The tropical analogue of polar cones

THE TR OPICAL ANALOGUE OF POLAR CONES ST ´ EPHANE GA UBER T AND RICARDO D. KA TZ Abstract. W e study the max-plus or tropical analogue of the notion of p olar: the polar of a cone represen ts the set of linear inequalities satisfied b y i ts elemen ts. W e e stablish an analogue of the bipolar theorem, which cha racterizes all the inequalities satisfied by the elemen ts of a tropical con ve x cone. W e derive this characterization from a new separation theorem. W e also establish v ariants of these results concerning systems of linear equalities. 1. Introduction Max-plus or tropical alg ebra r e fers to the analogue of cla ssical a lgebra obtained by considering the max-plus semiring R max , which is the set R ∪ {−∞} equipp ed with the addition a ⊕ b := max( a, b ) and the multiplication a ⊗ b := a + b . Max-plus c onvex c ones o r semimo dules are subsets V o f R n max stable by ma x-plus linear com binations, meaning that max( λ + x, µ + y ) ∈ V (1) for all x, y ∈ V and λ, µ ∈ R max . Her e, λ + x denotes the vector with entries λ + x i and the “max” of tw o v ectors is understo o d entrywise. Max-plus c onvex sets ar e subsets V of R n max which satisfy (1) for all x, y ∈ V and λ, µ ∈ R max such that max( λ, µ ) = 0. Max-plus c o nv e x sets and co nes hav e b een s tudied by several author s with dif- ferent motiv atio ns . They w ere in tro duced b y K. Z immermann [Zim7 7] when s tudying discrete opti- mization problems. Another motiv ation arose from the use b y Maslov [Mas8 7] of a max- plus ana- logue of the sup erp ositio n principle, whic h a pplies to the solutions of a stationa ry Hamilton-Jaco bi equation whose Hamiltonia n is conv ex in the adjoint v aria ble. Hence, the space of solutions has a structure analogous to that of a linear space. This was one of the motiv ations fo r the developmen t of the Idemp otent Analy- sis [MS92, K M97] which includes the study , by Litvinov, Mas lov, and Shpiz [LMS0 1] of the idemp otent spaces, of which max-plus cones are specia l cases. See [LS07] for a recent developmen t. A third motiv ation c omes from the a lgebraic appro ach to discrete event systems initiated by Cohen, Dub ois, Quadra t a nd Viot [CDQV85]. The spaces that a ppea r in the study of max-plus linear dynamical systems (reachable and observ a ble spac es) are naturally equipp ed with str uctures of max-plus c ones [CGQ99]. This motiv ated 2000 Mathematics Subje ct Classific ation. prim ary: 52A01, secondary: 16Y60, 46A55. Key wor ds and phr ases. Max-plus semiring, max-plus conv exity , tropical con ve xity , extremal con vex ity , B -conv exity , duality , s eparation theorem, F ark as Lemma, semimo dules, idemp oten t spaces. The fir s t author was partially supported by the join t CNRS-RFBR gr an t n umber 05-01-02807. 1 2 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ the study of max-plus cones or se mimo dules by Cohen, Quadrat, and the first author [CGQ01, CGQ04]. The results of [CGQ04] w ere co mpleted in a work with Singer [CGQ S05]. A development by the second author [Kat07] of the work on discrete even t s ystems includes a max-plus cone based synthesis of cont ro lle rs. See also [GK04]. The theory of max-plus conv ex cones, or tropical conv ex sets, has recently bee n developed in relation to tropical geometry . The tropical analogues of poly- top es were considered by Develin and Sturmfels [DS04 ], who established a re- mark a ble connection with phylogenetic ana ly sis, s howing that tree metrics may b e thought of in terms of tropical p o ly top es. T ropica l co nv e x it y was further studied by Joswig [Jo s 05] a nd Develin and Y u [D Y07]. See [JSY07] for a new dev elopment. Another in terest in the max-plus analog ues of conv ex co nes comes from abstract conv ex a na lysis [Sin97], as can be seen in the w ork of Nitica and Singer [NS07 a, NS07b, NS07c]. Indep endently , Briec, Horv ath and Rubinov [B H04, BHR05] intro- duced and studied the notion of B -conv exity which is another name for max-plus conv exity . See also [AR06]. These motiv atio ns led to the pro of of max-plus analog ues o f classical results such as: Hahn-Banach theorem either in analytic (extensio n o f linea r forms) o r geo- metric (separation) form [Zim77, SS92, CGQ 0 1, DS04, CGQ04, BHR05, CGQS0 5], Minko wski theorem (genera tio n of a closed conv ex set by its extreme p oints) [Hel88, GK06, BSS07, GK07], and Helly and Carath´ eo dory t yp e theorems [BH04, GS07, GM08]. How ever, so me duality r esults are still missing, b eca use the idemp o tency of addition creates difficulties which are absent in the class ical theory . F or example, in the classica l theory there exists a bijectiv e corr esp ondence betw een the facets of a conv ex co ne and the extreme rays of its p olar cone. This kind o f pr op erties, rela ting int erna l a nd external repre s entations of conv ex cones, a re not yet w ell under sto o d in the ma x-plus case. Indeed, classical duality theory r elates a co nv ex cone with the set of linear in- equalities that its p oints satisfy . In the ma x -plus setting, the p olar of a max-plus conv ex co ne V ⊂ R n max can be defined as V ◦ = { ( f , g ) ∈ ( R n max ) 2 | max i ( f i + x i ) ≤ max j ( g j + x j ) , ∀ x ∈ V } . Conv ersely , we may co ns ider the set o f p oints satisfying a given set of inequalities. This leads us to define, for all W ⊂ ( R n max ) 2 , a “dua l” pola r cone: W ⋄ = { x ∈ R n max | max i ( f i + x i ) ≤ max j ( g j + x j ) , ∀ ( f , g ) ∈ W } . It follows from the sepa ration theor em o f [Zim77, SS92, CGQS0 5] that a closed max-plus conv ex co ne V is characterized by its p ola r cone: ( V ◦ ) ⋄ = V . It is natural to ask which subsets W of ( R n max ) 2 arise a s p ola rs of cones, or equiv a- lent ly , whic h subsets are of the for m W = ( U ⋄ ) ◦ for some U ⊂ ( R n max ) 2 . The main result of this pa per is the following analogue o f the bipola r theorem: Theorem 1 (Bipo lar) . A subset W ⊂ ( R n max ) 2 is the p olar of a c one if, and only if, the fol lowing c onditions ar e satisfie d: (i) W is a close d max-plus c onvex c one, (ii) f ≤ g = ⇒ ( f , g ) ∈ W , (iii) ( f , g ) ∈ W , ( g , h ) ∈ W = ⇒ ( f , h ) ∈ W . THE TROPICAL ANALOGUE OF POLAR CONES 3 This follo ws from T he o rem 1 0 below. W e deduce this result fr om a new separa- tion theorem, Theorem 8, whic h is in some s ense a dual of the s eparation theorem of [CGQS05]. W e sha ll fir st establish similar results when the scalars form a complete s emir- ing which is reflexive in the sense of [CGQ04]. This applies in pa rticular to the completion of the max-plus semir ing R max . Thanks to completeness and reflexivity prop erties, we shall obtain the separation theore m b y alg ebraic and or der theoreti- cal a rguments, relying o n re s iduation theory (Galois c onnection in lattices). Then, we shall der ive the results in the case of the max- plus s emiring by a p erturbation argument. The notion of po la r is illustrated in Figures 1 and 2 below. Fig ure 1 repre sents a max-plus co nvex cone V a nd e ig ht max-plus linear inequalities comp osing a set W such tha t V = W ⋄ . F or visibility rea sons, w e hav e depicted co mpletely only one of thes e inequalities, which is also shown separa tely . Detailed ex planations of the constructions of Figure 1 c an be found in Example 3. W e shall show that if W is a finite subset o f ( R n max ) 2 , representing a finite set of linear inequalities, then there are in gener al elemen ts in the tra nsitive clo sure of W which are not linea r combinations of W . This means that there are so me inequalities whic h can b e logically deduced from the finite set o f ineq ualities max i ( f i + x i ) ≤ max j ( g j + x j ) , ( f , g ) ∈ W but which cannot b e obtained by a linea r combination of these inequalities. Hence, Theorem 1 shows that there is no dire c t analogue o f F ark as lemma. This is related to the difficulties in defining the faces of tropical po lyhedra, due to the obstructions men tioned in [GK07] and [DY07]. Some pro p er ties concerning ma x -plus convex cones ar e analogo us to the case of cones in classica l conv exity . Ho wev er, max-plus co nv ex cones also hav e s ome features whic h ar e reminiscent of classical linear spaces. This is b ecause in max - plus alg e bra, an inequality max i ( f i + x i ) ≤ max j ( g j + x j ) can alwa ys be rewr itten as an equality , max i (max( f i , g i ) + x i ) = max j ( g j + x j ). This po in t of view leads to define [GP97] the or thogonal of a max- plus con vex cone V , V ⊥ = { ( f , g ) ∈ ( R n max ) 2 | max i ( f i + x i ) = max j ( g j + x j ) , ∀ x ∈ V } . Such an o rthogona l is a congruence of semimo dules, in the sense that will b e defined in Section 3. Like in the case of linear inequa lities, w e may a lso c o nsider the set of po int s satisfying a given set of linear equalities: for W ⊂ ( R n max ) 2 define W ⊤ = { x ∈ R n max | max i ( f i + x i ) = max j ( g j + x j ) , ∀ ( f , g ) ∈ W } . As a co nsequence of the separation theorem of [Zim77, SS92, CGQS05] we have ( V ⊥ ) ⊤ = V , if V is a closed cone, a res ult whic h w as already stated in [GP 97] in the pa rticular case of finitely generated cones. W e shall prove, as a v ariant o f o ur main theorem, the following bio rthogona l theor em. Like the bipola r theorem ab ove (Theorem 1), it also follo ws from Theorem 10. Theorem 2 (Biorthogona l) . A subset W ⊂ ( R n max ) 2 is the ortho gonal of a c one if, and only if, the fol lowing c onditions ar e satisfie d: (i) W is a close d max-plus c onvex c one, 4 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ (ii) ( f , f ) ∈ W , ∀ f ∈ R n max , (iii) ( f , g ) ∈ W = ⇒ ( g , f ) ∈ W , (iv) ( f , g ) ∈ W , ( g , h ) ∈ W = ⇒ ( f , h ) ∈ W . Congruences are used in [LGK L08] to solve so me observ ability pro blems for max- plus linear discrete even t s ystems, i.e. to reco nstruct so me infor mation on the state of the s ystem from o bserv a tions. The pr esent bior thogonality theorem is used there to construct dyna mic obs e rvers: in fact, cones are simpler to manipulate than co n- gruences, due to the absence of “doubling” of the dimension, duality allows us to reduce algo r ithmic problems concerning congruences to algor ithmic problems con- cerning cones. The duality res ults of the present pap er may also be useful in the recent applicatio n of max-plus p olyhedra to s tatic analysis by a bstract interpreta- tion [A GG08]. 2. Preliminaries In this section we recall s ome notions and results concerning semimo dules over idempo tent semirings which we shall need throug hout this pa pe r. W e refer the reader to [BCOQ92, CGQ04, BJ 7 2] for more background. A monotone map F : T → S betw een tw o ordered sets ( T , ≤ ) and ( S, ≤ ) is said to b e r esiduate d if for e a ch s ∈ S the least upp er b ound of the set { t ∈ T | F ( t ) ≤ s } exists and b e longs to it. When F is residuated, the map F ♯ : S → T defined b y F ♯ ( s ) := ∨{ t ∈ T | F ( t ) ≤ s } , wher e ∨ Q deno tes the least upper b ound o f Q , is called the r esidual of F . W e s ay that F is c ontinuous if F ( ∨ Q ) = ∨ F ( Q ) for all Q ⊂ T , whe r e F ( Q ) := { F ( t ) | t ∈ Q } . The ter m contin uous refer s to the Scott top ology [GHK + 03], which is a (non- Hausdorff ) topo logy well adapted to the study of completed or dered algebraic structures. When T and S are complete o rdered sets, meaning that any of their s ubs ets admits a le ast upp er b ound, it can b e shown that F : T → S is residuated if, and only if, it is contin uo us (see [BJ7 2, Th. 5.2] or [BCOQ92, Th. 4.50]). Recall tha t any idemp otent semiring ( K , ⊕ , ⊗ , ε, e ), where ε and e repr esent the neutral elements for addition ⊕ and multiplication ⊗ resp ectively , can b e equipp ed with the natur al orde r relation: a ≤ b ⇐ ⇒ a ⊕ b = b , for which ∨ { a, b } = a ⊕ b . W e say that an idemp otent semiring K is c omplete if it is complete as a na turally ordered set, and if the left and right m ultiplications, L a : K → K , L a ( b ) = a ⊗ b and R a : K → K , R a ( b ) = b ⊗ a , are contin uous . Given a semiring ( K , ⊕ , ⊗ , ε K , e ), a right K -semimo du le is a comm utative monoid ( X, ˆ ⊕ ), with neutral element ε X , equipp ed with a ma p X × K → X , ( x, λ ) → x · λ , (right action), which sa tisfies: ( x ˆ ⊕ z ) · λ = x · λ ˆ ⊕ z · λ , x · ( λ ⊕ µ ) = x · λ ˆ ⊕ x · µ , x · ( λ ⊗ µ ) = ( x · λ ) · µ , x · ε K = ε X , ε X · λ = ε X , x · e = x , for a ll x, z ∈ X a nd λ, µ ∈ K . Henceforth, the semimo dule addition will be denoted by ⊕ (instead of ˆ ⊕ ), like the semiring addition, and we will use concatenatio n to denote b oth the multiplication o f K and the right action. Also for simplicity , we will de no te by ε b o th the zero element ε K of K and the zer o element ε X of X . Left K -se mimo dules are defined dually . A subsemimo dule o f X is a subset V ⊂ X such that xλ ⊕ z µ ∈ V for a ll x, z ∈ V a nd λ, µ ∈ K . Note that a right (or left) K -semimo dule ( X , ⊕ ) is necessar ily idemp otent when K is idemp otent. In this case , X is said to b e c omplete , if it is complete a s a natur a lly or dered set, a nd if for all THE TROPICAL ANALOGUE OF POLAR CONES 5 u ∈ X a nd λ ∈ K , the ma ps R λ : X → X , z 7→ z λ a nd L u : K → X , µ 7→ uµ , are bo th contin uous . Then, we can define u \ x := L ♯ u ( x ) = ∨{ µ ∈ K | u µ ≤ x } and x/λ := R ♯ λ ( x ) = ∨{ z ∈ X | z λ ≤ x } , for a ll x, u ∈ X and λ ∈ K . Note that by definition, we have u ( u \ x ) ≤ x and ( x/λ ) λ ≤ x . A subsemimo dule V o f X is complete, if ∨ Q ∈ V for all Q ⊂ V . It is convenien t to recall the following univ ersal s eparation theorem for complete semimo dules. Theorem 3 ([CGQ04, Th. 8]) . L et V ⊂ X b e a c omplete subsemimo du le. A ssume that x 6∈ V . Then, v ∈ V = ⇒ v \ P V ( x ) = v \ x and x \ P V ( x ) 6 = x \ x , wher e P V ( x ) := ∨{ v ∈ V | v ≤ x } . Example 1 . The set R n max is a r ig ht R max -semimo dule when it is equipp ed with the usua l op era tions: ( x ⊕ z ) i := x i ⊕ z i and ( xλ ) i := x i λ for all x, z ∈ R n max and λ ∈ R max . Since R max is not a complete semiring, it will b e conv enient to consider the c omplete d ma x-plus semiring R max , which is obtained by adjoining the elemen t + ∞ to R max . In o rder to state s e paration theor ems with the usua l sca la r pro duct form, we next recall the notion of dual pair of [CGQ04], which is analogous to the one that arises in the theory of top o logical vector spa ces (see for ex ample [Bou64]). As usual, a map b etw ee n right (res p. left) K - semimo dules is right (res p. left) line ar if it pre serves finite sums and commutes with the action of sca lars. W e call pr e- dual p air for a complete idemp otent semiring K , a complete right K -semimo dule X together with a complete left K -s emimo dule Y equipp ed with a bracket h· | ·i from Y × X to K , s uch that, for all x ∈ X and y ∈ Y , the maps y 7→ h y | x i a nd x 7→ h y | x i are resp ectively left and rig ht linear , and contin uo us. W e shall deno te by ( Y , X ) this pre-dual pair. W e s ay tha t Y sep ar ates X if ( ∀ y ∈ Y , h y | x i = h y | x ′ i ) = ⇒ x = x ′ , and that X sep ar ates Y if ( ∀ x ∈ X , h y | x i = h y ′ | x i ) = ⇒ y = y ′ . A pre-dual pair ( Y , X ) such that X s e parates Y and Y separa tes X is called dual p air . When X is a complete right K -s emimo dule, we call opp osite semimo dule of X the left K -semimo dule X op whose set of elements is X equipp ed with the addition ( x, u ) 7→ ∧ { x, u } and the left action ( λ, x ) 7→ x/λ . Her e, ∧ { x, u } is the gr eatest low er b ound of { x, u } fo r the natura l order of X . The fact that X op is a complete left K -semimo dule follows fro m prope r ties of the r esidual map x/λ (see [CGQ0 4] for details). In particular , the co mplete idemp otent semiring K can be thought of as a righ t K -semimo dule, and so the same construction and no tation apply to K . Theorem 4 ([CGQ04, Th. 2 2]) . L et X b e a c omplete right K -semimo dule. Then, the semimo dules X op and X form a dual p air for the br acket X op × X → K op , ( y , x ) 7→ h y | x i = x \ y . 6 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ Given a pre-dual pair ( Y , X ) for K and an arbitrar y element ϕ ∈ K , consider the maps: X → Y , x 7→ − x = ∨{ y ∈ Y | h y | x i ≤ ϕ } , Y → X , y 7→ y − = ∨{ x ∈ X | h y | x i ≤ ϕ } . Note tha t if the dual pair for K op of Theorem 4 is used, then the suprema in the definition ab ove are actually infima (in terms of the order of K ) and the inequalities are reversed. The element s of X and Y of the form y − and − x , r esp ectively , ar e called close d . F or λ ∈ K , we define λ − := λ \ ϕ = ∨{ µ ∈ K | λµ ≤ ϕ } and − λ := ϕ/λ = ∨{ µ ∈ K | µλ ≤ ϕ } . A complete idemp otent semiring K is r eflexive if there exists ϕ ∈ K such that − ( λ − ) = λ and ( − λ ) − = λ for all λ ∈ K . W e shall need the following prop osition. Prop ositi on 1 ([CGQ04, Prop. 32 and Pro p. 27]) . If K is r eflex ive and ( Y , X ) is a pr e-dual p air for which Y sep ar ates X , t hen, al l t he elements of X ar e close d. Mor e over, for al l x, u ∈ X it is satisfie d: u \ x = h − x | u i − . (2) Example 2 . The co mpleted max-plus semiring R max is reflexive. If we tak e ϕ = 0, then − λ = λ − = − λ for all λ ∈ R max . If R n max is equipp e d with the brack et h y | x i := ⊕ i y i x i , then (2) b eco mes u \ x = − ( ⊕ i u i ( − x i )) for all u, x ∈ R n max . The results a b ove are rela ted to the max-plus analogues of the repre sentation the- orem for linear forms and the analytic form of the Hahn-Ba na ch theo r em of [CGQ04] which extend the corresp onding results of [LMS01]. F or a detailed presentation and examples of pr e-dual pairs a nd reflexiv e semir ings, we refer the rea der to [CGQ0 4]. 3. Sep ara tion Theorems Given a rig ht K - semimo dule X , a subset W of X 2 is called a pr e-c ongruenc e if it is a subsemimo dule s uch that ( f , f ) ∈ W , ∀ f ∈ X and ( f , g ) ∈ W , ( g , h ) ∈ W = ⇒ ( f , h ) ∈ W . If in a ddition a pre-congruence W verifies ( f , g ) ∈ W = ⇒ ( g , f ) ∈ W , then W is said to b e a c ongruenc e . In other words, a pre-co ngruence (resp. co n- gruence) is a pre-o r der (resp. eq uiv alence) relation on X which has a semimo dule structure when it is thoug ht of as a subset of the semimo dule X 2 . A pre-co ngruence W which sa tisfies the prope r ty f ≤ g = ⇒ ( f , g ) ∈ W , (3) is called a p olar c one . R emark 1 . When W is a pre-congr ue nce , note that (3) is equiv alent to f ≤ g , ( g , h ) ∈ W = ⇒ ( f , h ) ∈ W . (4) THE TROPICAL ANALOGUE OF POLAR CONES 7 Indeed, cle arly (4 ) implies (3) b ecause ( g , g ) ∈ W for all g ∈ X . Conv ersely , since W is a pre-co ngruence, it follows that ( f , g ) ∈ W and ( g , h ) ∈ W imply ( f , h ) ∈ W , and so (3) implies (4). The definition ab ov e is motiv ated b y the classical no tion of p olar cones. Recall that if V ⊂ R n is a (classical) con vex cone , then its p olar is defined as { f ∈ R n | X i f i x i ≤ 0 , ∀ x ∈ V } , where f i x i denotes the usual multiplication of the real num b ers f i and x i . The direct extension of this definition to X = R n max , i.e. { f ∈ R n max | ⊕ i f i x i ≤ ε, ∀ x ∈ V } , where f i x i denotes f i ⊗ x i = f i + x i , the usual addition o f real num b er s, is not conv enient be c a use this set is usually trivial independently of V ⊂ R n max . Howev er, as it was men tioned in the intro duction, the notion o f p o la r can be extended to the max-plus alge br a framework if we co nsider pairs o f vectors instead of individua l vectors, i.e. if we define the p olar of V by (5) W := { ( f , g ) ∈ ( R n max ) 2 | ⊕ i f i x i ≤ ⊕ j g j x j , ∀ x ∈ V } . Observe that an y s et of this form is a p olar cone of X 2 in the s e nse defined ab ove. How ever, no t a ll p o lar cones hav e this for m (i.e. are the p olar of a cone V ), take for example W = { ( f , g ) ∈ ( R max ) 2 | g = ε = ⇒ f = ε } . Example 3 . Let K = R max and X = R 3 max . Consider the semimo dule V ⊂ X generated b y the vectors a = (1 , 0 , 2) T , b = (1 , 1 , 0) T and c = (2 , 4 , 0) T , i.e. the semimo dule comp osed o f all the max- plus linear co m binations of these three vec- tors. This semimo dule is represented by the b ounded dark gray reg ion o f Figur e 1 together with the segmen ts joining the points a and c to it. In this figur e, ev ery element of V with finite en tries is pr o jected or thogonally onto a ny plane orthogo nal to the main diagona l (1 , 1 , 1) T of R 3 . Note that vectors that are prop ortiona l in the max-plus sense are sent to the sa me point, so this g ives a faithful image of V . If we define the se mimo dule W ⊂ X 2 by (5), then it ca n b e check ed tha t W is a p olar cone of X 2 . In o ther words, this p olar cone is comp osed of a ll the pairs of vectors ( f , g ) such that the corresp onding max-plus ha lf-space { x ∈ R 3 max | ⊕ i f i x i ≤ ⊕ j g j x j } contains V . F or example, if we take f = (2 , ε , ε ) T and g = ( ε, 0 , 3) T , then ( f , g ) ∈ W be cause a , b , c and hence V a re contained in the ha lf- space { x ∈ R 3 max | 2 x 1 ≤ x 2 ⊕ 3 x 3 } , whic h is represented b y the unbounded light gray region of Figure 1. In this case there exist eight max-plus linear inequalities ⊕ i f i x i ≤ ⊕ j g j x j which satisfy that V is the intersection of the cor resp onding half-spaces. These inequali- ties, whic h a re asso ciated with elements of W as explained ab ov e, are the following: x 2 ≤ 2 x 1 , x 2 ≤ 4 x 3 , x 3 ≤ 2 x 2 , x 3 ≤ 1 x 1 , x 1 ≤ 1 x 2 , x 1 ≤ 2 x 3 , 1 x 1 ≤ 1 x 2 ⊕ x 3 and 2 x 1 ≤ x 2 ⊕ 3 x 3 . The b oundaries of the corre s po nding eight half-spaces have been represented in Figure 1. Only the half-spa ce corr esp onding to the last ineq uality has b een depicted completely (no te that this inequa lity is the one ass o ciated with the pair o f v ector s ( f , g ) ∈ W consider ed above). Since the purp ose of this section is to esta blish sepa r ation theorems for complete congruences and po lar cones, in the rest of this section we will assume that K is a complete idempotent semiring and that X is a complete rig ht K -semimo dule. 8 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ a x 2 x 1 V x 3 c b x 2 x 1 x 3 2 x 1 ≤ x 2 ⊕ 3 x 3 V Figure 1. The semimo dule V , a suppo rting ha lf-space, and the po lar of V . Given a complete pre-congr uence W ⊂ X 2 , for g ∈ X , we set ˆ g := ∨{ f ∈ X | ( f , g ) ∈ W } . (6) Observe that ( ˆ g , g ) ∈ W b eca use W is complete. Moreov er, since ( g , g ) ∈ W , it follows that g ≤ ˆ g for all g ∈ X . When W is a cong ruence, ˆ g is nothing but the ma ximal element in the equiv a - lence class of g modulo W . Ther efore, in this ca se we have the following obvious prop erties, whic h hold for all g , f ∈ X , g ≤ ˆ g = ˆ ˆ g , (7) ( f , g ) ∈ W ⇐ ⇒ ˆ f = ˆ g . (8) W e shall use the fact that f ≤ g = ⇒ ˆ f ≤ ˆ g . (9) Indeed, if f ≤ g then g = f ⊕ g , and if h ∈ X is such that ( h, f ) ∈ W , w e ha ve ( h ⊕ g , g ) = ( h ⊕ g , f ⊕ g ) = ( h, f ) ⊕ ( g , g ) ∈ W be c ause W is a pre-congruence. It follows tha t ˆ g ≥ h ⊕ g ≥ h . Since this ho lds for all h ∈ X suc h tha t ( h, f ) ∈ W , we deduce that ˆ g ≥ ˆ f , which sho ws (9). THE TROPICAL ANALOGUE OF POLAR CONES 9 Lemma 1. L et W ⊂ X 2 b e a c omplete pr e-c ongruenc e. Then, ( f , g ) ∈ W = ⇒ g \ ˆ h ≤ f \ ˆ h for al l h ∈ X . Pr o of. Let λ := g \ ˆ h , so that ˆ h ≥ g λ , or equiv alently ˆ h = ˆ h ⊕ g λ . Since W is a pre-congr uence, fro m ( ˆ h ⊕ f λ, ˆ h ) = ( ˆ h ⊕ f λ, ˆ h ⊕ g λ ) = ( ˆ h, ˆ h ) ⊕ ( f , g ) λ ∈ W a nd ( ˆ h, h ) ∈ W , we deduce that ( ˆ h ⊕ f λ, h ) ∈ W and so f λ ≤ ˆ h ⊕ f λ ≤ ˆ h . Hence, f \ ˆ h ≥ λ , i.e. f \ ˆ h ≥ g \ ˆ h .  Now it is p ossible to prov e the following separ ation theorem for c omplete po lar cones. Theorem 5. L et W ⊂ X 2 b e a c omplete p olar c one. Assume that s, t ∈ X ar e such that ( s, t ) 6∈ W . Then, ther e exists x ∈ X such that ( f , g ) ∈ W = ⇒ g \ x ≤ f \ x and t \ x 6≤ s \ x . R emark 2 . In other words, Theorem 5 says that the s e t M := { ( f , g ) ∈ X 2 | g \ x ≤ f \ x } sep ar ates the complete polar cone W a nd the p oint ( s, t ): W ⊂ M and ( s, t ) 6∈ M . Pr o of of The or em 5. W e will show that the asser tion of the theo r em holds with x := ˆ t . In the firs t place, note that if x = ˆ t , then by Lemma 1 the first ass e r tion of the theorem is satisfied. Assume that t \ ˆ t ≤ s \ ˆ t . Then, w e g et s ( t \ ˆ t ) ≤ s ( s \ ˆ t ) ≤ ˆ t . Since ˆ t ≥ t , we hav e t \ ˆ t ≥ e , and so s ≤ ˆ t . Thus, from s ≤ ˆ t and ( ˆ t, t ) ∈ W , it follows that ( s, t ) ∈ W b ecause W is a p olar cone, which contradicts o ur assumption.  In the case of co mplete congruences, w e hav e the following separation theorem. Theorem 6. L et W ⊂ X 2 b e a c omplete c ongruenc e. Assume that s , t ∈ X ar e such that ( s, t ) 6∈ W . Then, ther e exists x ∈ X such that ( f , g ) ∈ W = ⇒ f \ x = g \ x and s \ x 6 = t \ x . Pr o of. Since ( s, t ) 6∈ W , b y (8) w e have ˆ s 6 = ˆ t , hence , ˆ s 6≤ ˆ t and/o r ˆ t 6≤ ˆ s . Assume for instance that ˆ t 6≤ ˆ s . Then, we claim that s \ ˆ s 6 = t \ ˆ s . Indeed, if s \ ˆ s = t \ ˆ s , we get t ( s \ ˆ s ) = t ( t \ ˆ s ) ≤ ˆ s . Since ˆ s ≥ s , we hav e s \ ˆ s ≥ e , and so t ≤ ˆ s . F ro m (7) and (9), it follows that ˆ t ≤ ˆ ˆ s = ˆ s , which contradicts o ur assumption, s o this prov es our cla im. Finally , as ( f , g ) ∈ W implies that ( g , f ) ∈ W bec ause W is a congruence, b y Lemma 1 the assertion o f the theorem holds with x := ˆ s .  T o recov er separa tion theorems with the usual scalar pro duct for m, we need to consider reflexive semiring s . 10 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ Corollary 1. L et W ⊂ X 2 b e a c omplete c ongruenc e and ( Y , X ) b e a pr e-dual p air for a r eflexive semiring K such t hat Y sep ar ates X . I f ( s, t ) 6∈ W , then, t her e exists y ∈ Y such that ( f , g ) ∈ W = ⇒ h y | f i = h y | g i and h y | s i 6 = h y | t i . Pr o of. By Theorem 6 and Prop osition 1 it follo ws that ( f , g ) ∈ W = ⇒ h − x | f i − = h − x | g i − and h − x | s i − 6 = h − x | t i − , for s o me x ∈ X . Since K is reflexive, the map λ 7→ λ − is injectiv e, and thus w e hav e ( f , g ) ∈ W = ⇒ h − x | f i = h − x | g i and h − x | s i 6 = h − x | t i . Therefore, the assertion of the theo rem holds with y := − x .  The following cor ollary of Theorem 5 and Pr o p osition 1 can b e prov ed a long similar lines, using the fact that λ 1 ≤ λ 2 = ⇒ − λ 2 ≤ − λ 1 , λ − 2 ≤ λ − 1 for all λ 1 , λ 2 ∈ K . Corollary 2. L et W ⊂ X 2 b e a c omplete p olar c one and ( Y , X ) b e a pr e- dual p air for a r eflexive semiring K such t hat Y sep ar ates X . I f ( s, t ) 6∈ W , then, t her e exists y ∈ Y such that ( f , g ) ∈ W = ⇒ h y | f i ≤ h y | g i and h y | s i 6≤ h y | t i . Given a pre-dual pair ( Y , X ), w e define the following cor resp ondences b etw een subsemimo dules of X 2 and Y : W ⊂ X 2 7→ W ⋄ := { y ∈ Y | h y | f i ≤ h y | g i , ∀ ( f , g ) ∈ W } , V ⊂ Y 7→ V ◦ := { ( f , g ) ∈ X 2 | h y | f i ≤ h y | g i , ∀ y ∈ V } , and W ⊂ X 2 7→ W ⊤ := { y ∈ Y | h y | f i = h y | g i , ∀ ( f , g ) ∈ W } , V ⊂ Y 7→ V ⊥ := { ( f , g ) ∈ X 2 | h y | f i = h y | g i , ∀ y ∈ V } . Then, ta king Y = X op and h y | x i = x \ y , the universal separ ation theorem for complete semimo dules of Section 2 (se e also [CGQ 04, Th. 13]) implies that the following statements ar e equiv alent: (i) V ⊂ Y is a complete semimo dule, (ii) V = ( V ◦ ) ⋄ , (iii) V = ( V ⊥ ) ⊤ . Now, thanks to the previous separation theorems, w e can pro ve the dual result. THE TROPICAL ANALOGUE OF POLAR CONES 11 Theorem 7. L et ( Y , X ) b e a pr e-dual p air which satisfies the fol lowing pr op erty: If W ⊂ X 2 is a c omplete p olar c one ( r esp. c omplete c ongruenc e) and ( s, t ) 6∈ W , ther e ex ists y ∈ Y such that ( f , g ) ∈ W = ⇒ h y | f i ≤ h y | g i and h y | s i 6≤ h y | t i . Then, a su bsemimo dule W ⊂ X 2 is a c omplete p olar c one (r esp. c omplete c ongru- enc e) if, and only if, W = ( W ⋄ ) ◦ (r esp. W = ( W ⊤ ) ⊥ ) . Pr o of. W e only pr ov e the theo r em for p olar cones. The case of co ngruences is similar. Note that any set of the form V ◦ is a complete pola r co ne. This follows fro m the fact tha t the map x 7→ h y | x i is right linear and con tinuous for all y ∈ Y . Since the inclusion W ⊂ ( W ⋄ ) ◦ is straightforw ard, it suffices to show that ( W ⋄ ) ◦ ⊂ W . Assume that ( s, t ) 6∈ W . Then, there exists y ∈ Y suc h that h y | s i 6≤ h y | t i and h y | f i ≤ h y | g i for all ( f , g ) ∈ W . Since this means that y ∈ W ⋄ but h y | s i 6≤ h y | t i , w e conclude that ( s, t ) 6∈ ( W ⋄ ) ◦ .  Note that, by Theorems 5 and 6, the condition of the previous theorem is satisfied when Y = X op and h y | x i = x \ y , and, by Co rollarie s 1 and 2, it is also satisfied when ( Y , X ) is a pre-dual pair for a reflexive semiring K such tha t Y sepa rates X . F or any subset U of X 2 , we denote by po l( U ) (resp. cong( U )) the smallest complete pola r co ne (res p. co mplete congruence) co ntaining it. Then, we hav e the following corolla ry . Corollary 3. L et ( Y , X ) b e a pr e-dual p air which satisfi es the c ondition in The o- r em 7. Then, for any subset U of X 2 we have po l( U ) = ( U ⋄ ) ◦ (r esp. cong( U ) = ( U ⊤ ) ⊥ ). Pr o of. Since any set o f the form V ◦ is a complete po lar co ne, it follows that ( U ⋄ ) ◦ is a co mplete po lar cone whic h clearly contains U . Let W b e a complete pola r cone con taining U . Then, U ⊂ W = ⇒ W ⋄ ⊂ U ⋄ = ⇒ ( U ⋄ ) ◦ ⊂ ( W ⋄ ) ◦ , and by Theor em 7 we have ( W ⋄ ) ◦ = W . Therefore, ( U ⋄ ) ◦ is the smallest complete po lar cone c o ntaining U . The case of complete congruences can b e proved along the same lines .  Example 4 . When K = R max and X = R n max , note tha t if we consider the brack et h y | x i := ⊕ i y i x i , Corollary 2 can b e stated as follows: If W ⊂ ( R n max ) 2 is a complete pola r cone and s, t ∈ R n max are such that ( s, t ) 6∈ W , then, there exists y ∈ R n max such that ( f , g ) ∈ W = ⇒ ⊕ i y i f i ≤ ⊕ j y j g j and ⊕ i y i s i 6≤ ⊕ j y j t j . F or instance, consider the p ola r co ne W := V ◦ , wher e V is the s ubs emimo dule of R 3 max generated b y the vectors a = (2 , 0 , 3) T , b = (2 , 1 , 0) T and c = (2 , 4 , 0) T . This 12 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ x 1 x 2 x 3 c b − ˆ t V a 1 x 1 ≤ x 2 ⊕ 2 x 3 Figure 2. Illustra tion of the separ ation theorem for complete po - lar cones. semimo dule is repres ented by the bo unded dark gray region of Figure 2 tog ether with the segment joining the point a = (2 , 0 , 3) T to it. If we tak e s = (1 , ε, ε ) T and t = ( ε, 0 , 2) T , then ( s, t ) 6∈ W bec ause the ineq uality 1 x 1 ≤ x 2 ⊕ 2 x 3 is not satisfied by the element x = b = (2 , 1 , 0) T ∈ V . Let us compute the vector − ˆ t which, accor ding to Theor em 5 and Cor o llary 2, should separate ( s, t ) from W . By definition, ˆ t = ∨{ f ∈ R 3 max | ( f , t ) ∈ W } = ∨{ f ∈ R 3 max | V ⊂ H f } , where H f := { x ∈ R 3 max | ⊕ i f i x i ≤ ⊕ j t j x j } = { x ∈ R 3 max | f 1 x 1 ⊕ f 2 x 2 ⊕ f 3 x 3 ≤ x 2 ⊕ 2 x 3 } . Note that if we define I := { 1 ≤ i ≤ 3 | f i > t i } and J := { 1 , 2 , 3 } \ I , we hav e H f = { x ∈ R 3 max | ⊕ i ∈ I f i x i ≤ ⊕ j ∈ J t j x j } . Then, w e can hav e the following cases depe nding on the v ector f : - If f 2 ≤ 0 and f 3 ≤ 2, then H f = { x ∈ R 3 max | f 1 x 1 ≤ x 2 ⊕ 2 x 3 } . Note that the inequality f 1 x 1 ≤ x 2 ⊕ 2 x 3 is satisfied by the g enerator s a , b and c of V if, a nd only if, f 1 ≤ 0. Therefore, since H f is a semimo dule, it follows that V ⊂ H f if, and only if, f 1 ≤ 0, which sho ws that ˆ t ≥ (0 , 0 , 2) T . - If f 2 > 0 and f 3 ≤ 2, then H f = { x ∈ R 3 max | f 1 x 1 ⊕ f 2 x 2 ≤ 2 x 3 } . Note that (2 , 4 , 0) T never b elongs to H f bec ause this w ould imply that f 2 ≤ − 2. Then, in this case, V is nev er co ntained in H f bec ause the vector c do es not be lo ng to H f . THE TROPICAL ANALOGUE OF POLAR CONES 13 - If f 2 ≤ 0 and f 3 > 2, then H f = { x ∈ R 3 max | f 1 x 1 ⊕ f 3 x 3 ≤ x 2 } . Note that (2 , 0 , 3 ) T 6∈ H f bec ause f 3 > 2. Therefor e, like in the previous ca se, the set H f never contains V b ecause in particular it do es no t co nt ain the vector a . - If f 2 > 0 and f 3 > 2, the elements o f H f cannot hav e finite second and third ent ries . Then, in this cas e, V is never contained in H f . Therefore, we co nclude tha t ˆ t = (0 , 0 , 2) T and so − ˆ t = − ˆ t = (0 , 0 , − 2) T . Since − ˆ t = ( − 2 ) b ⊕ ( − 4) c ∈ V , it follows that h − ˆ t | f i ≤ h − ˆ t | g i for all ( f , g ) ∈ W . How ever, h − ˆ t | s i = 1 > 0 = h − ˆ t | t i and th us − ˆ t separates ( s, t ) fro m W . A natural way to define a congruenc e W is to take a contin uous rig ht linea r map F from X to a complete right K -semimo dule Z a nd to define W = ker F := { ( u, v ) ∈ X 2 | F ( u ) = F ( v ) } . The n, since F is contin uous, W is c omplete and it satisfies ˆ u = ∨{ v ∈ X | F ( v ) = F ( u ) } = ∨{ v ∈ X | F ( v ) ≤ F ( u ) } = F ♯ ( F ( u )) , for all u ∈ X . Conv ersely , it can b e chec ked that every complete congr uenc e W arises in this way . It suffices to take for F the ca nonical mor phism from X to its quotient by the equiv a lence relation W . The details are left to the reader. R emark 3 . A basic sp ecial cas e is when X = R n max and F is a cont inuous (rig ht ) linear ma p fro m X to R p max , so that F can b e represent ed by a p × n matrix A = ( a ij ) with entries in R max , mea ning that the i -th co ordinate of F is g iven b y F i ( x ) = max j ( a ij + x j ). The residual of F is the min-plus linear map represented by the matrix A tr a nsp osed and changed of sign ( − a j i ), F ♯ j ( y ) = min i ( − a ij + y i ), with the conv ention that + ∞ is a bsorbing for the (usual) addition. Hence, the maximal representative ˆ u = F ♯ ◦ F ( u ) is determined b y a simple “min-max” combinatorial formula. 4. Sep ara tion theorems f or the max-plus semiring In the preceding section, the semimo dules, p olar cones, a nd congruences were required to be complete, which is the s ame as being closed in the Scott top o logy . This to po logy is a dapted when the underlying s emiring is the co mpleted max- plus semiring, R max (w e refer the reader to [Aki99, AS03] for a discussion of this top ology within the max-plus s e tting). How ever, in man y applications, the s emiring of in terest is rather the non completed max -plus semiring R max , and the top ology of choice is the standard o ne, which can b e defined by the metric d ( a, b ) := | exp( a ) − exp( b ) | . Thus, instead of co mplete pre- congruences ov er R n max , we now deal with pre-congr uences ov er R n max that are closed (in the induced pro duct top olog y), and we would like to find separa tion theorems in the spirit of Theorem 5 involving closed sepa r ating sets. W e ma y of course co nsider the restriction to ( R n max ) 2 of the separating set constructed in the pr o of of this theorem, i.e . { ( f , g ) ∈ ( R n max ) 2 | g \ ˆ t ≤ f \ ˆ t } , but it can b e c heck ed that this set is not closed as so on as ˆ t has a − ∞ co ordina te (see the example b elow). In this section, w e apply a pertur bation technique to derive separatio n theorems whic h are appro priate for clo sed pola r cones and congruences ov er R n max . Example 5 . Consider again the p olar cone W = V ◦ and the pa ir of vectors ( s, t ) of Example 4, but assume that we add the vector ( e, ε, ε ) T to the set of genera tors of 14 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ V . Then, it c a n b e check ed that with this modifica tion we hav e ˆ t = t = ( ε, 0 , 2) T . Therefore, the r e striction to ( R 3 max ) 2 of the set that separates ( s, t ) from W (given by Theorem 5) is { ( f , g ) ∈ ( R 3 max ) 2 | (+ ∞ ) f 1 ⊕ f 2 ⊕ ( − 2) f 3 ≤ (+ ∞ ) g 1 ⊕ g 2 ⊕ ( − 2) g 3 } = { ( f , g ) ∈ ( R 3 max ) 2 | g 1 6 = ε } ∪ { ( f , g ) ∈ ( R 3 max ) 2 | f 1 = g 1 = ε, f 2 ⊕ ( − 2) f 3 ≤ g 2 ⊕ ( − 2) g 3 } which is not a closed s ubset of ( R 3 max ) 2 . Inspired b y the pr evious section, if W ⊂ ( R n max ) 2 is a clo sed pr e-congr uence, for g ∈ R n max we set ¯ g := sup { f ∈ R n max | ( f , g ) ∈ W } ∈ R n max . (10) Unlik e the case of complete pre-co ngruences, obser ve that ¯ g may hav e some entries equal to + ∞ , so ( ¯ g , g ) need not belong to W . Howev er , since ( g , g ) ∈ W , we s till hav e g ≤ ¯ g for all g ∈ R n max . W e shall us e the fact tha t g ≤ f = ⇒ ¯ g ≤ ¯ f , (11) which can be proved as in the case of complete pr e -congruence s. Due to the fa ct that ( ¯ g , g ) do es not necess arily b elong to W , we cannot use the vector ¯ g in the same wa y we did it b efore. Therefore, we need a per turbation argument in which the following simple construction will be very imp orta nt. F o r z ∈ R n max and β ∈ R max , w e define the v ector z β ∈ R n max by: z β i =  β if z i = + ∞ , z i otherwise. In other words, z β is obtained fro m z by replacing its + ∞ entries by β . W e denote by e i the i -th unit vector, i.e. the vector defined by (e i ) i := e a nd (e i ) j := ε if j 6 = i . W e shall need the follo wing lemma. Lemma 2. L et W ⊂ ( R n max ) 2 b e a c ongruenc e and f , g ∈ R n max b e such t hat ( f , g ) ∈ W . Then, ( g , g ⊕ e i γ ) ∈ W for al l γ ≤ f i . Pr o of. If γ ≤ f i , since W is a congruence and ( f , g ) ∈ W , w e hav e ( f , g ⊕ e i γ ) = ( f ⊕ e i γ , g ⊕ e i γ ) = ( f , g ) ⊕ (e i , e i ) γ ∈ W . Hence, ( g , g ⊕ e i γ ) ∈ W b e c ause ( g , f ) ∈ W , ( f , g ⊕ e i γ ) ∈ W and W is a congr uence.  Then we hav e the follo wing corollary . Corollary 4 . L et W ⊂ ( R n max ) 2 b e a close d c ongruenc e or a close d p olar c one. Then, for al l g ∈ R n max , if β ≥ max i ( g i ) we have ( ¯ g β , g ) ∈ W . Pr o of. Let us firs t co nsider the case where W is a closed cong ruence. If ¯ g i = + ∞ , there exists f ∈ R n max such that ( f , g ) ∈ W and f i > β . T he n, by Lemma 2 we hav e ( g , g ⊕ e i β ) ∈ W . Now assume that ¯ g i < + ∞ . Then, if γ < ¯ g i , there exis ts f ∈ R n max such that ( f , g ) ∈ W and f i > γ . Thus, by Lemma 2 w e have ( g , g ⊕ e i γ ) ∈ W . Since this holds for all γ ∈ R max such that γ < ¯ g i , it follows that ( g , g ⊕ e i ¯ g i ) ∈ W because W is closed. THE TROPICAL ANALOGUE OF POLAR CONES 15 Consequently , as W is a semimodule, we have ( ¯ g β , g ) = (( ⊕ { i | ¯ g i < + ∞} g ⊕ e i ¯ g i ) ⊕ ( ⊕ { i | ¯ g i =+ ∞} g ⊕ e i β ) , g ) = ( ⊕ { i | ¯ g i < + ∞} ( g ⊕ e i ¯ g i , g )) ⊕ ( ⊕ { i | ¯ g i =+ ∞} ( g ⊕ e i β , g )) ∈ W . The case o f pola r cones is ea s ier (indeed the asse rtion holds for any β ∈ R max ). If ¯ g i = + ∞ , there exists f ∈ R n max such that ( f , g ) ∈ W and f i > β . Then, as e i β ≤ f and ( f , g ) ∈ W , it follo ws that (e i β , g ) ∈ W , b eca use W is a p olar cone. Assume now that ¯ g i < + ∞ . Then, if α < ¯ g i , there exists f ∈ R n max such that ( f , g ) ∈ W a nd f i > α . Therefore, as e i α ≤ f a nd ( f , g ) ∈ W , we hav e (e i α, g ) ∈ W . Since this holds for all α ∈ R max such that α < ¯ g i , we deduce that (e i ¯ g i , g ) ∈ W , bec ause W is closed. Finally , a s W is a semimo dule, we g et ( ¯ g β , g ) = (( ⊕ { i | ¯ g i < + ∞} e i ¯ g i ) ⊕ ( ⊕ { i | ¯ g i =+ ∞} e i β ) , g ) = (( ⊕ { i | ¯ g i < + ∞} (e i ¯ g i , g )) ⊕ ( ⊕ { i | ¯ g i =+ ∞} (e i β , g )) ∈ W , which completes the pro of of the corollary .  Like in the case of complete pre-congruence s , we hav e: Lemma 3. L et W ⊂ ( R n max ) 2 b e a close d c ongruenc e or a close d p olar c one. Then, ( f , g ) ∈ W = ⇒ g \ ¯ h ≤ f \ ¯ h for al l h ∈ R n max . Pr o of. When g \ ¯ h = −∞ the asser tion is obvious, so a ssume that g \ ¯ h 6 = −∞ . Let α < g \ ¯ h , so that ¯ h ≥ g α . Then, if β ≥ max i ( g i α ⊕ h i ), we hav e g α ≤ ¯ h β , a nd by Cor ollary 4 we know that ( ¯ h β , h ) ∈ W . Since W is a pre- congruence, fro m ( ¯ h β ⊕ f α, ¯ h β ) = ( ¯ h β ⊕ f α, ¯ h β ⊕ g α ) = ( ¯ h β , ¯ h β ) ⊕ ( f , g ) α ∈ W and ( ¯ h β , h ) ∈ W , it follows that ( ¯ h β ⊕ f α, h ) ∈ W , and so, f α ≤ ¯ h β ⊕ f α ≤ ¯ h . Therefore, we hav e f \ ¯ h ≥ α . Since this holds fo r a ll α ∈ R max such that α < g \ ¯ h , we conclude that g \ ¯ h ≤ f \ ¯ h .  The following lemmas will b e useful to prove the separa tion theorems for the max-plus semiring. Lemma 4. L et W ⊂ ( R n max ) 2 b e a close d pr e-c ongru enc e. Assume that s, t, h ∈ R n max ar e such t hat t ≤ h and s 6≤ ¯ h . Then, we have t \ ¯ h 6≤ s \ ¯ h . Pr o of. Assume that t \ ¯ h ≤ s \ ¯ h . Then, w e get s ( t \ ¯ h ) ≤ s ( s \ ¯ h ) ≤ ¯ h . Since t ≤ h , by (11) w e hav e t ≤ ¯ t ≤ ¯ h , and so e ≤ t \ ¯ h . There fo re, it follows that s = se ≤ s ( t \ ¯ h ) ≤ ¯ h , which is a contradiction.  Lemma 5. L et W ⊂ ( R n max ) 2 b e a close d c ongruenc e or a close d p olar c one. Assu m e that s, t ∈ R n max ar e such that s 6≤ ¯ t . Then, ther e exists h ∈ R n max with finite entries such that t ≤ h and s 6≤ ¯ h . Pr o of. Assume that s ≤ ¯ h for all h ∈ R n max with finite en tries tha t satisfy t ≤ h . Let { h k } k ∈ N ⊂ R n max be a decr easing sequence of vectors with finite entries suc h that lim k →∞ h k = t . Then, as by (11) the s e q uence { ¯ h k } k ∈ N ⊂ R n max is also decreasing, there exists z ∈ R n max such that lim k →∞ ¯ h k = z . Since s ≤ ¯ h k for a ll k ∈ N , we hav e s ≤ z . Note tha t we c a n assume, without loss o f g enerality , that ( ¯ h k ) i = + ∞ ⇐ ⇒ z i = + ∞ . If λ ≥ max i ( h 1 ) i , then b y C o rollar y 4 we know that (( ¯ h k ) λ , h k ) ∈ W for a ll k ∈ N . Ther efore, we g et ( z λ , t ) = lim k →∞ (( ¯ h k ) λ , h k ) ∈ W , 16 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ bec ause W is close d. Since this holds for all λ ∈ R max such that λ ≥ max i ( h 1 ) i , it follows that ¯ t ≥ z ≥ s , whic h contradicts our assumption.  As a co nsequence of the previous results w e obtain the separatio n theo rem for closed po lar cones. Theorem 8. L et W ⊂ ( R n max ) 2 b e a close d p olar c one. Assume that s, t ∈ R n max ar e such t hat ( s, t ) 6∈ W . Then, ther e exists y ∈ R n max such t hat ( f , g ) ∈ W = ⇒ ⊕ i f i y i ≤ ⊕ j g j y j and ⊕ i s i y i 6≤ ⊕ j t j y j . Pr o of. In the firs t place, we claim that s 6≤ ¯ t . Indee d, if s ≤ ¯ t , le t β ≥ max i ( s i ⊕ t i ). Then, s ≤ ¯ t β and by Coro llary 4 we hav e ( ¯ t β , t ) ∈ W . It follows that ( s, t ) ∈ W bec ause W is a po lar cone, which contradicts o ur ass umption. This pr ov es o ur claim. Since s 6≤ ¯ t , accor ding to Lemma 5 there exis ts h ∈ R n max with finite en tries such that t ≤ h and s 6≤ ¯ h . Therefor e, fro m Lemmas 3 and 4, it follows that ( f , g ) ∈ W = ⇒ g \ ¯ h ≤ f \ ¯ h and t \ ¯ h 6≤ s \ ¯ h . Then, a s u \ ¯ h = − ( ⊕ i u i ( − ¯ h i )) and ¯ h i ≥ h i > −∞ for all i , the assertio n of the theorem holds with y := − ¯ h .  In order to prove a sepa ration theorem for clo s ed co ngruences, w e need the following result. Lemma 6. L et W ⊂ ( R n max ) 2 b e a close d c ongruenc e. Then, for al l f , g ∈ R n max , the fol lowing pr op erties ar e satisfie d. ( f , g ) ∈ W ⇐ ⇒ ¯ f = ¯ g ; (12) f ≤ ¯ g = ⇒ ¯ f ≤ ¯ g . (13) Pr o of. (12) If ( f , g ) ∈ W , we ha ve { h ∈ R n max | ( h, f ) ∈ W } = { h ∈ R n max | ( h, g ) ∈ W } bec ause W is a congruence. Therefore , ¯ f = ¯ g . Conv ersely , a ssume that ¯ f = ¯ g . If λ ≥ ma x i ( f i ⊕ g i ), by Coro llary 4 it fo llows that ( ¯ f λ , f ) ∈ W and ( ¯ g λ , g ) ∈ W . Since ¯ f = ¯ g , we hav e ¯ f λ = ¯ g λ , and thus ( f , g ) ∈ W b ecaus e W is a c ongruence. (13) Let λ ≥ max i ( f i ⊕ g i ). Then, if f ≤ ¯ g we hav e f ≤ ¯ g λ , and by Corollar y 4 we know tha t ( ¯ g λ , g ) ∈ W . Therefor e , by (11) and (12), it follows that ¯ f ≤ ¯ g λ = ¯ g .  Theorem 9. L et W ⊂ ( R n max ) 2 b e a close d c ongruenc e. Assume that s, t ∈ R n max ar e such t hat ( s, t ) 6∈ W . Then, ther e exists y ∈ R n max such t hat ( f , g ) ∈ W = ⇒ ⊕ i f i y i = ⊕ j g j y j and ⊕ i s i y i 6 = ⊕ j t j y j . THE TROPICAL ANALOGUE OF POLAR CONES 17 Pr o of. Since ( s, t ) 6∈ W , b y (1 2) we hav e ¯ s 6 = ¯ t , hence, ¯ s 6≤ ¯ t and/or ¯ t 6≤ ¯ s . Assume for instance that ¯ s 6≤ ¯ t . The n, from (13) it follows that s 6≤ ¯ t . Therefor e, accor ding to Lemma 5 there exists h ∈ R n max with finite entries such that t ≤ h a nd s 6≤ ¯ h . Since ( f , g ) ∈ W implies ( g , f ) ∈ W beca us e W is a co ngruence, fro m Lemmas 3 and 4, we hav e ( f , g ) ∈ W = ⇒ f \ ¯ h = g \ ¯ h and t \ ¯ h 6 = s \ ¯ h . Then, taking into account that u \ ¯ h = − ( ⊕ i u i ( − ¯ h i )) and ¯ h i ≥ h i > −∞ for all i , the assertion of the theor em holds with y := − ¯ h .  Like in the previo us section, consider the following corr esp ondences b etw een subsemimo dules of ( R n max ) 2 and R n max : W ⊂ ( R n max ) 2 7→ W ⋄ := { x ∈ R n max | ⊕ i f i x i ≤ ⊕ j g j x j , ∀ ( f , g ) ∈ W } , V ⊂ R n max 7→ V ◦ := { ( f , g ) ∈ ( R n max ) 2 | ⊕ i f i x i ≤ ⊕ j g j x j , ∀ x ∈ V } , and W ⊂ ( R n max ) 2 7→ W ⊤ := { x ∈ R n max | ⊕ i f i x i = ⊕ j g j x j , ∀ ( f , g ) ∈ W } , V ⊂ R n max 7→ V ⊥ := { ( f , g ) ∈ ( R n max ) 2 | ⊕ i f i x i = ⊕ j g j x j , ∀ x ∈ V } . Then, by the separation theorem for closed semimo dules (see [Zim77, Th. 4 ], [SS92], see also [CGQS05, Th. 3 .14] for rec e nt improvemen ts), it follows that the following statements are equiv a lent: (i) V ⊂ R n max is a closed semimodule, (ii) V = ( V ◦ ) ⋄ , (iii) V = ( V ⊥ ) ⊤ . Now, as a co nsequence of Theorems 8 and 9 , we obta in the following dua l result. W e omit the pro of beca use it is similar to that of Theor em 7. Theorem 10. A semimo dule W ⊂ ( R n max ) 2 is a close d p olar c one (r esp. close d c ongruenc e) if, and only if, W = ( W ⋄ ) ◦ (r esp. W = ( W ⊤ ) ⊥ ) . As a corolla ry of the previous theore m, it follows that for any subset U of ( R n max ) 2 , ( U ⋄ ) ◦ (resp. ( U ⊤ ) ⊥ ) is the smallest clo sed pola r cone (resp. closed congruence) containing it. Example 6 . Consider the semimo dule V ⊂ R 3 max generated by the vectors a = (0 , 0 , 0 ) T , b = (0 , 1 , − 1) T and c = (0 , 2 , − 2) T . This semimo dule is r epresented by the b ounded dark g ray reg ion of Figure 3 together with the segments joining the po int s a a nd c to it. Then V ◦ is the solutio n set of a system o f homog eneous max- plus linear inequalities, and hence also of a system of ho mogeneous max-plus linear equations. More precisely , V ◦ = { ( f , g ) ∈ ( R 3 max ) 2 | ⊕ i f i x i ≤ ⊕ j g j x j , x = a, b , c } = { ( f , g ) ∈ ( R 3 max ) 2 | ⊕ i ( f i ⊕ g i ) x i = ⊕ j g j x j , x = a, b, c } . W e know that it is a finitely ge nerated s ubsemimo dule of R 6 max (see [BH84, Gau92, GP97]). Solving this system b y the elimination metho d (see [BH84] and [AGG08] 18 ST ´ EPHANE GAUB ER T AND RICARDO D. KA TZ x 1 V x 2 1 x 1 ≤ x 2 ⊕ 1 x 3 x 3 a c b x 1 x 2 V x 3 x 2 ⊕ 3 x 3 = 2 x 1 ⊕ 3 x 3 a c b Figure 3. V alid linear inequality and equa lit y for V . for recent improvemen ts), w e obtain that the polar cone V ◦ is the subsemimo dule of R 6 max generated by the columns of the following matrix         ε ε ε 1 ε 2 0 0 0 ε ε ε ε ε 0 ε ε ε 0 ε ε ε ε 0 ε ε ε ε ε 0 0 ε ε ε ε ε ε ε 0 ε ε ε 2 ε 0 ε ε ε ε ε 0 ε ε 0 ε ε ε 0 ε 0 ε 0 0 ε ε 0 ε ε 0 ε ε ε ε 1 4 3 ε 2 ε ε 0 ε ε 0         . The ordering o f the v ariables in the system of equa tions ab ove ha s be e n chosen such that, if we denote by f the vector whose en tries are the fir st three entries of a solution a nd by g the vector whose entries are the remaining three ent ries, then ( f , g ) ∈ V ◦ . F or instance, the fourth column of the matrix a b ov e corre s po nds to the pair of vectors f = (1 , ε, ε ) T and g = ( ε, 0 , 1) T . With each pair o f vectors ( f , g ) ∈ V ◦ is ass o ciated the linear inequality ⊕ i f i x i ≤ ⊕ j g j x j which is said to b e valid for V b eca us e it is satisfied by all its ele men ts. F or the previous pair of v ectors , we have the v alid inequality 1 x 1 ≤ x 2 ⊕ 1 x 3 , which is repr esented by the un b ounded light gray reg ion on the left-hand side of Figure 3. Analo gously , the congruence V ⊥ is the subsemimodule of R 6 max generated b y the columns of the following matrix         0 1 ε 0 2 ε ε ε ε ε ε ε 2 2 0 0 0 ε ε 0 0 0 ε ε 0 0 0 0 ε 0 ε 0 ε ε ε ε 0 ε ε ε 0 0 ε 4 3 ε ε 2 1 4 ε 3 ε 2 ε ε 0 ε ε ε 0 2 ε 2 0 ε 0 1 ε 2 ε 0 ε 0 ε ε 0 0 0 ε 0 ε ε 0 0 ε 0 0 ε 0 ε ε ε 0 ε ε 1 ε ε ε 4 3 ε 0 2 ε 4 ε 3 0 2 ε ε 0         . In this case, to s o lve the system of equations for V ⊥ , we have taken the same ordering of the v a riables as for V ◦ . Then, for example, the first column corr esp onds to the equalit y x 1 ⊕ x 2 = x 2 which is equiv alent to the v alid inequality x 1 ≤ x 2 (the seven th column of the matrix con taining the generator s of V ◦ ). Note that, as men tioned in the in tro duction, a ny v a lid inequa lity ⊕ i f i x i ≤ ⊕ j g j x j for V implies a v alid equality for V (meaning that it is sa tisfied by a ll its elemen ts), namely ⊕ i ( f i ⊕ g i ) x i = ⊕ j g j x j . 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