The concept of separation by hyperplanes is fundamental for convex geometry
and its tropical (max-plus) analogue. However, analogous separation results in
max-min convex geometry are based on semispaces. This paper answers the
question which semispaces are hyperplanes and when it is possible to
classically separate by hyperplanes in max-min convex geometry

Nitica, Viorel

Sergeev, Sergei

English

arXiv

The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry

University of Birmingham Research Portal

On hyperplanes and semispaces in max-min convex geometry

Sergeev, Sergeĭ

Digital Commons @ West Chester University

West Chester UniversityDigital Commons @ West Chester UniversityMathematics College of Arts & Sciences2010On hyperplanes and semispaces in max-minconvex geometryViorel NiticaWest Chester University of Pennsylvania, vnitica@wcupa.eduSergeĭ SergeevUniversity of BirminghamFollow this and additional works at: http://digitalcommons.wcupa.edu/math_facpubPart of the Algebraic Geometry CommonsThis Conference Proceeding is brought to you for free and open access by the College of Arts & Sciences at Digital Commons @ West ChesterUniversity. It has been accepted for inclusion in Mathematics by an authorized administrator of Digital Commons @ West Chester University. For moreinformation, please contact wcressler@wcupa.edu.Recommended CitationNitica, V., & Sergeev, S. (2010). On hyperplanes and semispaces in max-min convex geometry. Kybernetika, 46(3), 548-557. Retrievedfrom http://digitalcommons.wcupa.edu/math_facpub/14K Y BE R NE T IK A — VO L UM E 4 6 ( 2 0 1 0 ) , NU MB E R 3 , P AGE S 5 4 8 – 5 5 7ON HYPERPLANES AND SEMISPACESIN MAX–MIN CONVEX GEOMETRYViorel Nitica and Serge˘ı SergeevThe concept of separation by hyperplanes and halfspaces is fundamental for convexgeometry and its tropical (max-plus) analogue. However, analogous separation results inmax-min convex geometry are based on semispaces. This paper answers the question whichsemispaces are hyperplanes and when it is possible to “classically” separate by hyperplanesin max-min convex geometry.Keywords: tropical convexity; fuzzy algebra; separationClassification: 52A01, 52A30, 08A721. INTRODUCTIONConsider the set B = [0, 1] endowed with the operations⊕ = max,∧ = min. This is awell-known distributive lattice, and like any distributive lattice it can be consideredas a semiring equipped with addition ⊕ and multiplication ⊗ := ∧. Importantly,both operations are idempotent, a⊕a = a and a⊗a = a∧a = a, and closely relatedto the order: a⊕ b = b⇔ a ≤ b⇔ a∧ b = a. For standard literature on lattices andsemirings see e. g. [1] and [9].We consider Bn, the cartesian product of n copies of B, and equip this cartesianproduct with operations of taking componentwise ⊕: (x⊕y)i := xi⊕yi for x, y ∈ Bnand i = 1, . . . , n, and scalar ∧-multiplication: (a ∧ x)i := a ∧ xi for a ∈ B, x ∈ Bnand i = 1, . . . , n. Thus Bn is considered as a semimodule over B [9]. Alternatively,one may think in terms of vector lattices [1].A subset C of Bn is said to be max-min convex, (or briefly convex), if the relationsx, y ∈ C,α, β ∈ B, α⊕ β = 1 imply α ∧ x ⊕ β ∧ y ∈ C. Here and everywhere in thepaper we assume the priority of ∧ over ⊕. If x, y ∈ Bn, the set[x, y] := {α ∧ x⊕ β ∧ y ∈ Bn|α, β ∈ B, α⊕ β = 1}= {max(min(α, x),min(β, y)) ∈ Bn|α, β ∈ B,max (α, β) = 1}, (1)is called the max-min segment (or briefly, the segment) joining x and y. Like in theordinary convexity in the real space, a set is max-min convex if and only if any twopoints are contained in it together with the max-min segment joining them. Themax-min segments have been described in [14, 16]. Other types of convex sets aremax-min semispaces, halfspaces and hyperplanes.Semispaces and Hyperplanes in max–min Convex Geometry 549One of the main motivations for the investigation of max-min convex sets is in thestudy of tropically convex sets, analogously defined over the semiring Rmax, which isthe completed set of real numbers R∪{−∞} endowed with operations of idempotentaddition a⊕ b := max(a, b) and multiplication a⊗ b := a+ b. Tropical convexity andits lattice-theoretic generalizations, pioneered in [17, 18], received much attentionand rapidly developed over the last decades [3, 4, 5, 6, 10, 12, 13]. Another source ofinterest comes from the matrix algebra developed over the max-min semiring, alsoknown as fuzzy algebra [2, 7, 8].In this article we continue the study of max-min convex structures started in[11, 14, 15]. We are interested in separation of max-min convex sets by max-minhyperplanes and semispaces.When z ∈ Bn, we call a subset S(z) of Bn a max-min semispace (or, briefly, asemispace) at z, if it is a maximal (with respect to set-inclusion) max-min convexset avoiding z; a subset S of Bn is called a semispace, if there exists z ∈ Bn suchthat S = S(z). We recall that in Bn there exist at most n + 1 semispaces at eachpoint, exactly n+1 at each finite point, and each convex set avoiding z is containedin at least one of those semispaces [15].Amax-min hyperplane (briefly, a hyperplane) is the set of all points x = (x1, . . . , xn)∈ Bn satisfying an equation of the forma1 ∧ x1 ⊕ . . .⊕ an ∧ xn ⊕ an+1 = b1 ∧ x1 ⊕ . . .⊕ bn ∧ xn ⊕ bn+1, (2)with ai, bi ∈ B for i = 1, . . . , n+ 1, where each side contains at least one term. Thecombinatorial structure of hyperplanes is described in [11]. If the equality in (2) isreplaced by a strict (resp. non-strict) inequality, then we obtain an open halfspace(resp. a closed halfspace). Note that any max-min closed halfspace is a max-minhyperplane (due to a⊕ b = b⇔ a ≤ b) but not conversely.One of the main applications of semispaces is in separation results: the family ofsemispaces is the smallest intersectional basis for the family of all convex sets in Bn[15]. Separation results by hyperplanes and halfspaces in the tropical convexity andlattice-theoretic generalizations are found in [3, 4, 5, 6, 10, 17, 18]. These results arevery optimistic: any point can be separated from a closed tropically convex set, andeven any two compact non-intersecting convex sets can be separated from each otherby two closed halfspaces [6]. In contrast, [11] contains a counterexample to separationof a point and a max-min convex set by max-min hyperplanes (equivalently, by max-min halfspaces).The main goal of this paper is to further clarify separation by hyperplanes in max-min algebra. The main result of this paper, Theorem 3.1, shows which closures ofsemispaces are hyperplanes and which are not. As a corollary, we obtain in what caseit is possible to separate a point from a closed max-min convex set by a hyperplane.2. THE STRUCTURE OF SEMISPACESWe recall the structure of the semispaces in Bn at an arbitrary point x0. We followclosely [15].550 V. NITICA AND S. SERGEEVWithout loss of generality we may assume that the coordinates (x01, . . . , x0n) ofthe point x0 are in decreasing order, that is:x01 ≥ · · · ≥ x0n. (3)The set {x01, . . . , x0n} admits a natural subdivision into ordered subsets such thatthe elements of each subset are either equal to each other or are in strictly decreasingorder, sayx01 = · · · = x0k1> · · · > x0k1+l1+1 = · · · = x0k1+l1+k2 . . .> x0k1+l1+k2+l2+1 = · · · = x0k1+l1+k2+l2+k3 > . . .> x0k1+l1+···+kp−1+lp−1+1 = · · · = x0k1+l1+···+kp−1+lp−1+kp> · · · > x0k1+l1+···+kp+lp(= x0n),(4)where we make the following conventions:1. k1 = 0 if and only if the sequence (4) starts with the strict inequality x01 > x02;in this case l1 6= 0 and the beginning of the sequence will be:x01 > · · · > x0l1> x0l1+1 = · · · = x0l1+k2 > . . .· · · > x0l1+k2+l2 > x0l1+k2+l2+1 = . . . ;(5)in particular, if (4) has only strict inequalities between its terms one has p = 1, k1 =0, l1 = n. When (4) has only equalities between its terms, one has p = 1, k1 = n, l1 =0.2. lp = 0 if and only if the sequence {x01, . . . , x0n} ends with equalities, that is,with x0n−1 = x0n; in this case, if p ≥ 2, the end of the sequence (4) will be. . . > x0k1+l1+...+kp−1+lp−1+1 = . . . = x0k1+l1+...+kp−1+lp−1+kp , (6)while if p = 1, the whole sequence will be x01 = . . . = x0k1(= x0n). In other words, wetake lp 6= 0 if and only if x0n−1 > x0n.Let us introduce the following notations:L0 = 0,K1 = k1, L1 = K1 + l1 = k1 + l1, (7)Kj = Lj−1 + kj = k1 + l1 + . . .+ kj−1 + lj−1 + kj (j = 2, . . . , p), (8)Lj = Kj + lj = k1 + l1 + . . .+ kj + lj (j = 2, . . . , p). (9)We observe that lj = 0 if and only if Kj = Lj and Lp = n.The following description of semispaces is taken from [15, Proposition 4.1]. Weneed to distinguish the case when the sequence (4) ends with zeros either/or beginswith ones, since some semispaces become empty in that case.Proposition 2.1. Let x0 = (x01, . . . , x0n) ∈ Bn, x01 ≥ . . . ≥ x0n, and let k1, l1, . . . , kp,lp, p be non-negative integers as above.Semispaces and Hyperplanes in max–min Convex Geometry 551a) If 0 < x0i < 1 for all i = 1, . . . , n, then there are n+ 1 semispacesS0(x0), S1(x0), . . . , Sn(x0) at x0, namely:S0(x0) = {x ∈ Bn|xi > x0i for some 1 ≤ i ≤ n}, (10)SKj+q(x0) = {x ∈ Bn|xKj+q < x0Kj+q, or xi > x0i for some Kj + q + 1 ≤ i ≤ n}(q = 1, . . . , lj ; j = 1, . . . , p) if lj 6= 0,(11)SLj−1+q(x0) = {x ∈ Bn|xLj−1+q < x0Lj−1+q, or xi > x0i for some Kj + 1 ≤ i ≤ n}(q = 1, . . . , kj ; j = 1, . . . , p if k1 6= 0, or j = 2, . . . , p if k1 = 0).(12)b) If there exists an index i ∈ {1, . . . , n} such that x0i = 1, but no index j suchthat x0j = 0, then the semispaces at x0 are S1(x0), . . . , Sn(x0) of part a).c) If there exists an index j ∈ {1, . . . , n} such that x0j = 0, but no index i suchthat x0i = 1, then the semispaces at x0 are S0(x0), S1(x0), . . . , Sβ−1(x0) of part a),whereβ := min{1 ≤ j ≤ n| x0j = 0}. (13)d) If there exist indices i, j ∈ {1, . . . , n} such that x0i = 1 and x0j = 0, then thesemispaces at x0 are S1(x0), . . . , Sβ−1(x0) of part a), where β is the number (13).Pictures of all types of semispaces in B2 are shown in Figure 1. The figure istaken from [15].3. MAIN RESULTSIf we take the topological closure of semispaces, all inequalities in (10) – (12) becomenon-strict. We denote such closures by Si(x0).We will also denoteDn = {n︷ ︸︸ ︷(a, . . . , a) | a ∈ B}. (14)This set will be called the diagonal of Bn.Next we investigate when the closures of semispaces are hyperplanes.Theorem 3.1. (Semispaces and Hyperplanes) Assume that x0 ∈ Bn satisfiesx01 ≥ x02 ≥ · · · ≥ x0n and let H := Si(x0) for i = 0, 1 . . . , n. The following statementsare equivalent.a) H takes either of the following forms for some a ∈ B:H+(a) = {x | xi ≥ a for some i = 1, . . . , n}, for a < 1,H−i (a) = {x | xi ≤ a}, for a > 0, i = 1, 2, . . . , n.(15)b) H = Si(y) for some y ∈ Dn.c) H is a hyperplane.552 V. NITICA AND S. SERGEEVa1a2b1b2c1c2d1S0 S1 S2Fig. 1. Semispaces in dimension 2.Semispaces and Hyperplanes in max–min Convex Geometry 553P r o o f . First we observe that a) and b) are equivalent. Indeed, H+(a) = S0(x)and H−i (a) = Si(x) where x = (a, . . . , a).We can representH+(a) = {x |n⊕i=1xi = a⊕n⊕i=1xi},H−i (a) = {x | xi = a ∧ xi}.(16)which shows a)⇒ c).It remains to show that the closure of a semispace that is not of the form (15)cannot be a hyperplane.Case 1. Consider S0(x0) where x0 /∈ Dn.If x0i = 0 for some i, then S0(x0) = Bn = H+(0). Hence we can assume x0i > 0for all i.Let y ∈ Bn be such that yi < x0i for all i, andx01 > y1 > x0n > yn. (17)We define z1 and z2 byz1i ={x01, if i = 1,yi, otherwise,z2i ={x0n, if i = n,yi, otherwise.(18)Obviously S0(x0) contains both z1 and z2, but it does not contain y = z1 ∧ z2. Ourgoal is to show that any hyperplane defined by (2) that contains z1 and z2 will alsocontain y. Equation (2) for z1, z2 and y reduces to, respectively,a1 ∧ x01 ⊕ an ∧ yn ⊕ α = b1 ∧ x01 ⊕ bn ∧ yn ⊕ β, (19)a1 ∧ y1 ⊕ an ∧ x0n ⊕ α = b1 ∧ y1 ⊕ bn ∧ x0n ⊕ β, (20)a1 ∧ y1 ⊕ an ∧ yn ⊕ α = b1 ∧ y1 ⊕ bn ∧ yn ⊕ β, (21)whereα =⊕i6=1,nai ∧ yi, β =⊕i6=1,nbi ∧ yi. (22)We need to show that (19) and (20) together imply (21). We do this by showingthat the minimum of left hand sides of (19) and (20) is always equal to the left handside of (21). By analogy, the same holds for the right hand sides.We first pull α out of the brackets using the distributivity law (b⊕ a)∧ (c⊕ a) =(b ∧ c)⊕ a:(a1 ∧ x01 ⊕ an ∧ yn ⊕ α) ∧ (a1 ∧ y1 ⊕ an ∧ x0n ⊕ α)= ((a1 ∧ x01 ⊕ an ∧ yn) ∧ (a1 ∧ y1 ⊕ an ∧ x0n))⊕ α.(23)It remains to show that(a1 ∧ x01 ⊕ an ∧ yn) ∧ (a1 ∧ y1 ⊕ an ∧ x0n)= a1 ∧ y1 ⊕ an ∧ yn.(24)554 V. NITICA AND S. SERGEEVIf a1, an are large enough then (17) impliesa1 ∧ x01 ≥ a1 ∧ y1 ≥ an ∧ x0n ≥ an ∧ yn, (25)and in this case it is easy to see that (24) holds, both sides being equal to a1 ∧ y1.Note that the first and the third inequalities always hold. The second inequalitymay not hold true, but then a1 ≤ y1, in which case a1 ∧ x01 = a1 ∧ y1. In this casewe use the distributivity again, and this transforms the left hand side of (24) to(a1 ∧ y1)⊕ (an ∧ yn ∧ x0n) = (a1 ∧ y1)⊕ (an ∧ yn), (26)which proves (24) and hence the claim for Case 1.Case 2. Consider Si(x0) where x0 /∈ Dn.Denoten(i) ={i+ 1, if Ks + 1 ≤ i ≤ Ls,Ks+1 + 1, if Ls + 1 ≤ i ≤ Ks+1,(27)where Ks, Ls are defined by (7) – (9).If n(i) = n+1 then Si(x0) = H−i (x0i ). If x0i = 1 or x0j = 0 for some j ≥ n(i) thenSi(x0) = Bn. Otherwise, we construct points y, z and v defined byyj ={1, if j = i,x0j , otherwise,zj ={0, if j ≥ n(i),x0j , otherwise,vj =1, if j = i,0, if j ≥ n(i),x0j , otherwise.(28)It is clear that y and z belong to Si(x0) but v does not. Our goal will be to showthat if a hyperplane defined by (2) contains y and z then it also contains v. Equality(2) reduces in the cases of y, z and v respectively toai ⊕⊕s≥n(i)(as ∧ x0s)⊕ α = bi ⊕⊕s≥n(i)(bs ∧ x0s)⊕ β, (29)ai ∧ x0i ⊕ α = bi ∧ x0i ⊕ β, (30)ai ⊕ α = bi ⊕ β, (31)whereα =⊕s6=i, s<n(i)(as ∧ x0s)⊕ an+1, β =⊕s6=i, s<n(i)(bs ∧ x0s)⊕ bn+1. (32)We need to show that (29) and (30) imply (31). Assume by contradiction thatai ⊕ α 6= bi ⊕ β. Then there exists s ≥ n(i) such that (29) equals as ∧ x0s or bs ∧ x0simplying that x0s ≥ ai⊕ bi⊕α⊕β. But then x0i ≥ x0s ≥ ai⊕ bi⊕α⊕β, and equation(30), which is assumed to hold, is the same as (31), a contradiction. The proof ofCase 2 is complete and the theorem is proved. Semispaces and Hyperplanes in max–min Convex Geometry 555Remark 3.2. We recall that in the tropical (max-plus) convex geometry the closureof any semispace is a hyperplane [12].The key ingredient in the proof of Theorem 3.1 is the construction of exampleswhere a point cannot be separated from a closed semispace by a hyperplane. Theproof of Theorem 3.1 shows that such examples can be constructed for any dimensionand for any semispace except for (15) which are precisely the hyperplanes. The prooffor the case of S0(x0) (Case 1) also shows that such examples can be constructedfor any point outside the diagonal. We conclude the following.Corollary 3.3. (Non-separation by Hyperplanes) Let x ∈ Bn and x /∈ Dn.Then there exists a closed max-min convex set C ⊆ Bn such that x cannot beseparated from C by a hyperplane.Simple counterexamples to separation by hyperplanes in dimension two have beenobtained by one of the authors [11]: as shown on Figure 2, it actually suffices to takecertain max-min segments [14, 16]. The convex set C = [z1, z2] cannot be separatedby hyperplanes from the point x = z1 ∧ z2.z1z2xCFig. 2. Forbidden separation.Such examples can be extended cylindrically to any dimension, which is preciselythe geometric idea of the proof of Theorem 3.1.On the other hand, as the semispaces taken at a diagonal point are hyperplanes, itis possible to separate a point on the diagonal from a closed convex set “classically”.Corollary 3.4. (Diagonal Separation by Hyperplanes) Let x ∈ Bn and x ∈Dn. Then any closed max-min convex set C ⊆ Bn such that x /∈ C, can be separatedfrom x by a hyperplane.P r o o f . Since x 6∈ C, there is a semispace S at x containing C [15, Theorem 5.1].If S = S0(x), then for any y ∈ C there exists 1 ≤ i ≤ n such that yi > xi. Dueto compactness of C, there is δ > 0 such that above inequalities can be replaced byyi ≥ xi + δ. For x+ δ = (x1 + δ, . . . , xn+ δ), this implies C is included in S0(x+ δ).By Theorem 3.1 S0(x+ δ) is a hyperplane. Moreover it does not contain x.556 V. NITICA AND S. SERGEEVIf S = Si(x), then yi < xi for any y ∈ C. Due to compactness of C, there isδ > 0 such that above inequality can be replaced by yi ≤ xi − δ. This implies C isincluded in Si(x− δ). By Theorem 3.1 Si(x− δ) is a hyperplane. It avoids x. ACKNOWLEDGEMENTThis research was supported by NSF grant DMS-0500832 (Viorel Nitica), EPSRC grantRRAH12809, RFBR grant 08-01-00601 and joint RFBR/CNRS grant 05-01-02807(Serge˘ıSergeev).(Received April 14, 2010)REFERENCES[1] G. Birkhoff: Lattice Theory. American Mathematical Society, Providence, RI 1993.[2] K. Cechla´rova´: Eigenvectors in bottleneck algebra. Linear Algebra Appl. 175 (1992),63–73.[3] G. Cohen, S. Gaubert, J. P. Quadrat, and I. Singer: Max-plus convex sets andfunctions. In: Idempotent Mathematics and Mathematical Physics (G. Litvinov andV. Maslov, eds.), AMS, Providence 2005, pp. 105–129. E-print arXiv:math/0308166.[4] M. Develin and B. Sturmfels: Tropical convexity. Documenta Math. 9 (2004), 1–27.E-print arXiv:math/0308254.[5] S. Gaubert and R. Katz: Max-plus convex geometry. In: Lecture Notes in ComputerScience 4136, Springer, New York 2006. pp. 192–206.[6] S. Gaubert and S. Sergeev: Cyclic projectors and separation theorems in idempotentconvex geometry. J. Math. Sci. 155 (2008), 6, 815–829. E-print arXiv:math/0706.3347.[7] M. Gavalec: Periodicity in Extremal Algebras. Gaudeamus, Hradec Kra´love´ 2004.[8] M. Gavalec and J. Pla´vka: Strong regularity of matrices in general max-min algebra.Linear Algebra Appl. 371 (2003), 241–254.[9] J. Golan: Semirings and Their Applications. Kluwer, Dordrecht 2000.[10] G. L. Litvinov, V.P. Maslov, and G. B. Shpiz: Idempotent functional analysis: analgebraic approach. Math. Notes 69 (2001), 5, 758–797.[11] V. Nitica: The structure of max-min hyperplanes. Linear Algebra Appl. (2009),doi:10.1016/j.laa.2009.08.022.[12] V. Nitica and I. Singer: Max-plus convex sets and max-plus semispaces. I. Optimiza-tion 56 (2007), 171–205.[13] V. Nitica and I. Singer: Max-plus convex sets and max-plus semispaces. II. Optimiza-tion 56 (2007), 293–303.[14] V. Nitica and I. Singer: Contributions to max-min convex geometry. I. Segments.Linear Algebra Appl. 428 (2008), 7, 1439–1459.[15] V. Nitica and I. Singer: Contributions to max-min convex geometry. II. Semispacesand convex sets. Linear Algebra Appl. 428 (2008), 8–9, 2085–2115.[16] S.N. Sergeev: Algorithmic complexity of a problem of idempotent convex geometry.Math. Notes 74 (2003), 6, 848–852.Semispaces and Hyperplanes in max–min Convex Geometry 557[17] K. Zimmermann: A general separation theorem in extremal algebras. Ekonom.-Mat.Obzor 13 (1977), 179–201.[18] K. Zimmermann: Convexity in semimodules. Ekonom.-Mat. Obzor 17 (1981), 199–213.Viorel Nitica, Department of Mathematics, West Chester University, PA 19383, USA, andInstitute of Mathematics, P.O. Box 1-764, Bucharest. Romaniae-mail: vnitica@wcupa.eduSerge˘ı Sergeev, School of Mathematics, University of Birmingham, Edgbaston, BirminghamB15 2TT. United Kingdom.e-mail: sergiej@gmail.com

summary:The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry

Institute of Mathematics AS CR, v. v. i.

KybernetikaViorel Nitica; Sergei SergeevOn hyperplanes and semispaces in max–min convex geometryKybernetika, Vol. 46 (2010), No. 3, 548--557Persistent URL: http://dml.cz/dmlcz/140768Terms of use:© Institute of Information Theory and Automation AS CR, 2010Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czKYBERNET IKA — VOLUME 4 6 ( 2 0 1 0 ) , NUMBER 3 , PAGES 5 4 8 – 5 5 7ON HYPERPLANES AND SEMISPACESIN MAX–MIN CONVEX GEOMETRYViorel Nitica and Sergĕı SergeevThe concept of separation by hyperplanes and halfspaces is fundamental for convexgeometry and its tropical (max-plus) analogue. However, analogous separation results inmax-min convex geometry are based on semispaces. This paper answers the question whichsemispaces are hyperplanes and when it is possible to “classically” separate by hyperplanesin max-min convex geometry.Keywords: tropical convexity; fuzzy algebra; separationClassification: 52A01, 52A30, 08A721. INTRODUCTIONConsider the set B = [0, 1] endowed with the operations⊕ = max,∧ = min. This is awell-known distributive lattice, and like any distributive lattice it can be consideredas a semiring equipped with addition ⊕ and multiplication ⊗ := ∧. Importantly,both operations are idempotent, a⊕a = a and a⊗a = a∧a = a, and closely relatedto the order: a⊕ b = b ⇔ a ≤ b ⇔ a∧ b = a. For standard literature on lattices andsemirings see e. g. [1] and [9].We consider Bn, the cartesian product of n copies of B, and equip this cartesianproduct with operations of taking componentwise ⊕: (x⊕y)i := xi⊕yi for x, y ∈ Bnand i = 1, . . . , n, and scalar ∧-multiplication: (a ∧ x)i := a ∧ xi for a ∈ B, x ∈ Bnand i = 1, . . . , n. Thus Bn is considered as a semimodule over B [9]. Alternatively,one may think in terms of vector lattices [1].A subset C of Bn is said to be max-min convex, (or briefly convex), if the relationsx, y ∈ C,α, β ∈ B, α⊕ β = 1 imply α ∧ x ⊕ β ∧ y ∈ C. Here and everywhere in thepaper we assume the priority of ∧ over ⊕. If x, y ∈ Bn, the set[x, y] := {α ∧ x⊕ β ∧ y ∈ Bn|α, β ∈ B, α⊕ β = 1}= {max(min(α, x),min(β, y)) ∈ Bn|α, β ∈ B,max (α, β) = 1}, (1)is called the max-min segment (or briefly, the segment) joining x and y. Like in theordinary convexity in the real space, a set is max-min convex if and only if any twopoints are contained in it together with the max-min segment joining them. Themax-min segments have been described in [14, 16]. Other types of convex sets aremax-min semispaces, halfspaces and hyperplanes.Semispaces and Hyperplanes in max–min Convex Geometry 549One of the main motivations for the investigation of max-min convex sets is in thestudy of tropically convex sets, analogously defined over the semiring Rmax, which isthe completed set of real numbers R∪{−∞} endowed with operations of idempotentaddition a⊕ b := max(a, b) and multiplication a⊗ b := a+ b. Tropical convexity andits lattice-theoretic generalizations, pioneered in [17, 18], received much attentionand rapidly developed over the last decades [3, 4, 5, 6, 10, 12, 13]. Another source ofinterest comes from the matrix algebra developed over the max-min semiring, alsoknown as fuzzy algebra [2, 7, 8].In this article we continue the study of max-min convex structures started in[11, 14, 15]. We are interested in separation of max-min convex sets by max-minhyperplanes and semispaces.When z ∈ Bn, we call a subset S(z) of Bn a max-min semispace (or, briefly, asemispace) at z, if it is a maximal (with respect to set-inclusion) max-min convexset avoiding z; a subset S of Bn is called a semispace, if there exists z ∈ Bn suchthat S = S(z). We recall that in Bn there exist at most n + 1 semispaces at eachpoint, exactly n+1 at each finite point, and each convex set avoiding z is containedin at least one of those semispaces [15].Amax-min hyperplane (briefly, a hyperplane) is the set of all points x = (x1, . . . , xn)∈ Bn satisfying an equation of the forma1 ∧ x1 ⊕ . . .⊕ an ∧ xn ⊕ an+1 = b1 ∧ x1 ⊕ . . .⊕ bn ∧ xn ⊕ bn+1, (2)with ai, bi ∈ B for i = 1, . . . , n+ 1, where each side contains at least one term. Thecombinatorial structure of hyperplanes is described in [11]. If the equality in (2) isreplaced by a strict (resp. non-strict) inequality, then we obtain an open halfspace(resp. a closed halfspace). Note that any max-min closed halfspace is a max-minhyperplane (due to a⊕ b = b ⇔ a ≤ b) but not conversely.One of the main applications of semispaces is in separation results: the family ofsemispaces is the smallest intersectional basis for the family of all convex sets in Bn[15]. Separation results by hyperplanes and halfspaces in the tropical convexity andlattice-theoretic generalizations are found in [3, 4, 5, 6, 10, 17, 18]. These results arevery optimistic: any point can be separated from a closed tropically convex set, andeven any two compact non-intersecting convex sets can be separated from each otherby two closed halfspaces [6]. In contrast, [11] contains a counterexample to separationof a point and a max-min convex set by max-min hyperplanes (equivalently, by max-min halfspaces).The main goal of this paper is to further clarify separation by hyperplanes in max-min algebra. The main result of this paper, Theorem 3.1, shows which closures ofsemispaces are hyperplanes and which are not. As a corollary, we obtain in what caseit is possible to separate a point from a closed max-min convex set by a hyperplane.2. THE STRUCTURE OF SEMISPACESWe recall the structure of the semispaces in Bn at an arbitrary point x0. We followclosely [15].550 V. NITICA AND S. SERGEEVWithout loss of generality we may assume that the coordinates (x01, . . . , x0n) ofthe point x0 are in decreasing order, that is:x01 ≥ · · · ≥ x0n. (3)The set {x01, . . . , x0n} admits a natural subdivision into ordered subsets such thatthe elements of each subset are either equal to each other or are in strictly decreasingorder, sayx01 = · · · = x0k1 > · · · > x0k1+l1+1 = · · · = x0k1+l1+k2 . . .> x0k1+l1+k2+l2+1 = · · · = x0k1+l1+k2+l2+k3 > . . .> x0k1+l1+···+kp−1+lp−1+1 = · · · = x0k1+l1+···+kp−1+lp−1+kp> · · · > x0k1+l1+···+kp+lp(= x0n),(4)where we make the following conventions:1. k1 = 0 if and only if the sequence (4) starts with the strict inequality x01 > x02;in this case l1 6= 0 and the beginning of the sequence will be:x01 > · · · > x0l1 > x0l1+1 = · · · = x0l1+k2 > . . .· · · > x0l1+k2+l2 > x0l1+k2+l2+1 = . . . ;(5)in particular, if (4) has only strict inequalities between its terms one has p = 1, k1 =0, l1 = n. When (4) has only equalities between its terms, one has p = 1, k1 = n, l1 =0.2. lp = 0 if and only if the sequence {x01, . . . , x0n} ends with equalities, that is,with x0n−1 = x0n; in this case, if p ≥ 2, the end of the sequence (4) will be. . . > x0k1+l1+...+kp−1+lp−1+1 = . . . = x0k1+l1+...+kp−1+lp−1+kp , (6)while if p = 1, the whole sequence will be x01 = . . . = x0k1(= x0n). In other words, wetake lp 6= 0 if and only if x0n−1 > x0n.Let us introduce the following notations:L0 = 0,K1 = k1, L1 = K1 + l1 = k1 + l1, (7)Kj = Lj−1 + kj = k1 + l1 + . . .+ kj−1 + lj−1 + kj (j = 2, . . . , p), (8)Lj = Kj + lj = k1 + l1 + . . .+ kj + lj (j = 2, . . . , p). (9)We observe that lj = 0 if and only if Kj = Lj and Lp = n.The following description of semispaces is taken from [15, Proposition 4.1]. Weneed to distinguish the case when the sequence (4) ends with zeros either/or beginswith ones, since some semispaces become empty in that case.Proposition 2.1. Let x0 = (x01, . . . , x0n) ∈ Bn, x01 ≥ . . . ≥ x0n, and let k1, l1, . . . , kp,lp, p be non-negative integers as above.Semispaces and Hyperplanes in max–min Convex Geometry 551a) If 0 < x0i < 1 for all i = 1, . . . , n, then there are n+ 1 semispacesS0(x0), S1(x0), . . . , Sn(x0) at x0, namely:S0(x0) = {x ∈ Bn|xi > x0i for some 1 ≤ i ≤ n}, (10)SKj+q(x0) = {x ∈ Bn|xKj+q < x0Kj+q, or xi > x0i for some Kj + q + 1 ≤ i ≤ n}(q = 1, . . . , lj ; j = 1, . . . , p) if lj 6= 0,(11)SLj−1+q(x0) = {x ∈ Bn|xLj−1+q < x0Lj−1+q, or xi > x0i for some Kj + 1 ≤ i ≤ n}(q = 1, . . . , kj ; j = 1, . . . , p if k1 6= 0, or j = 2, . . . , p if k1 = 0).(12)b) If there exists an index i ∈ {1, . . . , n} such that x0i = 1, but no index j suchthat x0j = 0, then the semispaces at x0 are S1(x0), . . . , Sn(x0) of part a).c) If there exists an index j ∈ {1, . . . , n} such that x0j = 0, but no index i suchthat x0i = 1, then the semispaces at x0 are S0(x0), S1(x0), . . . , Sβ−1(x0) of part a),whereβ := min{1 ≤ j ≤ n| x0j = 0}. (13)d) If there exist indices i, j ∈ {1, . . . , n} such that x0i = 1 and x0j = 0, then thesemispaces at x0 are S1(x0), . . . , Sβ−1(x0) of part a), where β is the number (13).Pictures of all types of semispaces in B2 are shown in Figure 1. The figure istaken from [15].3. MAIN RESULTSIf we take the topological closure of semispaces, all inequalities in (10) – (12) becomenon-strict. We denote such closures by Si(x0).We will also denoteDn = {n︷ ︸︸ ︷(a, . . . , a) | a ∈ B}. (14)This set will be called the diagonal of Bn.Next we investigate when the closures of semispaces are hyperplanes.Theorem 3.1. (Semispaces and Hyperplanes) Assume that x0 ∈ Bn satisfiesx01 ≥ x02 ≥ · · · ≥ x0n and let H := Si(x0) for i = 0, 1 . . . , n. The following statementsare equivalent.a) H takes either of the following forms for some a ∈ B:H+(a) = {x | xi ≥ a for some i = 1, . . . , n}, for a < 1,H−i (a) = {x | xi ≤ a}, for a > 0, i = 1, 2, . . . , n.(15)b) H = Si(y) for some y ∈ Dn.c) H is a hyperplane.552 V. NITICA AND S. SERGEEVa1a2b1b2c1c2d1S0 S1 S2Fig. 1. Semispaces in dimension 2.Semispaces and Hyperplanes in max–min Convex Geometry 553P r o o f . First we observe that a) and b) are equivalent. Indeed, H+(a) = S0(x)and H−i (a) = Si(x) where x = (a, . . . , a).We can representH+(a) = {x |n⊕i=1xi = a⊕n⊕i=1xi},H−i (a) = {x | xi = a ∧ xi}.(16)which shows a)⇒ c).It remains to show that the closure of a semispace that is not of the form (15)cannot be a hyperplane.Case 1. Consider S0(x0) where x0 /∈ Dn.If x0i = 0 for some i, then S0(x0) = Bn = H+(0). Hence we can assume x0i > 0for all i.Let y ∈ Bn be such that yi < x0i for all i, andx01 > y1 > x0n > yn. (17)We define z1 and z2 byz1i ={x01, if i = 1,yi, otherwise,z2i ={x0n, if i = n,yi, otherwise.(18)Obviously S0(x0) contains both z1 and z2, but it does not contain y = z1 ∧ z2. Ourgoal is to show that any hyperplane defined by (2) that contains z1 and z2 will alsocontain y. Equation (2) for z1, z2 and y reduces to, respectively,a1 ∧ x01 ⊕ an ∧ yn ⊕ α = b1 ∧ x01 ⊕ bn ∧ yn ⊕ β, (19)a1 ∧ y1 ⊕ an ∧ x0n ⊕ α = b1 ∧ y1 ⊕ bn ∧ x0n ⊕ β, (20)a1 ∧ y1 ⊕ an ∧ yn ⊕ α = b1 ∧ y1 ⊕ bn ∧ yn ⊕ β, (21)whereα =⊕i6=1,nai ∧ yi, β =⊕i6=1,nbi ∧ yi. (22)We need to show that (19) and (20) together imply (21). We do this by showingthat the minimum of left hand sides of (19) and (20) is always equal to the left handside of (21). By analogy, the same holds for the right hand sides.We first pull α out of the brackets using the distributivity law (b⊕ a)∧ (c⊕ a) =(b ∧ c)⊕ a:(a1 ∧ x01 ⊕ an ∧ yn ⊕ α) ∧ (a1 ∧ y1 ⊕ an ∧ x0n ⊕ α)= ((a1 ∧ x01 ⊕ an ∧ yn) ∧ (a1 ∧ y1 ⊕ an ∧ x0n))⊕ α.(23)It remains to show that(a1 ∧ x01 ⊕ an ∧ yn) ∧ (a1 ∧ y1 ⊕ an ∧ x0n)= a1 ∧ y1 ⊕ an ∧ yn.(24)554 V. NITICA AND S. SERGEEVIf a1, an are large enough then (17) impliesa1 ∧ x01 ≥ a1 ∧ y1 ≥ an ∧ x0n ≥ an ∧ yn, (25)and in this case it is easy to see that (24) holds, both sides being equal to a1 ∧ y1.Note that the first and the third inequalities always hold. The second inequalitymay not hold true, but then a1 ≤ y1, in which case a1 ∧ x01 = a1 ∧ y1. In this casewe use the distributivity again, and this transforms the left hand side of (24) to(a1 ∧ y1)⊕ (an ∧ yn ∧ x0n) = (a1 ∧ y1)⊕ (an ∧ yn), (26)which proves (24) and hence the claim for Case 1.Case 2. Consider Si(x0) where x0 /∈ Dn.Denoten(i) ={i+ 1, if Ks + 1 ≤ i ≤ Ls,Ks+1 + 1, if Ls + 1 ≤ i ≤ Ks+1,(27)where Ks, Ls are defined by (7) – (9).If n(i) = n+1 then Si(x0) = H−i (x0i ). If x0i = 1 or x0j = 0 for some j ≥ n(i) thenSi(x0) = Bn. Otherwise, we construct points y, z and v defined byyj ={1, if j = i,x0j , otherwise,zj ={0, if j ≥ n(i),x0j , otherwise,vj =1, if j = i,0, if j ≥ n(i),x0j , otherwise.(28)It is clear that y and z belong to Si(x0) but v does not. Our goal will be to showthat if a hyperplane defined by (2) contains y and z then it also contains v. Equality(2) reduces in the cases of y, z and v respectively toai ⊕⊕s≥n(i)(as ∧ x0s)⊕ α = bi ⊕⊕s≥n(i)(bs ∧ x0s)⊕ β, (29)ai ∧ x0i ⊕ α = bi ∧ x0i ⊕ β, (30)ai ⊕ α = bi ⊕ β, (31)whereα =⊕s6=i, s<n(i)(as ∧ x0s)⊕ an+1, β =⊕s6=i, s<n(i)(bs ∧ x0s)⊕ bn+1. (32)We need to show that (29) and (30) imply (31). Assume by contradiction thatai ⊕ α 6= bi ⊕ β. Then there exists s ≥ n(i) such that (29) equals as ∧ x0s or bs ∧ x0simplying that x0s ≥ ai⊕ bi⊕α⊕β. But then x0i ≥ x0s ≥ ai⊕ bi⊕α⊕β, and equation(30), which is assumed to hold, is the same as (31), a contradiction. The proof ofCase 2 is complete and the theorem is proved. Semispaces and Hyperplanes in max–min Convex Geometry 555Remark 3.2. We recall that in the tropical (max-plus) convex geometry the closureof any semispace is a hyperplane [12].The key ingredient in the proof of Theorem 3.1 is the construction of exampleswhere a point cannot be separated from a closed semispace by a hyperplane. Theproof of Theorem 3.1 shows that such examples can be constructed for any dimensionand for any semispace except for (15) which are precisely the hyperplanes. The prooffor the case of S0(x0) (Case 1) also shows that such examples can be constructedfor any point outside the diagonal. We conclude the following.Corollary 3.3. (Non-separation by Hyperplanes) Let x ∈ Bn and x /∈ Dn.Then there exists a closed max-min convex set C ⊆ Bn such that x cannot beseparated from C by a hyperplane.Simple counterexamples to separation by hyperplanes in dimension two have beenobtained by one of the authors [11]: as shown on Figure 2, it actually suffices to takecertain max-min segments [14, 16]. The convex set C = [z1, z2] cannot be separatedby hyperplanes from the point x = z1 ∧ z2.z1z2xCFig. 2. Forbidden separation.Such examples can be extended cylindrically to any dimension, which is preciselythe geometric idea of the proof of Theorem 3.1.On the other hand, as the semispaces taken at a diagonal point are hyperplanes, itis possible to separate a point on the diagonal from a closed convex set “classically”.Corollary 3.4. (Diagonal Separation by Hyperplanes) Let x ∈ Bn and x ∈Dn. Then any closed max-min convex set C ⊆ Bn such that x /∈ C, can be separatedfrom x by a hyperplane.P r o o f . Since x 6∈ C, there is a semispace S at x containing C [15, Theorem 5.1].If S = S0(x), then for any y ∈ C there exists 1 ≤ i ≤ n such that yi > xi. Dueto compactness of C, there is δ > 0 such that above inequalities can be replaced byyi ≥ xi + δ. For x+ δ = (x1 + δ, . . . , xn + δ), this implies C is included in S0(x+ δ).By Theorem 3.1 S0(x+ δ) is a hyperplane. Moreover it does not contain x.556 V. NITICA AND S. SERGEEVIf S = Si(x), then yi < xi for any y ∈ C. Due to compactness of C, there isδ > 0 such that above inequality can be replaced by yi ≤ xi − δ. This implies C isincluded in Si(x− δ). By Theorem 3.1 Si(x− δ) is a hyperplane. It avoids x. ACKNOWLEDGEMENTThis research was supported by NSF grant DMS-0500832 (Viorel Nitica), EPSRC grantRRAH12809, RFBR grant 08-01-00601 and joint RFBR/CNRS grant 05-01-02807(SergĕıSergeev).(Received April 14, 2010)REFERENCES[1] G. Birkhoff: Lattice Theory. American Mathematical Society, Providence, RI 1993.[2] K. Cechlárová: Eigenvectors in bottleneck algebra. Linear Algebra Appl. 175 (1992),63–73.[3] G. Cohen, S. Gaubert, J. P. Quadrat, and I. Singer: Max-plus convex sets andfunctions. In: Idempotent Mathematics and Mathematical Physics (G. Litvinov andV. Maslov, eds.), AMS, Providence 2005, pp. 105–129. E-print arXiv:math/0308166.[4] M. Develin and B. Sturmfels: Tropical convexity. Documenta Math. 9 (2004), 1–27.E-print arXiv:math/0308254.[5] S. Gaubert and R. Katz: Max-plus convex geometry. In: Lecture Notes in ComputerScience 4136, Springer, New York 2006. pp. 192–206.[6] S. Gaubert and S. Sergeev: Cyclic projectors and separation theorems in idempotentconvex geometry. J. Math. Sci. 155 (2008), 6, 815–829. E-print arXiv:math/0706.3347.[7] M. Gavalec: Periodicity in Extremal Algebras. Gaudeamus, Hradec Králové 2004.[8] M. Gavalec and J. Plávka: Strong regularity of matrices in general max-min algebra.Linear Algebra Appl. 371 (2003), 241–254.[9] J. Golan: Semirings and Their Applications. Kluwer, Dordrecht 2000.[10] G. L. Litvinov, V.P. Maslov, and G. B. Shpiz: Idempotent functional analysis: analgebraic approach. Math. Notes 69 (2001), 5, 758–797.[11] V. Nitica: The structure of max-min hyperplanes. Linear Algebra Appl. (2009),doi:10.1016/j.laa.2009.08.022.[12] V. Nitica and I. Singer: Max-plus convex sets and max-plus semispaces. I. Optimiza-tion 56 (2007), 171–205.[13] V. Nitica and I. Singer: Max-plus convex sets and max-plus semispaces. II. Optimiza-tion 56 (2007), 293–303.[14] V. Nitica and I. Singer: Contributions to max-min convex geometry. I. Segments.Linear Algebra Appl. 428 (2008), 7, 1439–1459.[15] V. Nitica and I. Singer: Contributions to max-min convex geometry. II. Semispacesand convex sets. Linear Algebra Appl. 428 (2008), 8–9, 2085–2115.[16] S.N. Sergeev: Algorithmic complexity of a problem of idempotent convex geometry.Math. Notes 74 (2003), 6, 848–852.Semispaces and Hyperplanes in max–min Convex Geometry 557[17] K. Zimmermann: A general separation theorem in extremal algebras. Ekonom.-Mat.Obzor 13 (1977), 179–201.[18] K. Zimmermann: Convexity in semimodules. Ekonom.-Mat. Obzor 17 (1981), 199–213.Viorel Nitica, Department of Mathematics, West Chester University, PA 19383, USA, andInstitute of Mathematics, P.O. Box 1-764, Bucharest. Romaniae-mail: vnitica@wcupa.eduSergĕı Sergeev, School of Mathematics, University of Birmingham, Edgbaston, BirminghamB15 2TT. United Kingdom.e-mail: sergiej@gmail.com

On hyperplanes and semispaces in max–min convex geometry

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On hyperplanes and semispaces in max-min convex geometry

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