Idempotent convexity and algebras for the capacity monad and its submonads

IDEMPOTENT CONVEXITY AND ALGEBRAS F OR THE CAP A CITY MONAD AND ITS SUBMONADS OLEH NYKYFOR CHYN, DU ˇ SAN REPOV ˇ S Abstract. Idempoten t analogues of con ve xity are introduced. It is prov ed that the category of algebras for the capacit y monad i n the category of com- pacta is isomorphic to the category of (max , min)-idemp oten t biconv ex com- pacta and their biaffine maps. It is also shown th at the category of algebras for the monad of s up-measures ((max , mi n)-idempotent measures) is is omor phic to the category of (max , mi n)-idempoten t con ve x compacta and their affine maps. Introduction Monads (also called triples , [2, 8]) in top olo g ical categor ie s and alge br as for these monads are clo sely related to imp or tant ob jects of analy s is and top ologica l alg ebra. ´ Swirszcz [17] pr ov ed that algebra s and their morphisms for the probability mea- sure mona d are precisely convex compact maps of loca lly conv ex vector to po logical spaces a nd contin uous affine ma ps . By a re sult of Day (cf. Theorem 3.3 o f [6 ]), the category of algebra s for the filter monad in the ca tegory of sets is the ca teg ory of co nt inuous la ttices a nd their mappings that pr eserve directed joins and ar bitrary meets. Due to Wyler [19] algebras for the hype r space mona d are c o mpact Lawson semilattices. Zarichnyi [20] has shown tha t the categor y o f algebras for the s up er extension monad is isomor phic to the catego ry of compacta with (fixed) almo s t normal T 2 -subbase a nd their conv ex maps. W e will use a result of Radul [15] who in tro duced the inclusion hyperspa ce triple and proved that its algebras and their morphisms are in fact compact Lawson lattices and their complete homomorphisms. Unlik e probability (normed additive) meas ures whic h are a tra ditional ob ject of inv estigation by mea ns of categor ical top olog y , their non-a dditive analogues were paid less attention from this po int o f view. Meanwhile capacities (normed non- additive measures) that were introduced by Cho quet [4] and redis cov ered b y Sug eno under the name fuzzy measures hav e found numerous a pplications, e.g. in decision making under uncer taint y [7, 16]. One o f the mo st promising cla sses of non-additive measures is one of idempotent meas ures [1]. F or o ther imp ortant classe s of capac- ities and their topo lo gical prop erties see [3 ]. Upp er semicontin uous capa cities on compact spa c es were systematically s tudied in [14]. Date : May 2, 2022. 2000 Mathematics Subje ct Classific ation. Primary: 18B30,Secondary: 18C20,06B35,52A01. Key wor ds and phr ases. c ap acity functor, algebr a for a monad, i demp otent semimo dule, idem- p otent c onvexity. This research was supp or ted by the Slov enian Research Agency grant s P1-0292-0101, J1- 9643-0101 and BI- UA/09-10-002, and the State F und of F undamen tal Researc h of Ukraine gran t 25.1/099. 1 IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 2 Therefore it seems na tural to use methods of categ orical topolo gy to study non- additive meas ur es. Nykyforc hyn and Zar ichn y i [2 1] defined the ca pacity functor and the capacity monad in the categ ory of compacta, and prov ed basic top olo gical prop erties o f capacities on metriz a ble and non-metriza ble compacta. Tw o impo r- tant dual subfunctors o f the capacity functor, namely of ∪ -capacities (pos sibility measures) and of ∩ -ca pacities (necessity meas ures) were introduced in [9], and it was shown that they lead to submona ds of the capacity monad. The aim of this pap er is to describ e ca tegories of algebras fo r the capacit y mona d, for the mona ds of ∪ -capacities and of ∩ -capac ities, and to present internal relations of the capa c - it y monad a nd its submonads with idemp otent mathema tics a nd gene r alizations of conv exit y (in the form of join geometr y). 1. Preliminaries A c omp actum is a c o mpact Hausdorff top ologica l space. W e r e gard the unit segment I = [0 ; 1] as a subspace of the re al line with the natural top olog y . W e write A ⊂ cl B (resp. A ⊂ op B ) if A is a close d (resp. an o pen) subset o f a space B . F or a set X the iden tit y mapping X → X is denoted by 1 X . F or a compactum X we denote b y exp X the set o f all nonempty closed subsets of X with the Vietoris top olo gy . A base of this top olo g y consists o f all sets of the form h U 1 , U 2 , . . . , U n i = { F ∈ exp X | F ⊂ U 1 ∪ U 2 ∪· · ·∪ U n , F ∩ U i 6 = ∅ for all 1 6 i 6 n } , where n ∈ N and a ll U i ⊂ X are op en. The s pace exp X for a compactum X is a compactum as well. A nonempty clo sed subset F ⊂ exp X is ca lled an inclusion hyp ersp ac e if for all A, B ∈ exp X an inclusio n A ⊂ B and A ∈ F imply B ∈ F . The set GX o f all inclusion hyperspa ces is closed in exp(exp X ). F or mor e on e xp X and GX see [1 8]. W e reg ard any set S with an idemp otent, commutativ e and as s o ciative binary op eration ⊕ : S × S → S (with an additive denotation) a s an upp er s e milattice with the partia l order x 6 y ⇐ ⇒ y > x ⇐ ⇒ x ⊕ y = y and the pa irwise supremum x ⊕ y for x, y ∈ S . Similar ly , given an idemp otent, co mm utative and asso ciativ e op eration ⊗ : S × S → S (with a multiplicativ e denota tio n), we r egard S as a low er semilattice with the par tial order x 6 y ⇐ ⇒ y > x ⇐ ⇒ x ⊗ y = x a nd x ⊗ y being the infim um of x, y ∈ S . If t wo oper ations ⊕ , ⊗ : L × L → L are idemp otent, commutativ e and ass o cia- tive, and the distributive la ws and the laws o f absorption are v a lid, then L is a distributive lattice w.r.t. the par tial order x 6 y ⇐ ⇒ y > x ⇐ ⇒ x ⊕ y = y ⇐ ⇒ x ⊗ y = x , and x ⊕ y and x ⊗ y are the pair w is e supre mum and the pairwise infimum of x, y ∈ L . If f , g are functions with the same do main a nd v alues in a p oset, then b y f ∨ g and f ∧ g we also define their p oinwise s upr emum and infim um. If f is a function with v alues in a set L with an op eration “ ⊕ ” (or “ ⊗ ”), and α ∈ L , then ( α ⊕ f )( x ) = α ⊕ f ( x ) (r esp. ( α ⊗ f )( x ) = α ⊗ f ( x )) fo r any v alid argument x . An idemp otent semiring is a set R with binary op era tions ⊕ , ⊗ : R × R → R such that ( R , ⊕ ) is an ab elian mo noid with a neutral elemen t 0, “ ⊕ ” is idemp otent, i.e. a ⊕ a = a for all a ∈ R , ( R, ⊗ ) is a mono id with a neutral element 1 , the op eration “ ⊗ ” is distributive ov er “ ⊕ ” : a ⊗ ( b ⊕ c ) = ( a ⊗ b ) ⊕ ( a ⊗ c ) for a ll a, b, c ∈ R , and 0 ⊗ a = a ⊗ 0 = 0 for all a ∈ R . The most p opular idemp otent semiring is the tr opic al semiring ( R ∪ {−∞} , ⊕ , ⊗ ), where x ⊕ y = max { x, y } , x ⊗ y = x + y , which is the IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 3 basis of tr opic al mathematics [1 0]. A little less extensively s tudied is the idempo tent semiring ( R ∪ { ±∞} , ⊕ , ⊗ ), where x ⊕ y = max { x, y } , x ⊗ y = min { x, y } . W e will us e a semiring whic h is a lgebraica lly and top ologica lly iso mo rphic to it, but more conv enien t for our purp oses, namely ( I , ⊕ , ⊗ ) with x ⊕ y = max { x, y } , x ⊗ y = min { x, y } . In genera l, an y distributive lattice ( L, ⊕ , ⊗ ) with top and bo ttom elements is an idemp otent semiring. See [2, 1 2] for the definitions o f category , morphism, functor, natur al transfor ma- tion, monad, alg ebra for a monad, morphism of algebra s, tripleability and related facts. By 1 C we denote the iden tit y functor in a categor y C . Recall that all F - algebras for a fixed monad F and all their mo r phisms form a c ate gory of F -algebr as . It is prov ed in [1 8] that costructio ns exp a nd G can b e extended to functors in C omp that are functorial parts of monads. F or a c ontin uous map of compacta f : X → Y the maps ex p f : exp X → exp Y and Gf : GX → GY are defined by the formulae e xp f ( F ) = { f ( x ) | x ∈ F } , F ∈ exp X and Gf ( F ) = { B ⊂ cl Y | B ⊃ f ( A ) for some A ∈ F } , F ∈ GX . F or the inclusion hyp ersp ac e monad G = ( G, η G , µ G ) the c omp o nents η G X : X → GX and µ G X : G 2 X → GX of the unit and the multiplication a re defined as follows : η G ( x ) = { F ∈ exp X | x ∈ F } , x ∈ X and µ G X (F) = S { T A | A ∈ F } , F ∈ G 2 X . W e deno te by C omp the c ate gory of c omp acta tha t cons ists of a ll compacta and their contin uous mappings. If there is a natura l tra ns formation of one functor in C omp to another with a ll comp onents being topolog ical em be dding s, then the fir st functor is calle d a subfunct or of the latter [18]. Similarly an emb e dding of monads in C omp is a mo rphism of monads with all comp onents b eing topo logical embeddings. If there exists a n embedding of one monad in C omp into a no ther one, then the first monad is called a submonad of the latter. Now we present the main notions and results o f [21, 9] that concern c a pacities on compac ta, the capacity functor and the capacity mo nad. W e call a function c : e x p X ∪ { ∅ } → I a c ap acity o n a compactum X if the following three prope rties hold for all closed subsets F , G of X : (1) c ( ∅ ) = 0, c ( X ) = 1; (2) if F ⊂ G , then c ( F ) 6 c ( G ) (monotonicit y); (3) if c ( F ) < a , then there exists an op en set U ⊃ F such that G ⊂ U implies c ( G ) < a (upper semicontin uit y). W e extend a capacit y c to all o pen subsets in X by the fo r mula : c ( U ) = sup { c ( F ) | F ⊂ cl X , F ⊂ U } , U ⊂ op X . It is prov ed in [21] that the set M X of all ca pacities o n a compactum X is a compactum as well, if a to p o logy on M X is determined b y a subbase that consists of all sets of the form O − ( F, a ) = { c ∈ M X | c ( F ) < a } , where F ⊂ cl X , a ∈ R , and O + ( U, a ) = { c ∈ M X | c ( U ) > a } = { c ∈ M X | there exists a compactum F ⊂ U, c ( F ) > a } , where U ⊂ op X , a ∈ R . IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 4 The as s ignment M extends to the c ap acity functor M in the categor y of c om- pacta, if the map M f : M X → M Y for a co nt inuous map of compacta f : X → Y is defined b y the formula M f ( c )( F ) = c ( f − 1 ( F )) , where c ∈ M X , F ⊂ cl Y . This functor is the functorial part of the c ap acity monad M = ( M , η , µ ) that was desc r ib ed in [2 1]. Its unit a nd m ultiplication are defined by the formulae η X ( x ) = δ x where δ x ( F ) = ( 1 , if x ∈ F , 0 , if x / ∈ F , (a Dir a c measur e concentrated in x ) µX ( C )( F ) = sup { α ∈ I | C ( { c ∈ M X | c ( F ) > α } ) > α } , where x ∈ X , C ∈ M 2 X , F ⊂ cl X . W e call a capacity c ∈ M X a ∪ -c ap acity (also called sup-me asur e or p ossibili ty me asur e ) if c ( A ∪ B ) = ma x { c ( A ) , c ( B ) } for all A, B ⊂ cl X . A capacity c ∈ M X a ∩ -c ap acity (or ne c essity me asur e ) [9] if c ( A ∩ B ) = min { c ( A ) , c ( B ) } for all A, B ⊂ cl X . The sets of all ∪ -capa c ities a nd of all ∩ -ca pacities on a co mpactum X are denoted b y M ∪ X and M ∩ X . It is proved in [9] that M ∪ X and M ∩ X are clo sed in M X , M f ( M ∪ X ) ⊂ M ∪ Y and M f ( M ∩ X ) ⊂ M ∩ Y for an y contin uo us map of compacta f : X → Y , th us we obtain s ubfunctors M ∪ , M ∩ of the capa c ity functor M . Mor eov er, we get submo na ds M ∪ and M ∩ of the capacit y mona d M . Observe that for a ∪ -capacity c and a closed set F ⊂ X we hav e c ( F ) = max { c ( x ) | x ∈ F } , and c is completely determined by its v alues on singletons. Therefore w e often identify c with the upp er semicontin uous function X → I that sends each x ∈ X to c ( { x } ), a nd wr ite c ( x ) instea d of c ( { x } ). Con v ersely , each upper se mico nt inuous function c : X → I with max c = 1 determines a ∪ -capa city by the form ula c ( F ) = ma x { c ( x ) | x ∈ F } , F ⊂ cl X . A simila r, but a little mor e complicated obs erv atio n is v alid for ∩ -capacities. 2. Algebras for the monads of ∪ -cap a cities an d ∩ -cap acitie s Let an op eration ic : X × I × X → X be given for a set X . In the sequel we denote ic ( x, α, y ) b y x ⊕ ( α ⊗ y ) or simply by x ⊕ αy for the s a ke of shortness. W e call ic a n idemp otent c onvex c ombination of tw o points in X if the following equalities a r e v a lid for a ll x, y , z ∈ X , α, β ∈ I : 1) x ⊕ αx = x ; 2) ( x ⊕ αy ) ⊕ β z = ( x ⊕ β z ) ⊕ αy ; 3) x ⊕ α ( y ⊕ β z ) = ( x ⊕ αy ) ⊕ ( α ⊗ β ) z ; 4) x ⊕ 1 y = y ⊕ 1 x ; 5) x ⊕ 0 y = x . W e also call the set ∆ n ⊕ = { ( α 0 , α 1 , . . . , α n ) ∈ I n +1 | α 0 ⊕ α 1 ⊕ . . . ⊕ α n = 1 } the (idemp otent) n - dimensional ⊕ -s implex . Now, assuming 1)–5), for any co effi- cients ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊕ and elements x 0 , x 1 , . . . , x n ∈ X we define the idem- p otent c onvex c ombination of n + 1 p oints as follows (assume that α k = 1 for some IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 5 0 6 k 6 n ) : α 0 x 0 ⊕ α 1 x n ⊕ . . . ⊕ α n x n = ( . . . (( x k ⊕ α 0 x 0 ) ⊕ . . . ) ⊕ α k − 1 x k − 1 ) ⊕ α k +1 x k +1 ) ⊕ . . . ) ⊕ α n x n . Conditions 2),4) a ssure that the combination is well defined and do es not depend on the or der of summands. O bviously 1 x ⊕ αy = x ⊕ αy . By 5 ) summands with zero co efficients can b e dropp ed, and by 1) and 3), if tw o summands c o nt ain the same p oint, then a summand with a greater co efficient absorbs a summand with a less co efficient. Conditio ns 2),3) also imply a “ big ass o ciativ e law” : α 0 ( β 0 0 x 0 0 ⊕ . . . ⊕ β 0 k 0 x 0 k 0 ) ⊕ α 1 ( β 1 0 x 1 0 ⊕ . . . ⊕ β 1 k 1 x 1 k 1 ) ⊕ . . . ⊕ α n ( β n 0 x n 0 ⊕ . . . ⊕ β n k n x n k n ) = ( α 0 ⊗ β 0 0 ) x 0 0 ⊕ . . . ⊕ ( α 0 ⊗ β 0 k 0 ) x 0 k 0 ⊕ ( α 1 ⊗ β 1 0 ) x 1 0 ⊕ . . . ⊕ ( α 1 ⊗ β 1 k 1 ) x 1 k 1 ⊕ . . . ⊕ ( α n ⊗ β n 0 ) x n 0 ⊕ . . . ⊕ ( α n ⊗ β n k n ) x n k n , where x i j ∈ X , ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊕ , ( β i 0 , β i 0 , . . . , β i k i ) ∈ ∆ k i ⊕ for i = 0 , 1 , . . . , n . Prop erties 1)–4) imply that the op eratio n ∨ : X × X → X , x ∨ y = x ⊕ 1 y for all x, y ∈ X , is comm utative, a sso ciative and idemp otent, th us ( X , ∨ ) is an upp er semilattice with a partial or der x 6 y ⇐ ⇒ x ∨ y = y for which x ∨ y is a pairwise supremum o f x and y . If X is a compactum such that 6) for a neighborho o d U of an y element x ∈ X there is a neighborho o d V of x , V ⊂ U , suc h that y ⊕ 1 z ∈ V for a ll y , z ∈ V ; then each point of X has a lo c al base consisting of subsemilattices, and ( X , ∨ ) is a compact Lawson upp er semilattice [11]. W e will call a pair ( X , ic ) of a com- pactum X with idemp otent co nv ex combination ic that satisfies the prop erty 6 ) a (max , min) -idemp otent c onvex c omp actum . Theorem 2.1. L et X b e a c omp actum. Ther e is a one-to-one c orr esp ondenc e b etwe en c ontinuou s maps ξ : M ∪ X → X such that the p air ( X , ξ ) is an M ∪ -algebr a, and c ontinuous idemp otent c onvex c ombinations ic : X × I × X → X such that ( X, ic ) is a (max , min) -idemp otent c onvex c omp actum. If for a c ont inuous ic : X × I × X → X c onditions 1)–5) ar e valid, then 6) implies a str onger pr op erty : 6+) for a neighb orho o d U of any element x ∈ X ther e is a neighb orho o d V of x , V ⊂ U , such that y ⊕ αz ∈ V for al l y , z ∈ V , α ∈ I . Pr o of. Let ( X , ξ ) b e an M ∪ -algebra . Define the op er ation ic : X × I → X b y the formula ic ( x, α, y ) = ξ ( δ x ⊕ αδ y ). It is obvious that ξ is well-defined, contin uous and s atisfies 1), 4), 5). T o prov e 2), observe tha t b y the definition of a n a lg ebra for a monad w e obtain ( x ⊕ αy ) ⊕ β z = ξ ( δ ξ ( x ⊕ αy ) ⊕ β δ z ) = ξ ◦ M ∪ ξ ( δ δ x ⊕ αδ y ⊕ β δ δ z ) = ξ ◦ µ ∪ X ( δ δ x ⊕ αδ y ⊕ β δ δ z ) = ξ ( δ x ⊕ αδ y ⊕ β δ z ) = ξ ( δ x ⊕ β δ z ⊕ αδ y ) = ( x ⊕ β z ) ⊕ αy . Pro of of 3) is quite analogous. Thus the map ic is an idemp otent conv ex co mbina- tion of t wo p oints, and we consider idempo tent conv ex combinations of arbitra r y finite num ber of p oints to b e defined as describ ed ab ov e. Let U b e a neighborho o d o f x ∈ X . B y contin uity of ξ and the equality ξ ( δ x ) = x there is a neighbor ho o d ˜ U ⊂ M ∪ X o f δ x such that for a ll c ∈ ˜ U we hav e ξ ( c ) ∈ U . There also exists a neighbor ho o d ˜ V ∋ x s uch that for all y 0 , y 1 , . . . , y n ∈ ˜ V , IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 6 ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊕ we have α 0 δ y 0 ⊕ α 1 δ y 1 ⊕ . . . ⊕ α n δ y n ∈ ˜ U . It is straig ht forward to verify that the set V = { α 0 y 0 ⊕ α 1 y 1 ⊕ . . . ⊕ α n y n | n ∈ { 0 , 1 , . . . } , ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊕ , y 0 , y 1 , . . . , y n ∈ ˜ V } is a neighborho od of x requested b y 6+), w hich implies 6). Thus it is proved that an M ∪ -algebra ( X , ξ ) determines a contin uous op eration ic that satisfies conditions 1)–6). Now a ssume that we ar e given a compactum X and a cont inuous op eration ic : X × I × X → X th at satisfies conditions 1)– 6). Recall that X with the op eration ∨ : X × X → X , defined by the formula x ∨ y = x ⊕ 1 y , is a compa c t Lawson upper semilattice, therefore for all nonempty closed F ⊂ X there is sup F that dep ends o n F con tin uously w.r.t. Vietoris top ology [1 3]. Let c ∈ M ∪ X a nd c ( x 0 ) = 1 for some x 0 ∈ X . W e put ξ ( c ) = sup { x 0 ⊕ αx | x ∈ X, α 6 c ( x ) } . W e will prov e that ξ : M ∪ X → X is well defined (i.e. do es not depend on the choice of x 0 ) and con tin uous. F or each x ∈ X let g r ( x ) be the collection ( x ∨ y ) y ∈ X ∈ X X . Then the map of compacta g r : X → X X is contin uous and injective, ther e fore is an em b edding. The equality ξ ( c ) ∨ y = sup { x 0 ⊕ αx | x ∈ X , α 6 c ( x ) } ∨ y = sup { y ⊕ 1 x 0 ⊕ αx | x ∈ X , α 6 c ( x ) } = sup { ( y ⊕ 1 x 0 ) ∨ ( y ⊕ αx ) | x ∈ X , α 6 c ( x ) } = ( y ⊕ 1 x 0 ) ∨ sup { ( y ⊕ αx | x ∈ X , α 6 c ( x ) } = sup { ( y ⊕ αx | x ∈ X , α 6 c ( x ) } . holds for each y ∈ X , and the latter ex pression do es not depend on x 0 . This implies that g r ( ξ ( c )) and thu s ξ ( c ) are uniquely determined. Mor eov er, pr y ◦ g r ( ξ ( c )) is the supremum o f the image of the c losed set { ( x, α ) | x ∈ X , α ∈ I , α 6 c ( x ) } ⊂ X × I under the contin uous map that sends ( x, α ) to y ⊕ αx ∈ X . T aking into account that this set (the hyp o gr aph of the function c : X → I ) dep ends on c ∈ M ∪ X contin uo usly , we obtain that the corresp ondence c 7→ g r ( ξ ( c )) is contin uous, which implies co nt inuit y of ξ : M ∪ X → X . T o show that ( X, ξ ) is an M ∪ -algebra , we again assume c ( x 0 ) = 1 for a capa cit y c ∈ M ∪ X . Then y ⊕ αξ ( c ) = y ⊕ α sup { x 0 ⊕ β x | x ∈ X , β 6 c ( x ) } = sup { y ⊕ αx 0 ⊕ ( α ⊗ β ) x | x ∈ X , β 6 c ( x ) } = sup { ( y ⊕ αx 0 ) ∨ ( y ⊕ ( α ⊗ β ) x ) | x ∈ X , β 6 c ( x ) } = ( y ⊕ αx 0 ) ∨ sup { y ⊕ ( α ⊗ β ) x | x ∈ X, β 6 c ( x ) } = sup { y ⊕ ( α ⊗ β ) x | x ∈ X , β 6 c ( x ) } . holds for each y ∈ X , α ∈ I . The seco nd equality sign fo llows from an “infinite distributive law” y ⊕ α sup F = sup { y ⊕ αx | x ∈ F } , with F a nonempty subset of X . This law is first proved fo r finite F and then extended to infinite case b y contin uity of lo west upp er bounds. It is obvious that ξ ( δ x ) = x for a p oint x ∈ X , i.e. ξ ◦ η ∪ X = 1 X . W e choose a capacity C ∈ M 2 ∪ X and compare ξ ◦ M ∪ ξ ( C ) and ξ ◦ µ ∪ X ( C ). F or a point y ∈ X we IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 7 hav e y ∨ ( ξ ◦ M ∪ ξ ( C )) = sup { ( y ⊕ αx | x ∈ X , α 6 M ∪ ξ ( C )( x ) } = sup { ( y ⊕ αξ ( c ) | c ∈ M ∪ X , α 6 C ( c ) } = sup { sup { y ⊕ ( α ⊗ β ) x | x ∈ X, β 6 c ( x ) } , c ∈ M ∪ X , α 6 C ( c ) } = sup { ( y ⊕ αx | x ∈ X , c ∈ M ∪ X , α 6 min { C ( c ) , c ( x ) }} = y ∨ ( ξ ◦ µ ∪ X ( C )) . This implies ξ ◦ M ∪ ξ = ξ ◦ µ ∪ X , i.e. ( X , ξ ) is a M ∪ -algebra . T o prove that the c o rresp ondence “ M ∪ -algebra ↔ idemp otent conv ex co mbi- nation tha t satisfies 1)–6)” is one-to-one, ass ume that for some c o ntin uous ic : X × I × X satisfying 1)–6 ) there is a contin uous map ξ ′ : M ∪ X → X such that ( X , ξ ′ ) is a M ∪ -algebra and ic ( x, α, y ) = ξ ′ ( δ x ⊕ αδ y ) for all x, y ∈ X , α ∈ I . T her efore ξ ′ ( δ x ⊕ αδ y ) = ξ ( δ x ⊕ αδ y ) for the constructed ab ov e map ξ . Let 1 > α 1 > α 2 > 0 , x 0 , x 1 , x 2 ∈ X , then ξ ( δ x 0 ⊕ α 1 δ x 1 ⊕ α 2 δ x 2 ) = ξ ◦ µ ∪ X ( δ δ x 0 ⊕ α 1 δ δ x 1 ⊕ α 2 δ x 2 ) = ξ ◦ M ∪ ξ ( δ δ x 0 ⊕ α 1 δ δ x 1 ⊕ α 2 δ x 2 ) = ξ ( δ x 0 ⊕ α 1 δ ξ ( δ x 1 ⊕ α 2 δ x 2 ) ) = ξ ′ ( δ x 0 ⊕ α 1 δ ξ ′ ( δ x 1 ⊕ α 2 δ x 2 ) ) = · · · = ξ ′ ( δ x 0 ⊕ α 1 δ x 1 ⊕ α 2 δ x 2 ) . By induction in a similar manner we prove that ξ ( δ x 0 ⊕ α 1 δ x 1 ⊕ α 2 δ x 2 ⊕ . . . ⊕ α n δ x n ) = ξ ′ ( δ x 0 ⊕ α 1 δ x 1 ⊕ α 2 δ x 2 ⊕ . . . ⊕ α n δ x n ) for ar bitr ary integer n > 0. By contin uit y we deduce that ξ ( c ) = ξ ′ ( c ) for all c ∈ M ∪ X .  Let i c : X × I × X → X and ic ′ : X ′ × I × X ′ → X ′ be idemp otent convex combinations. W e s ay that a map f : ( X, ic ) → ( X ′ , ic ′ ) is affine if it pr eserves idempo ten t conv ex com bination, i.e. f ( ic ( x, α, y )) = ic ′ ( f ( x ) , α, f ( y )) for all x, y ∈ X , α ∈ I . Theorem 2.2. L et ( X, ξ ) , ( X ′ , ξ ′ ) b e M ∪ -algebr as, ic : X × I × X → X and ic ′ : X ′ × I × X ′ → X ′ b e the r esp e ctive idemp otent c onvex c ombinations. Then a c ontinu ous map f : X → Y is a morphism of M ∪ -algebr as ( X , ξ ) → ( X ′ , ξ ′ ) if and only if f : ( X, ic ) → ( X ′ , ic ′ ) is affine. Pr o of. Necessity . Let f : ( X , ξ ) → ( X ′ , ξ ′ ) b e a morphism of M ∪ -algebra s, x, y ∈ X , α ∈ I . Then f ( ic ( x, α, y )) = f ◦ ξ ( δ x ∨ αδ y ) = ξ ′ ◦ M f ( δ x ∨ αδ y ) = ξ ′ ( δ f ( x ) ∨ αδ f ( y ) ) = ic ′ ( f ( x ) , α, f ( y )) . Sufficiency . L e t f : ( X , ic ) → ( X ′ , ic ′ ) b e affine, then f ( x ∨ y ) = f ( x ) ∨ f ( y ) for all x, y ∈ X . Co nt inuit y of f implies that f preserves suprema of closed sets. F o r c ∈ M ∪ X w e c ho ose a point x 0 ∈ X such that c ( x 0 ) = 1, then M ∪ f ( x )( f ( x 0 )) = 1 . Therefore : ξ ′ ◦ M ∪ f ( c ) = sup { f ( x 0 ) ⊕ αx ′ | x ′ ∈ X ′ , α 6 M ∪ f ( c )( x ′ ) } = sup { f ( x 0 ) ⊕ αf ( x ) | x ∈ X , α 6 c ( x ) } = s up { f ( x 0 ⊕ αx ) | x ∈ X , α 6 c ( x ) } = f (sup { x 0 ⊕ αx | x ∈ X , α 6 c ( x ) } ) = f ◦ ξ ( c ) , and f is a morphism of M ∪ -algebra s.  IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 8 R emark 2 .3 . It is eas y to see that (max , min)-idemp otent conv ex compacta and their affine contin uo us maps co nstitute a ca tegory C onv max , min of (ma x , min)-idempo ten t conv ex co mpa cta that by the latter theor em is monadic (=tripleable) [17] ov e r the category o f compacta. Conv ex compacta ar e usually defined a s co mpact clos ed subsets of lo c ally co nv ex top ological vector spaces. T o obtain a similar description for (max , min)-idempotent conv ex compacta, we need some ex tr a definitions and facts. F or an idemp otent semiring [5] S = ( S, ⊕ , ⊗ , 0 , 1 ) a (left idempotent) S -semimodule is a set L with op erations ⊕ : L × L → L and ⊗ : S × L → L s uch that for all x, y , z ∈ L , α, β ∈ S : 1) x ⊕ y = y ⊕ x ; 2) ( x ⊕ y ) ⊕ z = x ⊕ ( y ⊕ z ); 3) there is an (obviously unique) element ¯ 0 ∈ L such that x ⊕ ¯ 0 = x for all x ; 4) α ⊗ ( x ⊕ y ) = ( α ⊗ x ) ⊕ ( α ⊗ y ), ( α ⊕ β ) ⊗ x = ( α ⊗ x ) ⊕ ( β ⊗ x ); 5) ( α ⊗ β ) ⊗ x = α ⊗ ( β ⊗ x ); 6) 1 ⊗ x = x ; 7) 0 ⊗ x = ¯ 0. W e adopt the usual conv ention and wr ite αx instead of α ⊗ x . Observe that these axioms imply α ¯ 0 = ¯ 0, x ⊕ x = x . Informa lly sp eaking, an idemp otent semimodule is a v ector spac e o ver an idempo tent semiring. If S = ( I , max , min , 0 , 1), w e will talk ab out a (max , min )- idemp otent semimo d- ule . In this case we define an op eratio n ic : L × I × L → L by the formula ic ( x, α, y ) = x ⊕ ( α ⊗ y ) ( ⊕ and ⊗ ar e fro m L ). It is easy to see that ic sat- isfies 1)–5). The combination α 0 x 0 ⊕ α 1 x 0 ⊕ . . . ⊕ α n x n of po int s x 0 , x 1 , . . . , x n is defined in an o bvious wa y a nd coincides with the des crib ed ab ove oper ation if ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊕ . A subse t A of a (max , min)-idemp otent semimo dule L is called c onvex if x ⊕ αy ∈ A whenev er x, y ∈ A , α ∈ I . A co nv e x subset A ⊂ L contains all idemp otent conv ex combin ations of its elements. Let a (max , min)-idemp otent semimo dule L b e a compactum, the op eratio ns ⊕ and ⊗ be contin uo us, and the topo logy on L satisfy an additional co ndition : 8) for a neighborho o d U o f any element x ∈ L there is a neighborho o d V of x , V ⊂ U , suc h that y ⊕ z ∈ V for all y , z ∈ V . Then we call ( L, ⊕ , ⊗ ) a c omp act L awson (max , min) -idemp otent semimo dule . By the above theore m L is a M ∪ -algebra , whic h implies 8+) for a neig hborho o d U of a ny element x ∈ L there is a neigh b orho o d V of x , V ⊂ U , suc h that y ⊕ αz ∈ V for all y , z ∈ V , α ∈ I . Thu s for ev ery p oint of L there is a local base that consis ts of co n vex neig hbor- ho o ds, and w e say that L is lo c al ly c onvex . The nature of a co mpactum X with an idempo tent conv e x combination that satisfies 1 )–6) is clarified by the following Theorem 2.4. A p air of a c omp actum X and a c ontinuous map ic : X × I × X → X is a (max , min) -idemp otent c onvex c omp actum if and only if X is a close d c onvex subset of a c omp act L awson (max , min) -idemp otent semimo dule ( L, ⊕ , ⊗ ) such that ic ( x, α, y ) ≡ x ⊕ αy | {z } in L . Pr o of. Sufficiency is obvious. T o prov e necessity , as sume that X is a co mpactum and a co ntin uous map ic : X × I × X → X satisfies conditions 1)–6). W e define an e quiv alence relation “ ∼ ” on X × I as follows : ( x 1 , a 1 ) ∼ ( x 2 , a 2 ) if y ⊕ a 1 x 1 = IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 9 y ⊕ a 2 x 2 for all y ∈ X . This relation is clo sed in ( X × I ) × ( X × I ), ther efore the quotient space X × I / ∼ , which we denote b y ¯ X , is a co mpa ct Hausdo rff space. W e also deno te by [( x, a )] the equiv a lence class of the pair ( x, a ). The map i : X → ¯ X that s e nds a point x ∈ X to [( x, 1)] is an embedding b ecaus e ( x 1 , 1) ∼ ( x 2 , 1) is po ssible only if x 1 = x 2 . W e define op erations ⊗ : I × ¯ X → ¯ X and ⊕ : ¯ X × ¯ X → ¯ X by the formulae α ⊗ [( x, a )] = [( x, α ⊗ a )] and [( x, a )] ⊕ [( y , b )] = ( [( x ⊕ by , a )] , a > b, [( y ⊕ ay , b )] , a 6 b. The element ¯ 0 = [( x, 0)] do es not depend on x and satisfies 3). Pro p erties 5), 6), 7) are obvious. V erification that ⊕ , ⊗ ar e well defined, contin uous and satisfy 1), 2), 4), 8), is more conv enient with a generaliza tion o f the mapping g r : X → X X that was defined in the pro of of the latter theo rem. T o avoid introducing extra denotations, we denote by g r ( x, α ), where x ∈ X , α ∈ I , the collectio n ( t ⊕ αx ) t ∈ X . Then the map g r : X × I → X X is contin uo us (but, as can be shown, not injectiv e). It is obvious that ( x 1 , α 1 ) ∼ ( x 2 , α 2 ) if and o nly if g r ( x 1 , α 1 ) = g r ( x 2 , α 2 ), thus we will identif y the image of the map g r with the q uo tient space ¯ X = X × I / ∼ , and g r with the quotient map. Let ¯ x, ¯ y , ¯ z be po ints in ¯ X , and ¯ x = g r ( x, a ) = ( x t ) t ∈ X , ¯ y = g r ( y , b ) = ( y t ) t ∈ X , ¯ z = g r ( z , c ) = ( z t ) t ∈ X . Observe that x ⊕ y = ( x t ∨ y t ) t ∈ , α ⊗ ¯ x = ( t ⊕ αx t ) t ∈ X , therefore ¯ x ⊕ ¯ y and α ⊗ ¯ x are uniq ue ly determined and contin uous w.r.t. ¯ x , ¯ y a nd α , ¯ x resp. Similar express ions ca n b e written for x ⊕ z and y ⊕ z , a nd 1),2) are easily se en. Next, α ⊗ ¯ x = ( t ⊕ αx t ) t ∈ X , α ⊗ ¯ y = ( t ⊕ αy t ) t ∈ X , thus ( α ⊗ ¯ x ) ⊕ ( α ⊗ ¯ y ) = (( t ⊕ αx t ) ∨ ( t ⊕ αx t )) t ∈ X = (( t ⊕ α ( x t ∨ αx t )) t ∈ X = α ⊗ ( ¯ x ⊕ ¯ y ) . Similarly ( α ⊗ ¯ x ) ⊕ ( β ⊗ ¯ x ) = (( t ⊕ αx t ) ∨ ( t ⊕ β x t )) t ∈ X = (( t ⊕ αx t ⊕ β x t )) t ∈ X = ( α ⊕ β ) ⊗ ¯ x, and condition 4) holds. Let G ⊂ ¯ X b e a clo s ed nonempty set, then G = g r ( F ) for s ome c lo sed F ⊂ X × I . There is ( x 0 , a 0 ) ∈ F s uch that a 0 = max { a | ( x, a ) ∈ F } . It is eas y to show that sup G in ¯ X is eq ual to [( x ′ , a 0 )] where x ′ = sup { x 0 ⊕ ax | ( x, a ) ∈ F } , thus the upper semilattice ¯ X is complete. It is also clearly seen that g r ( x ′ , a 0 ) = (sup { t ⊕ ax | ( x, a ) ∈ F } ) t ∈ X = (sup { x t | ( x t ) t ∈ X ∈ G } ) t ∈ X , therefore sup G dep ends on G contin uously w.r .t. Vietor is to po logy . It is a statement equiv alent to 8 ) [13].  As triples M ∪ and M ∩ are iso morphic through a na tur al transfo rmation κ defined in [9], and the ma p I → I tha t sends ea ch t to 1 − t is a n isomor phism of the idempo ten t semirings ( I , ⊕ , ⊗ , 0 , 1) and ( I , ⊗ , ⊕ , 1 , 0 ), by duality we immediately can state a n analogue o f Theor em 2 .1. Its pro of ca n be obtained by replacing M ∪ by M ∩ , ⊗ b y ⊕ , 1 by 0 , upp er semila ttices by lo wer ones, ∨ b y ∧ , sup b y inf , ∆ ⊕ by the (idemp otent) n -dimensional ⊗ - simplex ∆ n ⊗ = { ( α 0 , α 1 , . . . , α n ) ∈ I n +1 | α 0 ⊗ α 1 ⊗ . . . ⊗ α n = 0 } IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 10 and v ic e versa, where it is necessary . Thu s we define dual idemp otent c onvex c ombinations a nd (min , max)- idemp otent c onvex c omp acta that a re pr ecisely M ∩ - algebras . W e o mit ob vious details. O bserve that for a given M ∩ -algebra ( X , ξ ) the resp ective dual idemp otent conv ex combination ci : X × I × X → X is determined by the equality ci ( x, α, y ) = ξ ( δ x ∧ ( α ∨ δ y )). Conv ersely , the v alue ξ ( c ) for a ca pac- it y c ∈ M ∪ X (a ssuming that c ( X \ { x 0 } ) = 0) is equa l to ξ ( c ) = inf { ci ( x 0 , α, x ) | x ∈ X, α > c ( X \ { x ) } . It is easy also to formulate a na logues of Theorems 2.2,2.4. 3. Algebras for the cap a city monad In the sequel a (min , max) -idemp oten t bic onvex c omp actum is a compactum X with four o per ations ¯ ⊕ : X × X → X , ⊗ : I × X → X , ¯ ⊗ : X × X → X , ⊕ : I × X → X such that ( X , ¯ ⊕ , ¯ ⊗ ) is a Lawson la ttice, ( X , ¯ ⊕ , ⊗ ) is an ( I , ⊕ , ⊗ )-semimo dule, ( X, ¯ ⊗ , ⊕ ) is a n ( I , ⊗ , ⊕ )-semimo dule, the asso ciativ e laws ( α ⊕ x ) ¯ ⊕ y = α ⊕ ( x ¯ ⊕ y ), ( α ⊗ x ) ¯ ⊗ y = α ⊗ ( x ¯ ⊗ y ) and the distributive laws α ⊗ ( β ⊕ x ) = ( α ⊗ β ) ⊕ ( α ⊗ x ), α ⊕ ( β ⊗ x ) = ( α ⊕ β ) ⊗ ( α ⊕ x ) are v alid for a ll x, y ∈ X , α, β ∈ I . Theorem 3.1. L et X b e a c omp actum. Ther e is a one-to-one c orr esp ondenc e b etwe en : 1) c ontinu ou s maps ξ : M X → X su ch that the p air ( X , ξ ) is an M -algebr a; 2) quadruples ( ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) of c ontinu ou s op er ations ¯ ⊕ : X × X → X , ⊗ : I × X → X , ¯ ⊗ : X × X → X , ⊕ : I × X → X su ch that ( X , ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) is a (max , min ) - idemp otent bic onvex c omp actum; 3) quadruples ( ¯ ⊕ , ⊗ , p, m ) of c ontinuous m aps ¯ ⊕ : X × X → X , ¯ ⊗ : X × X → X , p, m : I → X such that a) ( X, ¯ ⊕ , ¯ ⊗ ) is a L awson lattic e; b) p : ( I , ⊕ ) → ( X , ¯ ⊕ ) is a morphism of upp er semilattic es that pr eserves a top element; c) m : ( I , ⊗ ) → ( X, ¯ ⊗ ) is a morphism of lower s emilattic es that pr eserves a b ottom element; d) for al l α, β ∈ I we have m ( α ) ⊗ p ( β ) = p ( α ⊗ β ) , m ( α ) ⊕ p ( β ) = m ( α ⊕ β ) . In the c ase 2) the fol lowing pr op erty of local biconvexit y holds : for a neighb or- ho o d U o f any element x ∈ X ther e is a neighb orho o d V of x , V ⊂ U , su ch that y ¯ ⊕ ( α ¯ ⊗ z ) ∈ V , y ¯ ⊗ ( α ¯ ⊕ z ) ∈ V fo r al l y , z ∈ V , α ∈ I . Pr o of. 1) → 3). Let ( X, ξ ) b e an M -algebr a. W e use the fact that G is a submonad of the capacity monad M . The comp onents of an embedding i G : G ֒ → M are of the for m i G X ( A )( F ) = ( 1 , if F ∈ A , 0 — otherwise . Therefore ( X, ξ ◦ i G X ) is a G -algebr a. Theor em 2 [15] states that for a G -algebra ( X, θ ) the op erations ¯ ⊕ : X × X → X , ⊗ : I × X → X defined b y the formulae x ¯ ⊕ y = θ ( η G X ( x ) ∩ η G X ( y )) and x ¯ ⊗ y = θ ( η G X ( x ) ∪ η G X ( y )) ar e such that ( X, ¯ ⊕ , ¯ ⊗ ) is a L awson lattice. W e a pply this theor em to θ = ξ ◦ i G X a nd obtain that X with the op erations x ¯ ⊕ y = ξ ( δ x ∨ δ y ) and x ¯ ⊗ y = ξ ( δ x ∧ δ y ) is a La wson lattice. W e denote b y ¯ 0 and ¯ 1 its least a nd gr eatest elements. Now we put p ( α ) = ξ ( δ ¯ 0 ∨ α ⊗ δ ¯ 1 ), m ( α ) = ξ ( δ ¯ 1 ∧ α ⊕ δ ¯ 0 ). It is obvious that p, m are co nt inuous and IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 11 p (1) = ¯ 0 ¯ ⊕ ¯ 1 = ¯ 1, m (0) = ¯ 1 ¯ ⊗ ¯ 0 = ¯ 0. Next, for all α, β ∈ I : p ( α ⊕ β ) = ξ ( δ ¯ 0 ∨ ( α ⊕ β ) ⊗ δ ¯ 1 ) = ξ ◦ µX ( δ δ ¯ 0 ∨ α ⊗ δ ¯ 1 ∨ δ δ ¯ 0 ∨ β ⊗ δ ¯ 1 ) = ξ ◦ M ξ ( δ δ ¯ 0 ∨ α ⊗ δ ¯ 1 ∨ δ δ ¯ 0 ∨ β ⊗ δ ¯ 1 ) = ξ ( δ p ( α ) ∨ δ p ( β ) ) = p ( α ) ¯ ⊕ p ( β ) . Similarly m ( α ⊗ β ) = m ( α ) ¯ ⊗ m ( β ) for all α, β ∈ I . W e also have m ( α ) ⊗ p ( β ) = ξ ( δ ξ ( δ ¯ 1 ∧ α ⊕ δ ¯ 0 ) ∧ δ ξ ( δ ¯ 0 ∨ β ⊗ δ ¯ 1 ) ) = ξ ◦ M ξ ( δ δ ¯ 1 ∧ α ⊕ δ ¯ 0 ∧ δ δ ¯ 0 ∨ β ⊗ δ ¯ 1 ) = ξ ◦ µX ( δ δ ¯ 1 ∧ α ⊕ δ ¯ 0 ∧ δ δ ¯ 0 ∨ β ⊗ δ ¯ 1 ) = ξ ( δ ¯ 0 ∨ ( α ⊗ β ) ⊗ δ ¯ 1 ) = p ( α ⊗ β ) , as well as m ( α ) ⊕ p ( β ) = m ( α ⊕ β ). 3) → 2). It is sufficien t to put α ⊗ x = m ( α ) ¯ ⊗ x , α ⊕ x = p ( α ) ¯ ⊕ x , a nd it is clear that all co nditions o f 2) are satisfied due to the commutativ e, a sso ciative and distributive la ws in ( X, ¯ ⊕ , ¯ ⊗ ). Observe also that, if m, p are de ter mined by an M -alg e bra ( X , ξ ) as descr ib e d ab ov e, then α ⊗ x = ξ ( δ ξ ( δ ¯ 1 ∧ α ⊕ δ ¯ 0 ) ∧ δ x ) = ξ ( δ ξ ( δ ¯ 1 ∧ α ⊕ δ ¯ 0 ) ∧ δ x ) = ξ ◦ M ξ ( δ δ ¯ 1 ∧ α ⊕ δ ¯ 0 ∧ δ δ x ) = ξ ◦ µX ( δ δ ¯ 1 ∧ α ⊕ δ ¯ 0 ∧ δ δ x ) = ξ ( δ ¯ 1 ∧ α ⊕ δ ¯ 0 ∧ δ x ) = ξ ◦ µX ( δ δ x ∧ α ⊕ δ ¯ 0 ∧ δ δ ¯ 1 ) = ξ ◦ M ξ ( δ δ x ∧ α ⊕ δ ¯ 0 ∧ δ δ ¯ 1 ) = ξ ( δ x ∧ α ⊕ δ ¯ 0 ) ¯ ⊗ ¯ 1 = ξ ( δ x ∧ α ⊕ δ ¯ 0 ) , and similar ly α ⊕ x = ξ ( δ x ∨ α ⊗ δ ¯ 1 ) for all x ∈ X , α ∈ I . In the sa me ma nner we can show that x ¯ ⊕ ( α ⊗ y ) = ξ ( δ x ∨ α ⊗ δ y ), x ¯ ⊗ ( α ⊕ y ) = ξ ( δ x ∧ α ⊕ δ y ) for all x, y ∈ X , α ∈ I . These formulae are the sa me that were used to define idem- po tent semico nv ex combinations and dual idemp otent semiconv ex co m binations in the pro ofs of Theore m 2.1 and the dual theor em. 2) → 1). Now let ( X , ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) b e a (min , max)-idemp o tent biconv e x compactum. If ic ( x, α, y ) = x ¯ ⊕ ( α ⊗ y ), ci ( x, α, y ) = x ¯ ⊗ ( α ⊕ y ), then it is obvious that ( X , ic ) is a (min , max)-idemp otent conv ex compactum and ( X , ci ) is a (max , min)-idemp otent conv ex compa c tum. Thus by T he o rem 2.1 and the dual theorem, if mappings ξ ∪ : M ∪ X → X and ξ ∩ : M ∩ X → X are defined by the formu lae ξ ∪ ( c ) = s up { x 0 ¯ ⊕ ( α ⊗ x ) | x ∈ X , α 6 c ( x ) } , c ∈ M ∪ X , x 0 ∈ X , c ( x 0 ) = 1 , and ξ ∩ ( c ) = inf { x 0 ¯ ⊗ ( α ⊕ x ) | x ∈ X , α > c ( X \{ x } ) } , c ∈ M ∩ X , x 0 ∈ X , c ( X \{ x 0 } ) = 0 , then the pair s ( X , ξ ∪ ) and ( X , ξ ∩ ) are resp. an M ∪ -algebra and a n M ∩ -algebra . In our case we ca n define ξ ∪ , ξ ∩ by s impler but equiv alent formulae (the second “ =” sign in e a ch equality is due to co mplete distributivity of a co mpact Lawson lattice) : ξ ∪ ( c ) = sup { c ( x ) ⊗ x | x ∈ X } = inf { c ( X \ A ) ⊕ sup A | A ⊂ cl X } , c ∈ M ∪ X , and ξ ∩ ( c ) = inf { c ( X \ { x } ) ⊕ x | x ∈ X } = sup { c ( X \ A ) ⊗ inf A | A ⊂ cl X } , c ∈ M ∩ X . If ξ , ξ ′ : M X → X are contin uous maps such that the pairs ( X , ξ ), ( X, ξ ′ ) a re M - algebras a nd ξ | M ∪ X = ξ ′ | M ∪ X = ξ ∪ , ξ | M ∩ X = ξ ′ | M ∩ X = ξ ∩ , then the t wo following IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 12 diagrams have to be commutativ e (we omit explicit no ta tions for re s trictions) : M ∪ M ∩ X µX / / M ∪ ξ ∩   M X ξ   M ∪ ξ ∪ / / X (*) M ∩ M ∪ X µX / / M ∩ ξ ∪   M X ξ ′   M ∩ ξ ∩ / / X (**) W e show that if C , C ′ ∈ M ∩ M ∪ X are such that µX ( C ) = µX ( C ), then ξ ∪ ◦ M ∪ ξ ∩ ( C ) = ξ ∪ ◦ M ∪ ξ ∩ ( C ′ ). Observe that µX ( C ) = µX ( C ) implies that for all A ⊂ cl X and α ∈ I the existence of c ∈ M ∩ X s uch that C ( c ) > α and c ( A ) > α is equiv alent to the ex is tence of c ′ ∈ M ∩ X such tha t C ′ ( c ′ ) > α and c ′ ( A ) > α . It is also obvious that the same s tatement is v alid fo r any o p en A ⊂ X . Thus : ξ ∪ ◦ M ∪ ξ ∩ ( C ) = sup { M ∪ ξ ∩ ( C )( x ) ⊗ x | x ∈ X } = sup {C ( c ) ⊗ ξ ∩ ( c ) | c ∈ M ∩ X } = sup {C ( c ) ⊗ s up { c ( X \ A ) ⊗ inf A | A ⊂ cl X } | c ∈ M ∩ X } = sup {C ( c ) ⊗ c ( X \ A ) ⊗ inf A | A ⊂ cl X , c ∈ M ∩ X } = sup { α ⊗ inf A | A ⊂ cl X , c ∈ M ∩ X , α ∈ I , α 6 C ( c ) , α 6 c ( X \ A ) } = sup { α ⊗ inf A | A ⊂ cl X , c ′ ∈ M ∩ X , α ∈ I , α 6 C ′ ( c ′ ) , α 6 c ′ ( X \ A ) } = · · · = ξ ∪ ◦ M ∪ ξ ∩ ( C ′ ) . An obvious dual statement is als o v alid. T aking into acco un t that by Theo- rem 8 [9 ] for a compactum X the equa lit y µ ( M ∩ M ∪ X ) = µ ( M ∪ M ∩ X ) = M X is v alid, and µ X | M ∩ M ∪ X : M ∩ M ∪ X → M X and µ X | M ∪ M ∩ X : M ∪ M ∩ X → M X are quotient maps as contin uous surjective maps o f c o mpacta, we obtain that the diagrams (*) and (**) uniquely determine cont inuous maps ξ , ξ ′ : M X → X . In the dia gram M 2 ∪ M ∩ X µ ∪ M ∩ X / / M 2 ∪ ξ ∩ $ $ J J J J J J J J J M ∪ µX   M ∪ M ∩ X M ∪ ξ ∩ $ $ J J J J J J J J J J µX   M 2 ∪ X µ ∪ X / / M ∪ ξ ∪   M ∪ X ξ ∪   M ∪ M X µX / / M ∪ ξ % % K K K K K K K K K K M X ξ % % K K K K K K K K K K K M ∪ X ξ ∪ / / X the top square and the side squar e s are commutativ e, a nd the leftmost vertical arrow is e pimorphic, therefor e the b ottom squa re comm utes as well. Using also dual arg ument s, we show that the t w o following diagra ms are commutativ e : M ∪ M X µX / / M ∪ ξ   M X ξ   M ∪ X ξ ∪ / / X M ∩ M X µX / / M ∩ ξ ′ ∪   M X ξ ′   M ∩ X ξ ∩ / / X IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 13 W e apply the functor M ∩ to the left diag r am and combine it with (**) : M 2 X M ξ % % J J J J J J J J J J M ∩ M ∪ M X µM X 7 7 p p p p p p p p p p p M ∩ M ∪ ξ / / M ∩ µX   µM X   M ∩ M ∪ X µX / / M ∩ ξ ∪   M X ξ ′   M ∩ M X M ∩ ξ / / µX ' ' N N N N N N N N N N N M ∩ X ξ ∩ / / X M 2 X µX / / M X ξ ′ 9 9 t t t t t t t t t t The restriction o f µM X to M ∩ M ∪ M X is an epimorphism, there fo re the commu- tativity of the outer c ontour imply that the left o f the t wo following diagra ms commutes. The right diagra m is commutativ e by dual ar g ument s. M 2 X µX / / M ξ   M X ξ ′   M X ξ ′ / / X M 2 X µX / / M ξ ′   M X ξ   M X ξ / / X Therefore in the diagr am M 3 X M µX / / µM X $ $ H H H H H H H H H M 2 ξ   M 2 X µX # # G G G G G G G G G M ξ ′   M 2 X µX / / M ξ   M X ξ   M 2 X M ξ ′ / / µX $ $ I I I I I I I I I M X ξ # # H H H H H H H H H M X ξ / / X the front square is commutativ e, whic h is the “harder part” of the definition of M - algebra. A pro o f of the “eas ie r part” ξ ◦ η X = 1 X is s traightforw ard. T hus ( X , ξ ) is a unique M -algebra s uch that x ¯ ⊕ ( α ⊗ y ) = ξ ( δ x ∨ ( α ⊗ δ y )) a nd x ¯ ⊗ ( α ⊕ y ) = ξ ( δ x ∧ ( α ⊕ δ y )) for all x, y ∈ X , α ∈ I . As a by-product we obtain that ξ = ξ ′ , i.e. defininions of ξ by the dia grams (*) and (**) ar e equiv alen t. T o prov e lo cal biconv exity , for a given neig hborho o d U of a p o int x by contin uit y of ξ and the equality ξ ( δ x ) = x w e cho ose a neighbor ho o d ˜ U ⊂ M ∪ X o f δ x such that for all c ∈ ˜ U we have ξ ( c ) ∈ U . There exists a neigh b orho o d ˆ U ∋ x suc h that for a ll x 0 , x 1 , . . . , x n ∈ ˆ U , ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊕ we have α 0 δ x 0 ∨ α 1 δ x 1 ∨ · · · ∨ α n δ x n ∈ ˜ U . Now we choose a neighborho od ˜ ˜ U ⊂ M ∩ X o f δ x such that for all c ∈ ˜ U we have ξ ( c ) ∈ ˆ U . There is a ls o a neighborho o d ˆ ˆ U ∋ x such that y 0 , y 1 , . . . , y n ∈ ˆ ˆ U , IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 14 ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊗ imply α 0 δ y 0 ∧ α 1 δ y 1 ∧ · · · ∧ α n δ y n ∈ ˜ ˜ U . Now we put ˜ V = { ( α ⊕ 0 y 0 ) ¯ ⊗ ( α 1 ⊕ y 1 ) ¯ ⊗ . . . ¯ ⊗ ( α n ⊕ y n ) | n ∈ { 0 , 1 , . . . } , ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊗ , y 0 , y 1 , . . . , y n ∈ ˆ ˆ U } , and the set V = { ( α 0 ⊗ y 0 ) ¯ ⊕ ( α 1 ⊗ y 1 ) ¯ ⊕ . . . ¯ ⊕ ( α n ⊗ y n ) | n ∈ { 0 , 1 , . . . } , ( α 0 , α 1 , . . . , α n ) ∈ ∆ n ⊕ , x 0 , x 1 , . . . , x n ∈ ˜ V } is a neighborho o d of x requested b y lo cal bico mm utativity .  F or (max , min)-idemp otent bico nvex co mpacta ( X , ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) a nd ( X ′ , ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) we say that a map f : X → X ′ is biaffine if it pr eserves idemp otent conv ex combination and the dual idemp otent convex combination, i.e. f ( x ¯ ⊕ ( α ⊗ y )) = f ( x ) ¯ ⊕ ( α ⊗ f ( y )), f ( x ¯ ⊗ ( α ⊕ y )) = f ( x ) ¯ ⊗ ( α ⊕ f ( y )) whenever x, y ∈ X , α ∈ I . Theorem 3.2. L et ( X , ξ ) , ( X ′ , ξ ′ ) b e M -algebr as and quadruples ( ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) of c ontinu ous op er ations b e determine d on X and X ′ by ξ and ξ ′ r esp. (in the sense of The or em 3.1).Then a c ontinuous map f : X → Y is a morphism of M ∪ -algebr as ( X, ξ ) → ( X ′ , ξ ′ ) if and only if ( X, ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) → ( X ′ , ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) is biaffine. Pr o of. Necessity . Let f b e a morphism o f algebra s. It was shown in the pro o f of the previo us theor em that the idempotent co nv ex combination and the dual idempo ten t conv ex combination of p oints x, y ∈ X a re determined by the formulae x ¯ ⊕ ( α ⊗ y ) = ξ ( δ x ∨ α ⊗ δ y ), x ¯ ⊗ ( α ⊕ y ) = ξ ( δ x ∧ α ⊕ δ y ) (in X ′ the s ame but ξ replaced with ξ ′ ). Then w e follow the line of the pro of o f Theorem 2 .2. Sufficiency . Let f b e biaffine. Then by Theorem 2.2 and a dual theorem f is a morphism of M ∪ -algebra s ( X , ξ | M ∪ X ) → ( X ′ , ξ ′ | M ∪ X ′ ) and a morphism of M ∩ - algebras ( X , ξ | M ∩ X ) → ( X ′ , ξ ′ | M ∩ X ′ ), i.e. the diagrams M ∪ X M ∪ f / / ξ | M ∪ X   M ∪ X ′ ξ ′ | M ∪ X ′   X f / / X ′ M ∩ X M ∩ f / / ξ | M ∩ X   M ∩ X ′ ξ ′ | M ∩ X ′   X f / / X ′ are co mm utative. Therefore the top face a nd the s ide face s of the dia gram M ∪ M ∩ X M ∪ M ∩ f / / M ∪ ( ξ | M ∩ X ) & & L L L L L L L L L L µX   M ∪ M ∩ X ′ M ∪ ( ξ ′ | M ∩ X ′ ) ′ & & M M M M M M M M M M µX ′   M ∪ M X M ∪ f / / ξ | M ∪ X   M ∪ M X ′ ξ ′ | M ∪ X ′   M X M f / / ξ & & M M M M M M M M M M M M X ′ ξ ′ & & N N N N N N N N N N N X f / / X ′ commute. The leftmost arrow µX : M ∪ M ∩ X → M X is an epimorphism, th us the bo ttom face comm utes as w ell, i.e. f is a morphism of M -a lgebras.  IDEMPOTENT CONVEXITY AND ALGEBRAS FOR THE CAP A CITY MONAD. . . 15 R emark 3.3 . The latter theorem implies that the catego ry B i C onv max , min of (max , min)- idempo ten t biconv ex compacta and their con tinuous biaffine maps is mona dic ov e r the ca teg ory of co mpacta. R emark 3.4 . Note that a bia ffine map f : ( X, ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) → ( X ′ , ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) not necessarily preser ves op erations ⊕ and ⊗ (altho ug h it prese rves ¯ ⊕ and ¯ ⊗ ). E.g ., let X = X ′ = I , ⊕ = ¯ ⊕ = max, ⊗ = ¯ ⊗ = min, f ( x ) = ma x { x, 1 2 } . Then f is biaffine, but f (0 ⊗ 1) = 1 2 6 = 0 ⊗ f (1) = 0. It is easy to show that a biaffine cont inuous map f : ( X , ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) → ( X ′ , ¯ ⊕ , ⊗ , ¯ ⊗ , ⊕ ) pr eserves ⊗ iff it preserves a bottom element, and it pr eserves ⊕ iff it preserves a top element. W e pr esent an example of (max , min)-idemp otent biconv ex compa cta. Let A b e a set and for each a ∈ A a non-decreas ing surjective map ϕ a : I → I is fixed. 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