x − 1 , el s e if ( x, y − 1 , z ) ∈ S then y := y − 1 , else z := z − 1 ; 3.2 i := i + 1 ; 3.3 B i := ( x, y , z ) ; until B i = (0 , 0 , 0) 4. R eturn the se quenc e B = B i , B i − 1 , ..., B 0 Rep eat Algorithm 3.5 for searc h order ( y , z , x ) and for searc h order ( z , x, y ). As a result, w e o bt a in three monotone increasing N 6 -paths [14] fro m (0 , 0 , 0) to ( x max , y max , z max ), denoted b y B Z , B X and B Y , resp ectiv ely . The length of an y of these c hains is equal to ( x max + y max + z max ). Theorem 3.6 Any n -step stair c ase C = B X ∨ B Y ∨ B Z . T o sho w this w e prov e a stronger statemen t. Change Algorithm 3 .5 in the following w ay . Let the XYZ-Boundary t rac- ing algorithm b egin fr o m some p o int ( x, y , z max ) ∈ C and return c hain H Z ; the YZ X-Boundary tracing algor ithm b egins from some p oin t ( x max , y , z ) ∈ C and returns chain H X ; and the ZXY-Boundary tracing algorit hm b egins from some p oin t ( x, y max , z ) ∈ C and returns chain H Y . Lemma 3.7 Any n -step stair c ase C = H X ∨ H Y ∨ H Z . Pro of. Let us pro ceed b y induction on n . F or n = 1 the XYZ-Boundary t r acing algorithm b eginning from the p oint ( x, y , z max ) ∈ C returns chain H Z that b egins from the p oint (0 , 0 , 0), mo v es to ( x min , y min , z max ), then to ( x min , y , z max ), and finishes at the p oin t ( x, y , z max ). So, this chain passes ov er all t he p oin ts o f P Z . In the same w a y , t he YZ X- Boundary tracing algorithm returns c hain H X that passes ov er a ll the p oin ts of P X , and the ZXY- Boundary tracing algorithm returns chain H Y that passes ov er all the p oin ts of P Y . Th us, C = H X ∨ H Y ∨ H Z . Assume that the prop o sition is correct for all k < n and prov e it f o r n . It is easy to see that the XYZ -Boundary tracing algorithm b egins from some p oin t ( x, y , z max ) ∈ C n , reache s the p oin t ( x n min , y n min , z n max ), mo ves do wn 11 to the p o in t ( x n min , y n min , z n − 1 max ) ∈ C n − 1 , and then contin ues to build c hain H Z . In the same w ay , the YZX-Boundary tracing algo rithm mo v es through the p oin t ( x n − 1 max , y n min , z n min ) ∈ C n − 1 , and the ZXY-Bo undar y tracing algo rithm mo v es through the p oin t ( x n min , y n − 1 max , z n min ) ∈ C n − 1 . The induction assumption implies C 1 ∪ C 2 ∪ ... ∪ C n − 1 ⊆ H X ∨ H Y ∨ H Z . It remains to sho w that C n ⊆ H X ∨ H Y ∨ H Z . T o t his end w e pro v e that P n X ⊆ H X ∨ H Y ∨ H Z , P n Y ⊆ H X ∨ H Y ∨ H Z , and P n Z ⊆ H X ∨ H Y ∨ H Z . W e ha v e already seen tha t the XYZ-Boundary tracing algor it hm returns c hain H Z that cov ers the part [( x n min , y n min , z n max ) , ( x n min , y n min , z n − 1 max )] from P n Z , and the remaining part [( x n min , y n min , z n − 1 max ) , ( x n min , y n min , z n min )] ⊆ C 1 ∪ C 2 ∪ ... ∪ C n − 1 . So P n Z ⊆ H X ∨ H Y ∨ H Z . It is easy to v erify tw o other cases. Finally , C = C 1 ∪ C 2 ∪ ... ∪ C n ⊆ ( H X ∨ H Y ∨ H Z ) ∨ ( H X ∨ H Y ∨ H Z ) ∨ ( H X ∨ H Y ∨ H Z ) = H X ∨ H Y ∨ H Z , sinc e eac h H is a c hain. On the other hand, H X ∨ H Y ∨ H Z ⊆ C , and so C = H X ∨ H Y ∨ H Z . Theorem 3.6 sho ws that it is enough to kno w only three maximal c hains B X , B Y , and B Z to describ e an n -step staircase. Corollary 3.8 An n-step stair c ase has c o nvex dimension at mos t 3 . Note that the three p oin ts (0 , 0 , 1), (0 , 1 , 0) a nd (1 , 0 , 0) cannot b e co v ered b y tw o chains only , since eac h suc h p oin t (m ultiset) cannot b e formed as a union of smaller m ultisets. So, there exist n -step staircases of conv ex dimension 3. Ho w eve r, the con v ex dimension of an arbitrary three-dimensional p oly- an timatroid ma y b e arbitrarily large [10]. Let S b e a set of po in ts: S = { ( x, y , z ) : (0 ≤ x, y ≤ N ) ∧ (0 ≤ z ≤ 1) ∧ ( z = 1 ⇒ x + y ≥ N ) } It is easy to c hec k that S is a three-dimensional p oly-an timat r oid. Con- sider N + 1 p oin ts ( x, y , 1) with x + y = N . Since eac h of these p oints cannot b e represen ted as a union of an y p oints from S with smaller co ordinates, the con v ex dimension of S is at least N + 1 [10]. In o rder to characterize the family of three-dimensional p oly-an tima t roids of con v ex dimension 3, consider a particular case of antimatroids called p oset antimatr oids [15]. A p oset an timatroid has as its feasible sets the lo w er sets of a p oset (partially ordered set). The p oset an timatroids can b e c haracter- ized as the unique an timatroids whic h are closed under intersec tion [15]. W e extend this definition to p oly-antimatroids. 12 Definition 3.9 A p oly-antimatr oid is c al le d a p oset p ol y-an timatr oid if it is close d under interse ction. No w, not e that n -step staircases are closed under intersec tio n to o. Indeed, consider t w o p oin ts A, B ∈ C . Let A ∈ C i , B ∈ C j , and i ≤ j . F rom Defini- tion 3.2 (b) it fo llo ws that min( x A , x B ) ≥ x i min . On the other hand, (Defini- tion 3.2 (d)), min( x A , x B ) ≤ x i max . So, A ∩ B = (min( x A , x B ) , min( y A , y B )) ∈ C i , i.e., A ∩ B ∈ C . So, our conjectures are as follo ws. Conjecture 3.10 A thr e e-dimensional p oset p oly-antimatr oid is a step stair- c ase. 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