Absolutely determined matrices

Mathematical Social Sciences 20 (1990) I-18 North-Holland

ABSOLUTELY DETERMINED MATRICES

Vladimir A. GURVICH Institute of Physics of the Earth, Academy of Science, Moscow 123810, U.S.S.R.

Leonid 0. LIBKIN 2nd Pugachevskaja ul.. d. 8, kor. 5, kv. 196, Moscow 107061, U.S.S.R.

Communicated by H. .Moulin Received 7 August 1989 Revised 11 January 1990

A matrix A is said to be absolutely determined if all its submatrices, including A itself, have saddle-points. In this paper the properties of absolutely determined matrices are studied. The one- to-one correspondence between symmetrical absolutely determined matrices and so-called quasilinear set functions is constructed. This correspondence is generalized to the functions on arbitrary finite semilattices. Some estimates on the number of absolutely determined matrices are established.

Key words: Absolutely determined matrix; quasiconvex, quasiconcave and quasilinear set functions; semilattice.

0. Introduction

0.1. Absolutely determined matrices

A real valued matrix A = /aJ is called determined if it has a saddle-point. In this case:

minmaxaii=maxminaii=ValA. (0.1) i j j i

Let us call a matrix A absolutely determined (AD) if every submatrix of A, including A itself, is determined.

Shapley (1955) proved that a matrix A is determined if all its 2x2 submatrices are determined. It is clear that Shapley’s condition is also sufficient (and of course necessary) for A to be an AD matrix (ADM). This criterion supplies us with a polynomial algorithm for checking the AD of time complexity C,(N)= O(N*), where N is a complexity of input.

In Section 1 some other criteria for AD and also an algorithm of checking AD with C,(N) = 0(N3”) will be given.

0165-4896/9O/f3.50 0 1990-Elsevier Science Publishers B.V. (North-Holland)

2 V.A. Gurvich, L.O. Libkin / Absolutely determined matrices

The classes of (0,l) ADMs and symmetrical ADMs (SADhls) will be studied especially, because the first class plays an essential assistant role, and the second one is important for further applications.

0.2. Quasiconvex analysis for set functions

Given a finite set W, consider its Boolean 2W as a semilattice (SL) with operation U . A subset S c 2W is called a subsemilattice (SSL) of 2w if it is closed with respect to this operation. SSLs will also be referred to as the convex subsets of 2W. Note that an intersection of convex subsets (SSLs) is convex (SSL) again. A SSL SC 2W will be called a separatory SSL (SSSL) if its complement, ZW\ S, is also a SSL. The notion of an SSSL is analogous to that of a half-space.

A real-valued mapping on Boolean f: 2w -, IR is called a set function. In accordance with general conceptions of Soltan (1984) let us call a set function f quasiconvex (QX) or, respectively, quasiconcave (QV) if its level sets {Hs, W [ f(H)lc) or, respectively, {Z-ZC W 1 f(H) zc are convex subsets (SSLs) for ) any constant c E R. A set function f is called quasilinear (QL) if it is both QX and QV.

The last notion was introduced and studied by Libkin and hluchnik (1988). The equivalent definitions for QX, QV and QL functions (QXF, QVF and QLF)

have also been obtained. Let us introduce the following functional inequalities:

f(H, U&) s max{f(Hi,),f(H2)), VHI,H2 E W, (0.2)

f(H, U4) 2 min{f(H1),fU&)), VHI,HZ E W. (0.3)

The functions satisfying (0.2) and (0.3) and (0.2), (0.3) simultaneously are, respectively, QX, QV and QL. It is clear that two given definitions are equivalent; see also Libkin and Muchnik (1988); Libkin, Muchnik and Schwartzer (1989).

The following three theorems, proved by Libkin and Muchnik (1988) (see also Subsection 3.1), are in the spirit of the traditional convex analysis and discover an important role played by SSSLs and QLFs. The first two theorems are concerned with separation and the representation of convex sets.

Theorem 1. For any pair of disjunctive convex sets (SSLs) S1, S2 C 2w (S, n S2 = 0) there exists a separating half-space (SSSL) S C_ 2 w: S, C_ S, S2 C 2 w \ S.

Theorem 2. Any convex set (SSL) can be represented as an intersection of half- spaces (SSSLs).

The third theorem is concerned with representation of QX and QV functions.

Theorem 3. Any QXF or, respectively, QVF can be represented as a maximum or, respectively, a minimum of QLFs.

V.A. Gurvich, L.O. Libkin / Absolutely determined matrices 3

Note that a consistent statement concerned with separation of the QXF from lesser QVF by some QLF is not true; see Subsection 3.2.

QX and QL programming problems are formulated as follows:

fo+min,J5ci, i= l,...,k,

where fo,...,fk are QXFs and QLFs, respectively. QX analysis has numerous applications in semilattice, monotonic systems

theories; see Libkin and Muchnik (1988) and Malishevski (1986). It has applications in general choice theory as well, mostly for the analysis of the choice functions satisfying the (Ex) condition (Moulin, 1984). The results can be found in Libkin (1988). The representation of QLFs given in the next section is important for these applications.

Remark 1. Note that inequalities (0.2) and (0.3) are valid obviously if H, = 0 or Hz= 0. Let us agree to treat set functions different only on the empty subset,

(01 E2w, as identical.

Remark 2. All the definitions and statements still remain correct if one changes the operation U to the operation ft ; see Subsection 3.3. Moreover, QX analysis can be developed for arbitrary SLs; see Subsection 3.4.

Remark 3. Note also that there is a connection between QX, QV and QL functions on SL and, respectively, submodular, supermodular and modular functions on lattices; see Lovbz (1983).

0.3. One-to-one correspondence between SADMs and QLFs

Fix a finite set W= { w,, . . . . w,} and consider the set functions f: 2w-+ [R and symmetrical n x n matrices A: Wx W+ 172 with rows and columns numerated by the same set of indices W.

For any QLF f: 2w+ IR define the symmetrical matrix Af: WX W+ R by the formula:

AJ= la& aii = aj;=f({Wi9Wj}), ij= l,..., n. (0.4)

For any nxn SADM A: Wx W-r iI? define the set function f: 2w-+ IR by the formula:

fA(H)=ValAH, VHG W, (0.5)

where An: Hx H+ fR is the diagonal submatrix of A: WX W+ IR corresponding to the subset HS W. The valuef(0) can be choosen arbitrarily in accordance with Remark 1.

Theorem 4. The mappings @ and w defined by formulae (0.4) and (0.5), respectively, are mutually inverse one-to-one correspondences between the sets of SADMs and QLFs.

4 V.A. Gurvich. L.O. Libkin / Absolutely derermined matrices

It follows at once from this theorem and formula (0.5) that a QLF is unambiguously determined by the values it takes on for l- and 2-element subsets of W; a QLF can take on for these subsets such and only such values that determine a SADM according to formula (0.4); a QLF can take no new values on other subsets of w.

Theorem 4 allows us also to establish a one-to-one correspondence between (0,l) SADMs and SSSLs; see Section 2.

1. Criteria for absolute determiness of matrices

1.1. (41) absolutely determined matrices

A matrix A = IlaJ, i= 1, . . . , k, j= 1, . . . , I, is called a (k x I) - (0, 1) matrix if aiiE (0, l} for all i, j.

Lemma 1. All the 2 x 2-(0,l) matrices are AD except

A matrix A is called monotonic if

(i, 5 iz , j, 5 j2, ai,i, = 1) =9 aid2 = 1.

A matrix a will be called monotonable if some permutation of rows and columns makes A monotonic.

It will be said that a row ai, (a column aj,) contains a row ai, (a column ajJ, if

Vj : Ui,j = 0 a UiJ = 0 (Vi : aii, = 0 I$ au2 = 0).

Theorem 5. The following properties of a (41) matrix A are equivalent: (a) A is ADM; (b) A is monotonable; (c) A does not contain the submatrices

(d) for any two rows one contains another; (e) for any two columns one contains another.

A sketch of the plications:

proof is given by the following chain of (almost) evident im-

C-4 s (cl = W A W = (4 v (e) = W =) (a).

V.A. Gurvich, L.O. Libkin / Absolutely determined matrices

1.2. Real-valued absolutely determined matrices

Lemma 2. The following properties of a 2 x 2 matrix A are equivalent: (a) A is determined;

(b) [alI ,a221 n [a12,a211 f 0; (c) max(al,,az2) L min(a12,azl); max(a,2,a21) 2 min(a,,,a2z).

The proof follows from the consideration of all the inequalities and equalities on four elements of the matrix A.

Given a kx I matrix A = /aiiII and a real number x, define the (k x I) - (0,l) matrix A(x)= Ilaii(x)ll by the formula:

aij(x) = 0, if aiiIx, i_ 1

1, if ati>x, - ,..., k, j= l,..., I.

Theorem 6. The following properties of a matrix A are equivalent: (a) A is ADM; (b) every 2 x 2 submatrix of A is determined; (c) A(x) is (41) ADM for any XE R; (d) A(x) is (0,l) ADMfor any x=aij, i= 1, . . . . k; j= 1, . . . . I.

Sketch of the proof. (c) j (a). Let (a) be not true and A’ be a nondetermined submatrix of A. Choose x between max min and min max of A’. Then A’(x) is also nondetermined and A(x) is not AD. Thus, (c) is not true. All the other implications follow from Theorem 5 and Lemma 2. 0

Shapley’s Theorem. (b) = (a) cannot be generalized for bimatrix games.

Example 1. The following 3 x 3 bimatrix game, a= (A’, A’), has no Nash equi- librium, but all nine 2x2 subbimatrices of a have at least one each:

A’ = Ilahll, ai, > ai, > a:, , ai > ai, > a:,, aj3 > ai, > a:, ,

A*= ia,$l, a:,>a:2>af,, ai3>ai,>az2, af2>af3>a:,.

A matrix A is called monotonic if

(i, I i2, jl S j,) * ai,j, I aitiz.

A matrix A will be called monotonable if some permutation of rows and columns makes A monotonic.

Any monotonable matrix is AD. But the opposite implication is not true, in con- trast to the case of (0,l) matrices.

Example 2. The following 3 x 3 matrix A is AD but not monotonable:

41 = 3, a12=a2] =a2,= 1, 013 = a3] = a23 = a32 = a33 = 2,


or more briefly:

2aij=3max2(i,j)-13max(i,j)+8, i,j= 1,2,3.

1.3. The algorithm of AD verification (sketch)

Check property (d) of Theorem 6. Order the elements of the matrix A. Let them take m different values xl <x2 < ... <x,,, (m 5 k x Z). Subdivide the verification into m-l steps, i=l,2 ,..., m - 1. By the ith step compute the (41) matrix A(xJ and verify (d) or (e) of Theorem 5, using the data and results of analogous verification for the matrix A(Xi_1) considered on the previous step.

The time complexity of such an algorithm can be reduced to C#V) = 0(14~‘~), where N= k x ix h and h is the number of bits required for representation of one element of the matrix A.

if A is a (0,l) matrix (m = 2), then CT(N) = O(N). If all the elements of A are different (m = k x I), then C,(N) = O(N. In N).

1.4. Symmetrical absolutely determined matrices

A square n x n matrix A = [aiill ig called symmetrical if

au=aji, for all i,j= l,..., n.

Lemma 3. A symmetrical 2 x 2 matrix A is determined iffy

min(al I, a22) 22 aI2 = a21 s max(q I, a22);

or, which is the same, iff:

(al 1 - 42W22 - a12) 5 0.

In this case, Val A = al2 = azl.

The proof follows from Lemma 2 and the symmetry of A.

Theorem 1. The following properties of a symmetrical matrix A are equivalent: (a) A is SADM; (b) any 2 x 2 submatrix of A is determined; (c) any diagonal submatrix of A is determined; (d) any diagonal 2 x 2 or 4 x 4 submatrix of A is determined.

A sketch of the proof is given below. The implications (a) S> (b)~(c) A(d), (c) 3 (d) are evident; (b) a (a) is proved by Shapley, see Theorem 6.

To prove that (d) a (b), split property (d) into two: (d’) any diagonal 2 x 2 submatrix of A is determined; (d”) any diagonal 4 x 4 submatrix of A is determined.

We first consider some consequences of (d’) and later add (d”). Assume without loss of generality that a,l ~a~~=-**~a,,,,. Then isj * att’atj=ajt’ajj.

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We classify 2 x 2 submatrices of A. Let a 2 x 2 submatrix be given by rows it, it and columns j, , j,. Assume without loss of generality that i, <iz, j, <jz, i, I j, .

We obtain eight cases: (Cl) i,<i2<jl<jz; (C2) i, = j, < iz = j2; (C3) i, < i2 = j, < jz; (C4) i, <j, < iz c j,; (CS) i, <j, < i2 = j,; (C6) i, = j, < i2 < j,;

(C7) i, <j, <j, < iz; (CS) i, = j, < jz < i2.

Lemma 4. It follows from (d’) that all the 2 x 2 submatrices of A, corresponding to the cases (C2)-(C8) (and also all the diagonal 3 x 3 submatrices of A) are determined.

Proof. For all seven cases the saddle-points can be found directly.

The case (Cl) is more difficult.

Lemma 5. If follows from (d’) and (d”) that the 2 x 2 submatrix (il, iz) x (j,, j,), where i, < i2 < j, <j, [corresponding to the case (Cl)] is determined.

Proof. Assume without loss of generality that

iI = 1, iz = 2, j, = 3, j, = 4.

Reason by contradiction. Let the 2 x 2 submatrix (1,2) x (3,4) be not determined. Then, according to Lemma 2, one of the following two cases is valid:

(Cl’) min(at4, azs) > max(at3, a24);

(Cl 7 max(a14, (123) < mln(q3, a24).

Prove that the diagonal 4 x 4 submatrix (1,2,3,4) x (1,2,3,4) is not determined. Consider the case (Cl’). Note that alz>a13; and also that

minh2,a22, a231 > a24 < all; mintal I, a12, q4) > 013 C a23.

Conclude that the points (1, l), (1,2), (1,3), (1,4), (2,3), (2,4) (and also the points symmetrical to them) cannot be saddle.

Consider the three remaining points: (3,3), (3,4), (4,4). Let (3.3) or (3,4) or (4,4) be a saddle-point, then respectively

a33 = a2, > al3 and a33 < a3], or

a34 2 014 > 013 and a23<a31,

or

a44 = aI4 > a24 and a, < a42.

Thus, any of these three assumptions leads to a contradiction. The case (Cl”) is considered in an analogous way. Thus, Lemma 5 and Theorem 7 are proved. Incidentally, one more property, (e), was added to the list (a)-(d) of Theorem 7.

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Proposition 1. The following property is equivalent to the properties (a)-(d): (e) any 2 x2 submatrix of A, corresponding to the case (Cl) or (C2), is

determined.

There are (2) diagonal and (;) overdiagonal submatrices corresponding to the cases (C2) and (Cl), respectively.

Yet one more property can be added to the list (a)-(e). Any symmetrical n x n-(0,1) matrix A = jaiijj induces a graph G(A) = ( W, E) with

the set of vertices W= { wl, . . . , w,} such that (wi, wj) E E if and only if aii= 1. Note that the obtained graph can contain loops but not multiple edges.

A graph will be called separatory if there exists a subset H of W such that

w~V WjEH*(Wi,Wi)EE;

Consider the following graph, Go= ( W”,Eo), with four vertices, W”= {w,, w2,

~3, wJ, and five edges E”= {(w,, wl),(w2, w,), (w,, w2),(wl, w~),(w~, w4)).

Proposition 2. The following property is equivalent to the properties (a)-(e): (f) for any real x the graph G(A(x)) is separatory and does not contain Go as a

subgraph.

Proof. The graph G(A(x)) is separatory if and only if (d’) holds and does not contain Go and if and only if, in addition, (d”) holds.

1.5. Skew-symmetrical absolutely determined matrices

A square n x n matrix A = laiil is called skew-symmetrical if aij = -aji for all i, j= l,..., n; in particular, aij= 0.

These matrices are more important for game theory than symmetrical ones. However, the following examples show that implications (c) =) (a), (d) =+ (c) and (d) w (b) of Theorem 7 do not hold for skew-symmetrical matrices.

Example 3. The skew-symmetrical n x n matrices, satisfying the conditions

n 2 3; aiiZO, if i<j; miNa12,a23) > 013 > 0,

are not AD; but all their diagonal submatrices are determined.

Example 4. The skew-symmetrical 3 x 3 matrix A, satisfying inequalities min(at2, aa)> 0 > a,3 is not determined, but all its diagonal 2 x 2 submatrices are determined.

V.A. Gurvich, L.O. Libkin / Absolurely determined matrices 9

2. One-to-one correspondence between symmetrical absolutely determined matrices

-and quasilinear functions

2.1. Proof of Theorem 4

Fix a finite set W and consider the mappings @ and v/ defined in Subsection 0.3 by the formulae (0.4) and (OS), respectively. By the definitions, $J transfers QLFs into symmetrical matrices and I,V transfers SADMs into set functions. It will be proved first that 4 and y are injective mappings into SADMs and QLFs, respectively. Thus, 9 and w are bijective because W is finite. Then it will be proved that @J and w are mutually inverse.

Let A: Wx W+ IR be an arbitrary symmetrical n x n matrix. Define the set functions fi and fy by:

f;(H) = max min AH, f;‘(H) = min max AH, H c W, (2.1)

where AH: HxH- IR is the diagonal submatrix of A: Wx W+ IR corresponding to the subset HG W.

Lemma 6. (Malishevski, 1986). The set functions &’ and &” are QX and QV, respectively.

Proof. Check for f; and f*” using the inequalities (0.2) an (0.3) directly. 0

Example 5. The matrices (A:) and (!A) demonstrate that fi is not QVF and fff is not QXF.

Lemma 7. The mapping v/ transfers SADMs into QLFs.

Proof. Let A be a SADM. Then w(A) =fA =fA=fA”. Thus fA is QXF and QVF simultaneously. Cl

Lemma 8. The mapping w is injective.

Proof. Let the SADMs A’ and A” be different. Then the QLFs t,u(A’) and y/(A”) differ on some I- or 2-element subset of W, according to formula (0.5) and Lemma 3. Cl

Lemma 9. The mapping @ transfers QLFs into SADMs.

Proof. Let f be QLF. Prove that the symmetrical matrix A = #(f) is AD. Reason by contradiction. Let A be not ADM. According to Theorem 6 there exists a nondetermined 2 x 2 submatrix of A. Let it be given by the rows i,, iz and the columns j,, jz. Note that i, #iz, j, +j,. Consider the subset He= {wi,, wi2, wj,, wj?}. Note that some elements may coincide, but there are no less than two and no more than

IO V.A. Gurvich, L.O. Libkin / Absolutely determined matrices

four elements in Ho. According to the formulae (0.2) and (0.3) and Lemma 2, we obtain:

f(HO) E lai,jl 9 (lij2 I n Iai, j:, ahJ, I = 0 *

Thus, Lemma 9 is proved by contradiction. 0

Lemma 10. Any QLF is unambiguously determined by values on l- and 2-element subsets of W and takes no other values.

Note that Lemma 9 means that these values are to generate a SADM according to formula (0.4).

Proof of Lemma 10. Let subset Hc, W contain more than two elements. Then H can be represented as a disjunctive union of non-empty subsets, H = H, U Hz U H3.

Let f be a QLF. Consider the values aii =f(Hi U Hi), i, j E { 1,2,3). Assume that a12~a13~a23. According to (0.2) and (0.3):

f(H) E Iai2, ai31 n 1% al31 = ai3.

Thus, f(H) =f(H, U H3). In any case, the value f(H) is determined by the values f(Hi UH2), f(H2 U H3) and f(H, U H3) and, moreover, coincide with the medium one. This proves the lemma. 0

Lemma 11. The mapping I$ is injective.

Proof. Let the QLFsf’andf” be different. Then according to Lemma 10 they differ on some l- or 2-element subset of W, and by (0.4) the matrices @(f’) and @(f”) are also different. 0

It follows from Lemmas 7, 8, 9, 11 that q4~ and v/ are bijective. Now the proof of Theorem 4 is concluded by the following:

Lemma 12. The bijections $I and w are mutually inverse.

The proof follows from Lemma 3 and from the definitions of @ and I,V by the formulae (0.4) and (0.5).

2.2. (0, I)-SADMS, (0, l)-QLFs and SSSLs

Fix a finite set W. A set function f: Zw+ (41) will be called a (0, l)-set function.

Any (0, I)-QLFfassigns the subset W)S~~, defined by <(f)=(HG W 1 f(H)=O). Any SSSL SC 2W assigns the characteristic set function c(S)=fs defined by

fs(H) = C 0, if HES,

1, if H$S, Hc W.

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Proposition 3. The mappings < and C are mutually inverse one-to-one corres-

pondence between the sets of (0, I)-QLFs and SSSLs.

The proof follows from the definitions. Consider again the mappings @: QLF + SADM and cy: SADM -+ QLF. Denote

their restrictions for (0, I)-QLFs and (0, I)-SADMs by r$’ and c’, respectively.

Theorem 8. The following diagram is commutative and all the mappings are bijective :

SSLSS + (0, I)-QLFs + (0, I)-SADMs.

The compositions c. qS and I+&. < are mutually inverse one-to-one correspondence between SSSLs and (0, I)-SADMs.

The proof follows from Proposition 3 and Theorem 4.

Remark 4. It was proved by Ashman and Ficker (1984) that lattices, topologies, rings and algebras on Boolean can be represented by square (41) matrices. The characterizations of these matrices were also found.

Theorem 8 supplements these results. It gives a representation of SSSLs by (0,l) matrices. Note that one cannot obtain the representation of SSLs in such a way. But according to Theorem 2 any family of SADMs represents some SSL of Boolean and conversely any SSL can be represented by a family of SADMs.

3. Quasiconvex analysis on Boolean and on arbitrary semilattices

3.1. Proof of Theorems 1, 2 and 3

Proof of Theorem 1. If St US, = 2w, then S=S, is the SSSL we look for. Let St U S, f 2W, then choose HE 2w \ (S, U S2). Denote by S; and S; the minimal SSLs containing the families St U {H} and S,U {H}, respectively. It is sufficient to prove that either S; fl S, = Q or S; fl St = 0. Then by induction we could obtain an SSSL since W is finite. Let, conversely, S; n S, # 0 and S; n St # 0. Then Wt E S, , Hz E S,: HI U HE S, and Hz U HE S, . The following equalities lead us to a contradiction:

H,UH,UH=(H,UH)UH,=(H,UH)UH,ES,nS2=@. 0

Proof of Theorem 2. Note that a single element is an SSL. Separate a given SSL S from each element {H} E 2w\ S. By some SSSL according to Theorem 1. Then the intersection of separating SSSLs equals S. Cl


Proof of Theorem 3 (sketch). Let a given QXF f take values xo<x, < a.- <xk. Consider the level SSLs:

S;={HC WJ f(H)<x;, i=o )...) k}.

For each i=O, 1, . . . . k represent the SSL Sj as an intersection of SSSLs S/, where j E 4 are some indices.

Define the two-valued QLFsJj by:

It follows from the definitions that

f=max(&jf;/‘I=O ,..., k,jEJ).

3.2. The example of non-separable QXF and QVF

The QXF f, and the QVF f2 will be constructed such that f, 2f2 (or fi >f2) and there is no separating QLF f: fi?f?f2.

Example 6. Let W= {w,, w2, We} and the set functions f, and f2 be given by the following formulae:

1, if H= {wl,w2},

f,(H) = 2, if H= (wJ} or H= W,

3, for all other H G W,

3, if H= {w~,wJ},

f,(H) = 2, if H={w,} or H= W,

1, for all other H C_ W.

It is clear that f, is QXF, f2 is QVF and fi zf2.

Suppose that there exists a QLF f such that fi If 1 f2. These inequalities imply

thatf({W,))E{53),f({Wj})E{I,2};f({ Wi,WZ))=I;f({W2,Ws))=3;andf(W)=2.

Then the QL off leads us to a contradiction:

1 =f(lwi, w2)) 2 min(f({w,)), f(Iw2))) =rf({w21) = 1,

3 =f({w2, w3)) 5 mWf(iw2)). f({+))) af(iw2)) = 3.

Note that the analogous example is impossible for (0, l)-set functions fi and f2.

For such functions the separation theorem is valid and is equivalent to Theorem 1. The inequality f, sf2 in Example 6 can be changed to a strict one.

Example 7. Let f; = 2f and f2’ = 2f2 - 1, then f{ is QXF, f; is QW, f\ >f2, and still

there is no separating QLF f: fi =f=fi.

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3.3. Replacement of U by f7

The Boolean ZW can also treated as the SL with operation fl . By substituting n for U in formulae (0.2) and (0.3), one obtains the definitions of tl -QXF, fl -QVF and fl -QLF instead of U -QXF, U -QVF and U -QLF considered above. Each set function f assigns a set function J according to the formula:

f(H) =f(W\ H), VH G W. (3.1)

Proposition 4. A set function f is U -QX or U -Q V or U -QL, respectively, iff the set function f is fl -QX or fl -Q V or fl -QL, respectively.

The proof follows from (0.2), (0.3) and (3.1). Thus, the involution f -$ enables us to transfer all the results obtained in this

paper for U-functions to n-functions.

3.4. Quasiconvex analysis on arbitrary finite semilattices

Consider an SL, i.e. the set S with binary join operation V, which is associative, commutative and idempotent (the last means that HVH= H for all HE S).

The partial order relation I is introduced by the formula:

H, I&, iff HIVH2 = Hz. (3.2)

The definitions of QX, QV and QL functions on SL can be obtained from analogous definitions in Subsection 0.2 by substituting S for 2W and V for U. Statement and proofs of Theorems 1, 2 and 3 also remain correct by the same substitution.

Remark 5. Infinite SLs also can be considered. The proofs of Theorems 1 and 2 remain exactly the same as before, but one needs to use the choice axiom. To prove Theorem 3 one needs to assume additionally that the considered QXF and QVF reach their infimum and supremum, respectively; see Libkin and Muchnik (1988) for details.

The correspondence of Section 2 between QLFs on Boolean and SADMs also can be generalized for arbitrary (finite) SLs, but in a more sophisticated way.

An element HES is called irreducible (Aigner, 1979) if it cannot be represented as a join of some other elements of S:

H=H,vH, 3 H=H, or H=H2. (3.3)

Proposition 5. Any element HE S can be represented as a join of irreducible ones. This representation is unique for all HE S if and only if SL (S;V) is isomorphic to

SL, F\ (01, u I.

14 V.A. Gurvich. L.O. Libkin / Absohrely derermined matrices

The proof follows from the definitions; see also Gurvich and Libkin (1989). Let {Hi, . . . . H,} be the set of all the irreducible elements of SL (S,V). The set

of indices ( 1, . . . , n} will be denoted by Z or Z(S, V).

The real-valued functions f: S + IR and symmetrical n x n matrices A: Ix I+ R

will be considered. For any function f: S-t R define the n x n matrix A: Z x I+ R by the formula:

A = @(f) = llqll, aii=f(H,VHj), i,jEI. (3.4)

If follows from (3.4) that aij= Uj;, ai;=f(Hi), i, jE Z, because the join operation v

is commutative and idempotent. Fix an arbitrary subset J c I. Denote by H, the join of all Hi, ie J, and by

AJ: J x J-+ IR the diagonal submatrix of A corresponding to J.

Theorem 9. Let f be QLF on a SL (S, v). Then the matrix A =@(f) is SADM and, moreover,

Val A, = f(H,), VJ s I. (3.5)

A sketch of the proof is given below.

Lemma 13. The following inequalities takes place:

max min AJ r f(H,) r min max A,. (3.6)

Proof. Let for every ic J ati reaches a minimal value on jie J, i.e. minjeJ aii=aU,. Since VieJ(HjV Hj,) = HJ, we have f(HJ>( maxieJ f(HiV Hj,) = max min A,.

Analogously, using QL we obtain f(H,) I min max A,. Thus (3.6) is proved. Taking into account that max min B = min max B for an arbitrary matrix B, we

have: max min A J = min max A, = Val A, = f(HJ). (3.7)

Formula (3.5) fohows immediately from (3.7). The AD of the matrix A follows from (3.7) and Theorem 7. The proof is complete. 0

This theorem demonstrates that only some special SADMs are in correspondence with the QLFs.

Given a SL (S, v), a SADM A: Z(S, V) x Z(S, V) -+ IR will be called fit (to SL (S, v))

if: HJ = HJ, j Val A, = Val A,, VJ,, J2 G Z(S, v). (3.8)

Consider the mapping Q defined by formula (3.4) and denote by @’ the restriction of @ on the set of QLFs.

Theorem 10. The mapping @’ establishes a one-to-one correspondence between the QLFs on SL (S, V) and the SADMs fit to this SL.

V.A. Gurvich. L.O. Libkin / Absolutely determined matrices 15

Proof. Letfbe a QLF, then G’(f) is a SADM fit to the SL according to Theorem 9. Conversely, let A be a SADM fit to S and Vf E I: f(H,) = Val A,. We have to

prove that f is a QLF. Let Jt,J*~_Iand H=H,,UH,,. If J=JtUJt, then H=H, and f(H) = Val A,, since A is fit to S. By Theorem 4, Val Ale [Val A,, Val AJz]. Hence f is a QLF. 0

Note that to check fitness is much more difficult than to check AD.

Proposition 6. All the SADMs A: I(S, V) xf(S, V)-r il? are fit to a SL S iff it is isomorphic to SL (2w \ {a}, U).

The proof follows from Theorem 10 and Proposition 5.

Proposition 7. All the functions f: S -+ I? on a SL S are QLFs if this SL is a chain.

The proof is obvious.

A SADM A: I(S, V) x I(S, V) -+ IT? is said to be 2-jit to a SL (S, V) if (3.8) holds for all 2-element subsets Jt, Jz G I(S, v). This definition can be formulated in the following more descriptive way.

Proposition 8. A SADM A is 2-fit to a SL S iffy

HiVHj=HkVH,qao=akl, Vi,j,k,lEI.

For some semilattices the notion of fitness is equivalent to that of 2-fitness. This can be proved for any SL with no more than three irreducible elements and also for any SL whose diagram is a tree.

Remark 6. Clearly, the join operation v can be changed to meet one A in accordance with Subsection 3.3.

3.5. Separation theorem for functions on a finite SL

We showed in Subsection 3.2 that the separation theorem does not hold for QXF and QVF on Boolean. Nevertheless, we can obtain this theorem for some SLs. The following result, proved by the second author, will be given here without the proof, which will appear elsewhere.

Let F(3) be a free SL with three generators [in fact, the SL (2w \ { 0 >, U ), where 1 WI = 31, and CO(~) be the SL of non-empty intervals of a three-element chain (it consists of six elements).

We say that a finite SL (S, V) satisfies the separation condition if, given a QXF fi and a QVF fi on SL such that fi >f2, there exists a QLF f such that fi >f>f2.

Theorem 11. A fmite SL satisJies the separation condition iff it does not contain a SSL isomorphic to F(3) and CO(~).


4. On the number of (O,l)-ADMs and SSSLs

4.1. The number of (0, l)-SADMs

Let a(n) stand for the number of (0, l)-SADMs of size n x n or, in other words, the number of SSSLs of the SL (2w \ {a}, U), where W contains n elements (see Theorem 8). Clearly, for each n the number a(n) is even. In this section we establish the lower and upper bounds for a(n).

Theorem 12. For every n the following inequalities hold:

2.n ( > 5a(n)I2”~n!-2nn!+2”+‘-2. n

Proof. Fix k, the number of elements of W belonging to a SSSL S G (2w \ { @ ), U }. Find the number c(k) of SSSLs satisfying 1 {w E W: {w} ES} 1 = k. Let wl, . . . , wk be those elements which belong to S. Thus, for a SADM A corresponding to S we have: Qii= 1 if isk and Uii= 0 if i>k. Hence, the matrix A is induced by its submatrix A’ generated by rows 1, . . . , k and columns k + 1, . . . , n. According to Theorem 5, we obtain, via some permutation of rows and columns, such a k x (n -k) matrix that its first row contains L, ones in the columns 1, . . . , II, the second row contains A2 ones in the columns 1, . . . , AZ, and so on. In addition, n-k?llzrlzz...r &rO. We can think of (A,, . . . , Ak) as being the monotonic mapping from the k chain (1<2<-.. ~k}tothen-k+1chain{O<1<~~~<n-k}.Ifm(n,k)standsfor the number of these mappings, we have:

m(n,k)sc(k)zz(m(n,k)-2)k!(n-k)!+2, (4.1)

since k! (n - k)! is the number of permutations of rows and columns of A’, and if A’ consists only of zero elements (or ones), after each permutation it remains the same.

Clearly, (4.1) holds if 0 c k c n. Furthermore, c(0) = c(n) = 1. According to Aigner (1979), the number of monotonic mappings from an r-

element chain to an m-element chain is:

(*+;-l). Hence

.

By (4.1) we have:

a(n) =

V.A. Gurvich, L.O. Libkin / Absolutely determined matrices 17

If k = 0 the right-hand part of (4.1) is 2 -n!, if k=n it is also 2-n!. Again by

(4.1):

The theorem is proved. Cl

Proposition 9. The number B(n) of SSSLs 0f(2~ \ (0}, U ), whose units coincide with W, satisfies the inequalities:

2n-1

( > +l-nIP(n)r2”-‘-n!-nn!+2”. n

The proof repeats the previous one with the additional condition L, = n -k (i.e. we have the monotonic mappings from a k - l-element chain to an n - k-element chain).

Proposition 10. The number c&m) of n xm - (91) SADMs satisfies the inequalities:

n+m

( > 5 a(n,m)r (n+m)! m

Proof. According to Theorem 5 and the proof of Theorem 12:

m(n,m+l)~a(n,m)~m(n,m+l)n!m!.

The equality

m(n,m+l) =

proves the proposition. El

Let y(n) be the number of SSLs of (2W \ { 0}, U ), 1 W ( = n.

Proposition 11. limn,, a(n)/y(n)=O.

Proof. Since nn!z2” (if n#l) we have a(n)12”.n!. If 4rG2W is an antichain, then 2W\{0}\ U({Hc W]HG;H~ for someHoEs)) is an SSL of (2W\{0}, U) (see Libkin and Muchnik, 1988). According to Sperner’s Theorem (see Aigner, 1979), y(n)r2(l”“RI), where [n/2] stands for the greatest integer lesser than n/2.

Using the Stirling’s formula we obtain for n--c 00 that ln(y(n))?O(2”) and ln(a(n))= O(n - In n). Hence, a(n) = o(y(n)). 0

18 V.A. Gurvich. L.O. Lib&in / Absolutely determined matrices

4.2. The number of non-equivalent (41) SADMs

Matrices of the same size are called equivalent if some permutations of rows and columns transfer one to another.

Proposition 12. There exist exactly 2” non-equivalent n xn-(0, 1) SADMs and (“‘,‘“) non-equivalent n x m-(0,1) ADMs.

Proof. The number of non-equivalent n x n-(0,1) SADMs is:

k = 0, . . ..n .

See the proof of Theorem 12. The second statement follows from Proposition 10. cl

Acknowledgements

The authors wish to express their gratitude to H. Moulin for the invitation to write this paper, and to I.B. Muchnik for helpful discussions.

References

M. Aigner. Combinatorial Theory (Springer-Verlag. New-York, 1979). C.J. Ashman and V. Ficker. Families of sets on a finite set, J. Austral. Math. Sot. A37 (1984) 405-412. V.A. Gurvich and L.O. Libkin, Absolutely determined matrices and quasilinear set-functions,

Automatica i Telemehanika 12 (1989) [in Russian]; English translation in J. Automation and Remote Control.

L.O. Libkin, The minimal sets of choice functions generating the main classes, Automatica i Telemehanika 12 (1988) 157-161 [in Russian].

L.O. Libkin and LB. Muchnik, Separatory subsemilattices and their properties, IMTA SZTAKI Korlemenyek 39 (1988) 83-92.

L.O. Libkin, I.B. Muchnik and L.V. Schwartzer, The quasilinear monotonic systems, Automatica i Telemehanika 9 (1989) 123-134 [in Russian]; English translation in J. Automation and Remote Control.

L. Lovasz, Submodular functions and convexity, in: A. Bachem, M. Grotshel and B. Korte, eds., Mathematical Programming: The State of the Art (Springer-Verlag. Berlin, 1983) pp. 235-257.

A.V. Malishevski, On the properties of set-functions in: Proceedings of the Seminar on Optimization and its Applications (Dushanbe, 1986) pp. 151-152 [in Russian].

H. Moulin, Choice functions over a finite set: A summary, Preprint N 82 (Inst. .Math. and its Appl., Univ. of Minnesota, 1984).

L.S. Shapley. A condition for the existence of saddle-points, RM-1958-PR. The RAND Corporation (1955).

V.P. Soltan, Introduction to the Axiomatic Theory of Convexity (Stiinca, Kishinev. 1984) [in Russian].

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