The dual geometry of Boolean semirings

THE DUAL GEOMETR Y OF BOOLEAN SEMIRINGS D ANIEL J. CLOUSE AND FE RNANDO GUZM ´ AN Abstract. It is well known that the v ariet y of B o olean semirings, which is generated b y the thr ee element semiring S , is dual to the categ ory of partially Stone spaces. W e place this duality in the context o f natural dualities. W e b egin by introducing a top olo g ical structure f S and obtain an optimal natural duality b etw een the quas i-v ariety I S P ( S ) a nd the catego ry I S c P + ( f S ). Then we construct an optimal a nd very small structure f S os that yields a strong duality . The geometry of some of the partially Stone spaces that take part in these dualities is pres en ted, a nd w e ca ll them ” ha iry cubes ” , as they a re n -dimensional cub es with unique incomparable cov er s for eac h elemen t of the cube. W e also o btain a p olynomia l representation for the elements o f the hair y cube. 1. Intro duction Extensions o f the concept o f a Bo olean Ring to include semirings has b een done in sev eral differen t directions. One source of divers it y are the differen t definitions of semiring . The other is ho w they get connected to Bo olean rings. W e will use the concept of semiring commonly used in formal languages and auto ma t a theory , that is, the only thing missing in order to b e a ring is the existe nce o f additiv e inv erses (see [3] and [6]). As in Guzman [4], w e will denote b y BS R the v ariet y generated by the tw o 2 -elemen t semirings, a nd will call it the v ariet y of Bo olean semirings. It turns out that this v ariet y is also generated by a 3-elemen t semiring with carrying set S = { 0 , h, 1 } , that w e denote S . The semiring S will pla y a crucial role in this pa p er. In [4], follo wing the ideas of Stone [7] in his now famous “Stone represen tation theorem” , a duality is established b et wee n the category B S R of Bo olean semirings and the category P S S of partially Stone spaces. On the other hand, Clark and D a vey [1] presen t a thorough study of natural dualities b etw een algebraic and top ological quasi-v arieties. It is t he goal of this pap er to place the dualit y from [4 ] in the muc h ric her con t ext of [1]. A structured top o logical space consists of X = h X ; G, H , R, T i where h X, T i is a topolo gical space, G is a set of finitary (total) op erations on X , H is a se t of finitary partial op erations on X and R is a set of finitary relations on X . The arities of the op erations, partial o p erations, and relations define the type of X . G iv en a finite discrete structured top ological space X , w e denote by I S c P + ( X ) the category of closed substructures of non- empt y pro ducts of copies of X . Date : Ja nu ary 28, 20 0 8. 2000 Mathematics Su bje ct Classific ation. Primar y:06D50, 08C15; Secondary:08 C05, 18 A40, 06E15 . Key wor ds and phr ases. Bo olean Semiring , Dualit y , Strong duality , Optimal duality . 1 2 D. J. CLOUSE AND F. GUZM ´ AN On one side of the duality w e will hav e the quasi-v ariety A = I S P ( S ) generated b y S . On the other side, we will ha v e the category of structured top olog ical spaces X = I S c P + ( f S ), generated by some appropriate structure f S = h S ; G, H , R, T i ha ving S = { 0 , h, 1 } as underlying set and T the discrete to p ology . W e b egin b y naming three binary relations on S : r 1 , r 2 , and r 3 , and prov e the following result. Theorem 1. The structur e f S = h S ; { r 1 , r 2 , r 3 } , T i yields a n optimal natur al duality on A . The pro of is in Section 2. The dualit y is optimal in the sense that if an y one of the relations w ere to b e deleted from the structure of f S , duality would b e lost. In the dualit y B S R ⇆ P S S , a Bo olean semiring A is mapped to the set of prime filters of A (recall t hat a Bo olean semiring can b e view ed as a s partially complemen ted distributiv e lattice. See [4]). F or finite A , ev ery prime filter is the upset of a join-irreducible elemen t. When X is a closed substructure of a finite p ow er of f S , w e would like to describe the join-irreducible elemen ts of X ( X , f S ). W e denote the set of all of them by X ( X , f S ) J . In Theorem 2 , join-irreducible elemen t s of X ( f S n , f S ) is giv en; see Section 2. In [2] a description of X ( X , f S ) J is giv en for any X ∈ X that is a closed substructure of a finite p ow er of f S . This will app ear in a subsequen t pap er. Here w e lay the foundatio n for those results, a description of t he meet semilattice X ( f S n , f S ) J , that we call the “ hairy cube ”. It consists of an n -dimensional cub e co v ered by “ hairs ”. More precisely , Theorem 3. The p oset X ( f S n , f S ) J c onsists o f two p arts: the “ base ” Y n which is an n -cub e and the “ hairs ” X ( f S n , f S ) J \ Y n , w h ich ar e p airwis e i n c omp ar able. Each element of the b ase is c ove r e d by a unique hair. Each hair c overs a unique b ase element. In Theorem 4 it is shown that these par t ial order prop erties of X ( f S n , f S ) J completely determine it as a partially Stone space. W e close Section 3 with a p olynomial represe n tat io n of the join- ir r educible elemen ts of X ( f S n , f S ). See Theorem 5 . In Section 4 we first establish that the duality in Theorem 1 is neither a full nor a strong dualit y . Then w e discuss wh y this is true and ho w tha t dualit y can easily b e upgraded to a strong duality , follo wing some of the ideas of [1]. Then w e sho w ho w to construct an optimal and v ery small structure f S os that yields a strong dualit y o n A . Despite the fact that S is not sub directly irreducible, Irr( S ) is 2, f S os consists of a single relation r 2 , and a single partial op eration λ 1 . Theorem 6. L et T denote the discr ete top olo gy and f S os = h S ; { r 2 } , { λ 1 } , T i . Then f S os yields an optimal str ong duality on A . Finally , w e discuss wh y the “Hairy Cub e” will p ersist in that strong duality . Corollary 6. L et X os = I S c P + ( f S os ) , then for any n ∈ N , X os ( f S n os , f S ) is the n-d imensional Hairy Cub e. The results in t his paper and in [2] greatly expand our understanding of the dual equiv - alence b etw een t he v ariety of Bo olean Semirings and the category of Partially Stone Spaces THE D UAL GEOMETR Y OF BOOLEAN SEMIRINGS 3 established in [4]. They also a complete a larg e initial step for in v estigating the strong duality w e establish b etw een A and X os . 1.1. Notation. Since some of the a rgumen ts in the pap er are of a n inductiv e nature, we need a con ve nien t notation to mov e bac k and forth b et wee n functions S n − 1 → S and functions S n → S . Giv en n ∈ N , Φ : S n → S , a ∈ S , and x ∈ S n − 1 w e denote b y Φ a the map Φ a : S n − 1 → S ( x 1 , . . . , x n − 1 ) 7→ Φ( a, x 1 , . . . , x n − 1 ) and b y Φ x the map Φ x : S → S a 7→ Φ( a, x 1 , . . . , x n − 1 ) Giv en ψ , ψ ′ , ψ ′′ : S n − 1 → S , w e denote by ( ψ , ψ ′ , ψ ′′ ) the map ( ψ , ψ ′ , ψ ′′ ) : S n → S ( x 1 , x 2 , . . . , x n ) 7→    ψ ( x 2 , . . . , x n ) if x 1 = 0 ψ ′ ( x 2 , . . . , x n ) if x 1 = h ψ ′′ ( x 2 , . . . , x n ) if x 1 = 1 Note that for any Φ : S n → S we ha ve Φ = (Φ 0 , Φ h , Φ 1 ). In particular, when n = 1 we ma y write an y Φ : S → S as a triple of elemen ts of S ; see, for example, Lemma 3. Since Φ a ( x ) = Φ x ( a ) for any x ∈ S n − 1 and a ∈ S , w e hav e Φ x = (Φ 0 ( x ) , Φ h ( x ) , Φ 1 ( x )). When ψ : S n − 1 → S is a term function of S on ( n − 1) v ariables, then Ψ = ( ψ , ψ , ψ ) is the same term function view ed as a term function on n v ariables (with the first one absen t). W e call Ψ the n -ary v ersion of ψ . F or 1 ≤ i ≤ n ∈ N we denote b y Π n i the i - t h pro jection map Π n i : S n → S ( x 1 , . . . , x n ) 7→ x i F or an y binary relation r ⊆ S 2 , w e denote b y r − 1 the inv erse relation { ( y , x ) | ( x, y ) ∈ r } . Giv en a structured top olog ical space X , and Y ∈ I S c P + ( X ), for a n y op eration, partial op eration or relation λ of the structure X , w e denote b y λ Y the corresponding op eration, partial o p eration or relation on Y . Definition 1. [4] A p artial ly c om plemente d distributive lattic e is a typ e h 0 , 0 , 0 , 2 , 2 i al- gebr a A = h A ; 0 , h, 1 , ∨ , ∧i such that h A ; 0 , 1 , ∨ , ∧i i s a b ounde d distributive lattic e and h [ h, 1]; h, 1 , ∨ , ∧i is a c o m plemente d distributive lattic e, i.e., a Bo ole an algebr a wher e [ h, 1] = { a ∈ A | h ≤ a ≤ 1 } . W e also wish to note here that it is sho wn in [4] that In an y partially complemen ted distributiv e lattice one can define a unary ( bar) op eration in terms of the complemen t o p eration ′ in [ h, 1]: x = ( x ∨ h ) ′ It satisfies tw o useful iden t it ies can b e defined as in the follow ing: 4 D. J. CLOUSE AND F. GUZM ´ AN Lemma 1. [4] Given a p artial ly c omplemen te d distributive lattic e h A ; 0 , h, 1 , ∨ , ∧i , the b ar op e r ation satisfies L1) x ∨ x = 1 , and L2) x ∧ x = x ∧ 1 . These t w o prop erties c ha racterize partia lly complemen ted distributiv e lattices. F rom these results it is deriv ed that BS R is dually equiv a len t to the cat ego ry of P artially Stone Spaces, P S S , and w e denote this dual equiv alence a s B S R ⇆ P S S . In BS R ⇆ P S S , the functor from B S R to P S S ta kes any partially complemen ted dis- tributiv e la t t ice A and maps it to pt(Idl( A )), the set of prime filters of A . These prime filters are difficult to characterize for arbitrary pow ers of S . F or example, if the cardinality o f the indexing set is at least coun tably infinite, ev ery prime filter has an infinite descending chain and the exis tence of of elemen ts of finite supp ort is unclear. Hence our curren t understanding of B S R inheren t in this represen tation is not en tirely satisfactory . 1.2. Natural Dualities. In this pap er w e will follow v ery closely the ideas of [1 ] for constructing natural dualities. The basic idea is to impo se on t he carrier S of the semiring S , the discrete top olo g y together with op erat ions, partial op era t ions and relations to form a dual top ological structure f S as the generator of the dual category X . M ore sp ecifically , X = I S c P + ( f S ) is the categor y of isomorphic copies, top o lo gically closed substructures of non-empt y pro ducts of copies of f S . F ollowing this construc tion, w e will ha ve a dual adjunction h D , E , e, ǫ i b etw een the categories A a nd X with t he man y desirable prop erties [1, 1.5.3]. One further prop ert y w e desire is that for any A ∈ A , A is isomorphic to E D ( A ) = X ( A ( A, S ) , f S ). If the dual adjunction h D , E , e, ǫ i satisfies this prop erty , it is called a dual represen tation of A in X . In t his case w e sa y that S yields a (natur al) duality o n A. If it is also true that for an y X ∈ X , X is isomorphic to D E ( X ) = A (( X , f S ) , S ), w e say that f S yields a f ull duality on A . Th us f S yields a full duality on A if it yields a duality on A whic h is a dual equiv alence. Finally , if f S yields a full duality on A and it is injective in X , f S is said to yield a strong dualit y on A . W e will construct three dualit ies, eac h one coming fr o m a differen t top ological structure. In all three of them the algebra side of the dualit y will b e A = I S P ( S ). The first top olo gical structure, f S , yields an optimal (na tural) duality X = I S c P + ( f S ) ⇆ A . The second one, f S s , yields a strong duality X s = I S c P + ( f S s ) ⇆ A . The third o ne, f S os , yields an o ptimal strong dualit y X os = I S c P + ( f S os ) ⇆ A . In the first dualit y , la b eling the appropriate contra v a rian t functors D a nd E , A is isomorphic to E D ( A ) = X ( A ( A, S ) , f S ). The situation is displa y ed in the diagr am b elow. X ⇆ D ,E A = I S P ( S ) ֒ → H S P ( S ) = BS R ⇆ P S S Similar remarks hold fo r the other tw o dualities. Moreo v er, in Corollar y 6 w e sho w that X os ( f S n os , f S ) = X ( f S n , f S ). Recall that for an y X ∈ I S c P + ( f S ), E ( X ) = X ( X , f S ) 6 S X , and D E ( X ) is the set of prime filters of X ( X , f S ). As a result, we desire that the structure placed on f S will be sufficien t so that w e can c haracterize the prime filters of X ( X , f S ), and thereby trace their images and the THE D UAL GEOMETR Y OF BOOLEAN SEMIRINGS 5 image o f A in P S S . In [4] it is shown that for a finite partially complemen ted distributive lattice L , the partially Sto ne space [ X , Y ] corr esp o nding to L under the B S R ⇆ P S S dua lity , has X = L J and Y = { x ∈ X | x ≤ h } . The top ology of this space is T = { φ ( I ) | I ∈ Idl( L ) } where φ ( I ) = { p ∈ X | p ∩ I 6 = ∅} for any I ∈ Idl( L ). W e call this top ology t he Stone Space top ology . In particular, when L = X ( f S n , f S ), w e identify the prime filters of X ( f S n , f S ) with its jo in-irreducible elemen ts. A ma j or p o rtion of this pap er is dev oted to c ha r a cterizing the join-irreducible elemen ts of X ( f S n , f S ); we denote the set o f suc h eleme n ts b y X ( f S n , f S ) J . 2. Est ablishing the Duality and Some F acts About Morphisms First of all, w e note that t ( x, y , z ) = xy + y z + xz is a ternary near-unanimity term on S . This prop ert y will allo w us to use many of t he results in [1] in the construction of the dual represen t a tions that we seek. W e further no t e that this prop erty implies that B S R is a congruence distributiv e v ar iety and finite pro ducts of S are sk ew-free. Arbitra r y pro ducts of S are know n to the authors not to b e sk ew-free, but the countere xample is o utside of the scop e o f this pap er. W e now define the fir st structure f S that w e will show yields a duality on A . W e will also sho w that t his dualit y is optimal, in the sense that if a n y single relat io n w ere to b e deleted from f S , duality would b e lost. Definition 2. Define the fol lowin g subsets of S 2 : r 1 = S 2 − { (1 , 0) } ; r 2 = S 2 − { ( h, 0 ) , (1 , 0) } ; r 3 = S 2 − { (0 , 1) , ( h, 1) , (1 , 0) , ( 1 , h ) } . L et f S = h S ; { r 1 , r 2 , r 3 } , T i wher e T is the discr ete top olo gy. The following result can b e sho wn by straightforw ard coun ting and closure calculations. Lemma 2. The fol lowing is the lattic e of sub algebr as of S 2 : The L attic e o f Sub algeb r as of S 2 S 2 i i i i i i i i i i i i i i U U U U U U U U U U U U U r 1 U U U U U U U U U U U i i i i i i i i i i i i i i r − 1 1 U U U U U U U U U U U U i i i i i i i i i i r 2 U U U U U U U U U U U r 1 ∩ r − 1 1 i i i i i i i i U U U U U U r − 1 2 i i i i i i i i r 2 ∩ r − 1 1 K K K K K K K K K K K K ( r 2 ∩ r − 1 1 ) − 1 s s s s s s s s s s s s r 3 i i i i i i i i i U U U U U U U U U r 2 ∩ r − 1 1 ∩ r 3 K K K K K K K K K K K K K K r 2 ∩ r − 1 2 ( r 2 ∩ r − 1 1 ∩ r 3 ) − 1 s s s s s s s s s s s s s s ∆ 6 D. J. CLOUSE AND F. GUZM ´ AN Consider no w the lattice of subalgebras o f S 2 coupled with the following facts: • if a morphism preserv es a binary relation r then it also preserv es r − 1 , and • if a morphism preserv es t w o k-ary relations r and s then it a lso preserv es their in tersection r ∩ s . F rom this w e see that if we use only the set of binary relatio ns r 1 , r 2 , and r 3 from Defi- nition 2 as structure, then the structure f S will entail all subalgebras of S 2 . Hence by the M-Shift Dualit y Lemma [1, 2.4.2] and the NU Duality Theorem [1, 2.3.4] we get the follo wing prop osition: Prop osition 1. f S yields a duality on A and f S is inje ctive in X . The next lemma determines the elemen ts o f X ( f S , f S ) 6 S S . The pro of is a simple v erifica- tion of which functions f : S − → S preserv es the relatio ns of f S . Lemma 3. The fol lowing is the lattic e X ( f S , f S ) : The L attic e X ( f S , f S ) • • • • • ? ? ? ? ? ? ? ? ? ? •                     • ? ? ? ? ? ? ? ? ? ? (0,0,0) (0,h,h) (h,h,h) (h,h,1) (1,1,1) (0,h,1) (1,1,h) Before w e in v estigate the lattice X ( f S n , f S ) J , we w ant to sho w that the duality yielded by f S on A is optimal. F rom the Predualit y Theorem [1, 1.5 .2] and the First Dua lity Theorem [1, 2.2.2] we get the follo wing lemma. Lemma 4. If f S ′ is a a structur e that yields duality on A , then the finitary term functions on S must b e exactly the morphisms f r om finite p owers of f S ′ into f S ′ . Ther efor e, fo r any n ∈ N X ′ ( f S ′ n , f S ′ ) = X ( f S n , f S ) . Theorem 1. The structur e f S = h S ; { r 1 , r 2 , r 3 } , T i yields a n optimal natur al duality on A . Pr o of. Using Lemmas 3 and 4 we can sho w that if we eliminate r 2 or r 3 from f S to form f S ′ , dualit y will b e lost. The map (1 , h, 1) preserv es r 1 and r 2 , but not r 3 . The map (0 , 0 , h ) preserv es r 1 and r 3 , but no t r 2 . No w let f S ′ = h S ; { r 2 , r 3 } , T i , X ′ = I S c P + ( f S ′ ) and supp ose t ha t f S ′ yields a dualit y on A . By the Dualit y and Entailmen t Theorem [1, 2 .4 .3], { r 2 , r 3 } must entail r 1 . By the THE D UAL GEOMETR Y OF BOOLEAN SEMIRINGS 7 En tailmen t Lemma [1 , 2.4.4 ], for an y X ⊆ S 2 and α ∈ X ′ ( X , f S ′ ), α mus t preserv e r 1 . L et X = { ( h, 0) , (0 , 1) } and define α : X − → f S b y α ( ( h, 0)) = 1 and α ((0 , 1)) = 0. Then α preserv es r 2 and r 3 , but no t r 1 . Therefore, w e cannot retain duality without r 1 .  Up on viewing Lemma 3, the Preduality Theorem and the F irst Duality Theorem, noting that h is a nullary op eration of S and recalling that relations are define d p oint wise, it is easy to see that Lemma 5 holds. This lemma giv es some recursiv e info rmation ab out X ( f S n , f S ), just enough fo r our needs. Lemma 5. L et n > 1 . (1) Given Φ ∈ X ( f S n , f S ) and x ∈ S n − 1 , w e have Φ x ∈ X ( f S , f S ) . (2) Given Φ ∈ X ( f S n , f S ) , an d a ∈ S , we have Φ a ∈ X ( f S n − 1 , f S ) , an d (a) Φ 0 ∨ (Φ 1 ∧ h ) = Φ h , h e n c e Φ 0 ≤ Φ h . (b) Φ 0 ∧ h ≤ Φ 1 . Mor e over, (c) I f Φ ≤ h then Φ h = Φ 1 . (d) I f Φ  h then fo r every x ∈ S n − 1 either Φ 0 ( x ) ≤ Φ h ( x ) ≤ Φ 1 ( x ) o r Φ 0 ( x ) = Φ h ( x ) = 1 and Φ 1 ( x ) = h . (3) Given ψ ∈ X ( f S n − 1 , f S ) , let Ψ = ( ψ , ψ , ψ ) b e the n -ary version of ψ . We have that Ψ ∧ h ∧ Π 1 , Ψ ∧ h ∧ Π 1 , Ψ ∧ Π 1 , Ψ ∧ Π 1 ∈ X ( f S n , f S ) , an d (a) Ψ ∧ h ∧ Π 1 = (0 , ψ ∧ h, ψ ∧ h ) (b) Ψ ∧ h ∧ Π 1 = ( ψ ∧ h, ψ ∧ h, ψ ∧ h ) (c) Ψ ∧ Π 1 = (0 , ψ ∧ h, ψ ) (d) Ψ ∧ Π 1 = ( ψ , ψ , ψ ∧ h ) W e now hav e enough results to b egin our w ork characterizing X ( f S n , f S ) J . W e begin with a simple but useful Corollary to Lemma 3. Corollary 1. The fol lowing d iagr a m is the p oset X ( f S , f S ) J : The Poset X ( f S , f S ) J • • ? ? ? ? ? ? ? ? ? ? •           • (0,h,h) (h,h,h) (0,h,1) (1,1,h) Prop osition 2. L et n > 1 . (1) I f ψ ∈ X ( f S n − 1 , f S ) J and Ψ = ( ψ , ψ , ψ ) is the n -ary version of ψ , then Ψ ∧ Π 1 = (0 , ψ ∧ h, ψ ) and Ψ ∧ Π 1 = ( ψ , ψ , ψ ∧ h ) ar e in X ( f S n , f S ) J . 8 D. J. CLOUSE AND F. GUZM ´ AN (2) I f Φ ∈ X ( f S n , f S ) J with Φ ≤ h , then ther e is ψ ∈ X ( f S n − 1 , f S ) J with ψ ≤ h s uch that Φ = (0 , ψ , ψ ) = Ψ ∧ Π 1 or Φ = ( ψ , ψ , ψ ) = Ψ ∧ Π 1 , w h er e Ψ = ( ψ , ψ , ψ ) is the n -ary version of ψ . (3) I f Φ ∈ X ( f S n , f S ) J with Φ  h , then ther e is ψ ∈ X ( f S n − 1 , f S ) J with ψ  h s uch that Φ = (0 , ψ ∧ h, ψ ) = Ψ ∧ Π 1 or Φ = ( ψ , ψ , ψ ∧ h ) = Ψ ∧ Π 1 , wher e Ψ = ( ψ , ψ , ψ ) is the n -ary version of ψ . Pr o of. 1. Assume ψ ∈ X ( f S n − 1 , f S ) is join- irreducible. Consider first the case Φ = (0 , ψ ∧ h, ψ ), and supp ose Φ = Γ ∨ ∆ with Γ , ∆ ∈ X ( f S n , f S ) a nd Γ , ∆ < Φ. W e ha v e Γ 0 = ∆ 0 = 0 , Γ h ∨ ∆ h = ψ ∧ h, Γ 1 ∨ ∆ 1 = ψ . Without loss of generalit y , we hav e Γ 1 = ψ and therefore Γ h < ψ ∧ h , i.e. there is x ∈ S n − 1 suc h that 0 ≤ Γ h ( x ) < ψ ( x ) ∧ h ≤ ψ ( x ) , h So, w e m ust ha ve Γ h ( x ) = 0. By Lemma 5.1, Γ x ∈ X ( f S , f S ) and b y Lemma 3 Γ x = (0 , 0 , 0 ) whic h give s 0 = Γ 1 ( x ) = ψ ( x ), a contradiction. The case Φ = ( ψ , ψ , ψ ∧ h ) can b e handled in the same w a y , except that instead of going from Γ 1 to Γ h , one g o es from Γ 0 to Γ1. 2. Assume Φ ∈ X ( f S n , f S ) is join- irreducible, with Φ ≤ h . By L emma 5.2 Φ h = Φ 1 ∈ X ( f S n − 1 , f S ), call it ψ , and 0 ≤ Φ 0 ≤ ψ . Moreo v er, ψ ≤ h . By Lemma 5 .3 w e ha v e (Φ 0 , Φ 0 , Φ 0 ) and (0 , ψ , ψ ) are in X ( f S n , f S ), a nd (Φ 0 , Φ 0 , Φ 0 ) ∨ (0 , ψ , ψ ) = (Φ 0 , ψ , ψ ) = Φ Therefore, either Φ = (0 , ψ , ψ ) or Φ = (Φ 0 , Φ 0 , Φ 0 ) = ( ψ , ψ , ψ ). The join irreducibilit y of Φ and Lemma 5.3 force ψ to b e jo in irreducible. 3. Assume Φ ∈ X ( f S n , f S ) is join- irreducible, with Φ  h . Using Lemma 5.2 ,3 w e get that (Φ 0 , Φ 0 , Φ 0 ∧ h ) and (0 , Φ 1 ∧ h, Φ 1 ) a re in X ( f S n , f S ). Using Lemma 5 .1 w e get (Φ 0 , Φ 0 , Φ 0 ∧ h ) ∨ (0 , Φ 1 ∧ h, Φ 1 ) = (Φ 0 , Φ 0 ∨ (Φ 1 ∧ h ) , (Φ 0 ∧ h ) ∨ Φ 1 ) = ( Φ 0 , Φ h , Φ 1 ) = Φ Therefore, either Φ = (Φ 0 , Φ 0 , Φ 0 ∧ h ) or Φ = (0 , Φ 1 ∧ h, Φ 1 ). In the first case, take ψ = Φ 0 , in the second case, take ψ = Φ 1 . Once aga in, the join irreducibilit y of Φ and Lemma 5.3, force ψ to b e join irreducible. The fa ct tha t Φ  h yields ψ  h .  F rom the previous pro p osition and Corolla r y 1, Corollary 2 follo ws b y induction. Corollary 2. If Φ ∈ X ( f S n , f S ) J then Φ ∧ h ∈ X ( f S n , f S ) J . Com bining the differen t parts of Prop osition 2 we get the follow ing theorem: Theorem 2. L et Φ ∈ X ( f S n , f S ) . (1) Φ is join irr e ducible if and only if Φ = (0 , ψ ∧ h, ψ ) or Φ = ( ψ , ψ , ψ ∧ h ) for some join irr e ducible ψ ∈ X ( f S n − 1 , f S ) , i.e. if and only i f Φ = Ψ ∧ Π 1 or Φ = Ψ ∧ Π 1 , wher e Ψ = ( ψ , ψ , ψ ) is the n -ary version of ψ . THE D UAL GEOMETR Y OF BOOLEAN SEMIRINGS 9 (2) When Φ ≤ h , Φ is join irr e ducible if and only if Φ = (0 , ψ , ψ ) or Φ = ( ψ , ψ , ψ ) f o r some join irr e ducible ψ ∈ X ( f S n − 1 , f S ) with ψ ≤ h . (3) When Φ  h , Φ is join irr e ducib l e if and only if Φ = (0 , ψ ∧ h, ψ ) or Φ = ( ψ , ψ , ψ ∧ h ) for some join irr e ducible ψ ∈ X ( f S n − 1 , f S ) with ψ  h . 3. The Poset and Pol ynomial Chara cteriza tion of the Join-irreducible Morphisms With Theorem 2 at o ur disp osal, w e can now pro ceed to o bt a in the p oset structure of X ( f S n , f S ) J . W e sho w that this p oset structure completely determines X ( f S n , f S ) J as a partially Stone Spa ce. Along the w ay we also obtain a p olynomial represen tation. W e are dealing with X ( f S n , f S ) J with its p oint wise partial order inherited fro m X ( f S n , f S ). This is the same as the o p en set partial order obtained from the Stone top o logy . When w e discuss the prop erties of the elemen ts of X ( f S n , f S ) J co ve ring, being cov ered or b eing incomparable, w e will b e considering t hem in the p oset X ( f S n , f S ) J , not in the partially complemen ted distributive lat tice X ( f S n , f S ). 3.1. The Base o f the Hairy Cube. Prop osition 3. The set Y n = { Φ ∈ X ( f S n , f S ) J | Φ ≤ h } is p oset isomorphic to 2 n . Pr o of. F o r n = 1 see figure of X ( f S , f S ) J giv en in Corollary 1. Assume no w that there exists a po set isomorphism η n − 1 : Y n − 1 − → 2 n − 1 . By Theorem 2.2 if Φ ∈ Y n , then w e kno w that Φ = (0 , ψ , ψ ) or Φ = ( ψ , ψ , ψ ) for some ψ ∈ X ( f S n − 1 , f S ) J . Define the following map: η n : Y n − → 2 n (0 , ψ , ψ ) 7→ (0 , η n − 1 ( ψ )) ( ψ , ψ , ψ ) 7→ (1 , η n − 1 ( ψ )) The fact tha t η n is bijectiv e follo ws immediately f r o m the f a ct that η n − 1 is. That η n and its in ve rse are order prese rving is clear from the definition a nd the fact that η n − 1 and its in v erse are order preserving.  3.2. The Cov ers. Prop osition 4. L et n ≥ 1 and Φ , Γ ∈ X ( f S n , f S ) J . (1) I f Φ , Γ  h and Φ 6 = Γ then Φ , Γ ar e in c om p ar able. (2) I f Φ ≤ h then ther e is a unique e Φ ∈ X ( f S n , f S ) J with e Φ  h , such that e Φ c ove rs Φ . Mor e over Φ = e Φ ∧ h . (3) I f Φ  h then it only c overs Φ ∧ h in X ( f S n , f S ) J . Pr o of. (1). T he case n = 1 is taken care of in Corollary 1. F or n > 1, Theorem 2.3 tells us that either Φ = (0 , φ ∧ h, φ ) or Φ = ( φ, φ, φ ∧ h ), and similarly Γ = (0 , γ ∧ h, γ ) or Γ = ( γ , γ , γ ∧ h ) for some φ, γ ∈ X ( f S n − 1 , f S ) J with φ, γ  h . Clearly (0 , φ ∧ h, φ ) and ( γ , γ , γ ∧ h ) are incomparable since φ  h . By induction (0 , φ ∧ h, φ ) and (0 , γ ∧ h, γ ) are 10 D. J. CLOUSE AND F. GUZM ´ AN incomparable; similarly , ( φ, φ, φ ∧ h ) and ( γ , γ , γ ∧ h ) are incomparable. (2). Once again, the case n = 1 is tak en care of in Corollary 1. F or n > 1, Theorem 2.2 tells us t ha t either Φ = (0 , ψ , ψ ) o r Φ = ( ψ , ψ , ψ ), for some ψ ∈ X ( f S n − 1 , f S ) J with ψ ≤ h . By induction, there is a unique e ψ ∈ X ( f S n − 1 , f S ) J with e ψ  h , suc h that e ψ cov ers ψ . Moreo ver, ψ = e ψ ∧ h . By Theorem 2.3, w e hav e that (0 , ψ , e ψ ) and ( e ψ , e ψ , ψ ) are in X ( f S n , f S ) J . Clearly , (0 , ψ , e ψ ) co v ers (0 , ψ , ψ ). That ( e ψ , e ψ , ψ ) co v ers ( ψ , ψ , ψ ) follows from the fact that neither ( ψ , e ψ , ψ ) nor ( e ψ , ψ , ψ ) are morphisms b y Lemma 5 .2 .d. This sho ws the existence part, b y taking e Φ = (0 , ψ , e ψ ) when Φ = (0 , ψ , ψ ) and e Φ = ( e ψ , e ψ , ψ ) when Φ = ( ψ , ψ , ψ ). In either case note that Φ = e Φ ∧ h . F o r uniquene ss, assume that Γ ∈ X ( f S n , f S ) J , with Γ  h , cov ers Φ. By Theorem 2.3, w e ha v e either Γ = (0 , γ ∧ h, γ ) or Γ = ( γ , γ , γ ∧ h ) fo r some γ ∈ X ( f S n − 1 , f S ) J with ψ  h . In the first case, it follo ws that Φ = (0 , ψ , ψ ) and γ co v ers ψ . By uniqueness of e ψ , w e m ust ha ve γ = e ψ , and Γ = (0 , ψ , e ψ ). In the second case, since (0 , ψ , ψ ) ≤ Γ implies ( ψ , ψ , ψ ) ≤ Γ we mus t hav e Φ = ( ψ , ψ , ψ ) and γ co ve rs ψ . Ag a in, b y uniqueness o f e ψ w e get γ = e ψ , and Γ = ( e ψ , e ψ , ψ ) . (3). By part 1, Φ can only co v er elemen ts of X ( f S n , f S ) J ∩ ↓ h , and part 2 yields the uniqueness. By Corollary 2, Φ ∧ h is in X ( f S n , f S ) J , and by part 2 it has a unique co v er e Φ  h . Therefore, w e mus t ha ve e Φ ≤ Φ. No w part 1 f o rces e Φ = Φ, so Φ co v ers Φ ∧ h .  W e can no w combine the results of Prop ositions 3 and 4 to describe the po set structure of X ( f S n , f S ) J . Theorem 3. The p ose t X ( f S n , f S ) J c on sists of two p arts: the “ base ” Y n which is an n -cub e and the “ hairs ” X ( f S n , f S ) J \ Y n , w h ich ar e p airwise inc omp ar able. Each element of the b ase is c ove r e d by a unique hair. Each hair c overs a unique b ase e lement. In Theorem 4 w e sho w that these pa r tial order prop erties of X ( f S n , f S ) J completely deter- mine it as a partially Stone space. Eve n though w e will only need the fact that X ( f S n , f S ) J is a p o set, we can actually see that it is a meet-semilattice. Corollary 3. X ( f S n , f S ) J is a me et-se m ilattic e. F or any Φ , Γ ∈ X ( f S n , f S ) J , Φ ∧ Γ = (Φ ∧ h ) ∧ (Γ ∧ h ) is a n element of the b ase of the hairy cub e. 3.3. The Pa rtially Stone Space Corresp onding to the Hairy Cub e. There is a w ell-know n duality b et w een T 0 Alexandro v spaces a nd partial orders. F or details see [5]. F or us, it will b e more con v enien t to use the opp osite par t ia l order and the opp osite (in terchange op en and closed) top olo gy . Here are the details. Giv en a p oset P , the set Λ = {↓ p | p ∈ P } forms a basis for a top ology o n P ; w e refer to it as the “ do wnset top ology ” (it is the oppo site of the “ Alexandrov top ology ”). Giv en a T 0 -space X , the follo wing defines a partial order on X : for x, y ∈ X , set x ≤ y if and only if ev ery op en subset of X that con tains y m ust also con tain x . W e refer to this as the “ op en set partial or der ” (it is the opp osite of the “ sp ecialization order ”). THE D UAL GEOMETR Y OF BOOLEAN SEMIRINGS 11 Just like in the Alexandro v duality w e get the following lemma: Lemma 6. (1) Supp ose that X is a T 0 A lexandr ov sp ac e, p artial ly or der e d with the op en set p ar- tial or der. The n the d ownset top olo gy in d uc e d by the p artial or der, and the original top olo gy on X ar e the same . (2) Supp ose that P is a p artial ly or der e d set, which is given the down set top olo gy. Then the op en set p artial or der and the original p artial or der ar e the same . Mor e over, a function b etwe en p osets i s or der pr eserving i f and only i f it is c o n tinuous as a map b etwe en T 0 A lexandr ov sp ac es. Note that any P a r tially Stone Space is T 0 , and ev ery finite space is Alexandro v. Recall t ha t for a finite partially complemen ted distributiv e la t t ice L , the partially Stone space [ X, Y ] corresp onding to L under the B S R ⇆ P S S dualit y has X = L J and Y = { x ∈ X | x ≤ h } . F rom the definition of the Stone top ology on X and the previous lemma w e get the follo wing corollary: Corollary 4. L et L b e a fini te p artial ly c omplem ente d distributive lattic e. L et L J b e the set of join-irr e ducible elements of L , with the p artial or der in herite d fr om L . L et [ X , Y ] b e the Partial ly Stone Sp ac e c orr esp onding to L under the duality B S R ⇆ P S S . (1) The downset top olo gy on L J and the Stone top olo gy on X ar e the same. (2) The p artial or der in L J and the op e n set p artial or der in X ar e the sa m e. W e no w c haracterize those partially Stone spaces that corresp ond to the Hairy Cub e. Theorem 4. L et [ X , Y ] b e a Partial ly Stone Sp ac e p artial ly or d er e d with the op en set p artial or de r such that the fol lowing hold: (1) Y is p oset isomorphic to 2 n ; (2) The elements of X − Y ar e p airwise inc om p ar able . (3) Every y ∈ Y has a unique c over x ∈ X − Y ; (4) Every x ∈ X − Y c overs only one y ∈ Y ; Then [ X , Y ] , and the Partial ly Stone Sp ac e c orr esp ondin g to X ( f S n , f S ) under the B S R ⇆ P S S duality, ar e Partial ly Stone Sp ac e home om orphic. Pr o of. It is clear fr om Theorem 3 that X and X ( f S n , f S ) J are p oset isomorphic, under an isomorphism η : X → X ( f S n , f S ) J that maps Y onto Y n . By Lemma 6, η is a homeomorphism of top olog ical spaces. The only a dditional fact neede d is that η | Y is coheren t. This fo llo ws from the fact that Y a nd Y n are finite.  3.4. Poly nomials. Recall from L emma 4 that X ( f S n , f S ) J is just t he set of n - a ry term functions on the algebra S whic h are join-irreducible. The follo wing result sho ws that these can be o bta ined using only t he ∧ op eratio n and unary op eration in tro duced in Lemma 1 . When writing term functions for a n algebra, pro jection maps are usually called “ v ariables ”, and denoted b y low er case letters. Giv en a v ariable p i , with 1 ≤ i ≤ n , and ǫ i ∈ { 0 , 1 } w e define 12 D. J. CLOUSE AND F. GUZM ´ AN p ǫ i i =  p i if ǫ i = 0 p i if ǫ i = 1 Theorem 5. L et Φ ∈ X ( f S n , f S ) J . (1) When Φ  h it c an b e uniquely written as a p olynomia l of the form Φ = n ^ i =1 p ǫ i i (2) When Φ ≤ h it c an b e uniquely written as a p olynomia l of the form Φ = n ^ i =1 p ǫ i i ! ∧ h Pr o of. When n = 1 the statemen t follows fro m Corollary 1. If n > 1 recall the map η in Prop osition 3. W e will sho w that taking ǫ = η (Φ ∧ h ), prov es existenc e. By Theorem 2.1 there is φ ∈ X ( f S n − 1 , f S ) J , and we ha v e tw o cases to consider. In the first case, Φ = ( 0 , φ ∧ h, φ ), η (Φ ∧ h ) = η (0 , φ ∧ h, φ ∧ h ) = (0 , η ( φ ∧ h )) , and Φ = (0 , h, 1) ∧ ( φ, φ, φ ) = p 1 ∧ φ. In the second case, Φ = ( φ, φ, φ ∧ h ), η (Φ ∧ h ) = η ( φ ∧ h, φ ∧ h, φ ∧ h ) = (1 , η ( φ ∧ h )) , and Φ = (1 , 1 , h ) ∧ ( φ , φ, φ ) = p 1 ∧ φ. In either case, Φ ≤ h if and only if φ ≤ h . By induction w e hav e: when Φ  h, Φ = p ǫ 1 1 ∧ φ = p ǫ 1 1 ∧ n ^ i =2 p η ( φ ∧ h ) i = n ^ i =1 p η (Φ ∧ h ) i ; when Φ ≤ h, Φ = p ǫ 1 1 ∧ φ = p ǫ 1 1 ∧ n ^ i =2 p η ( φ ∧ h ) i ∧ h = n ^ i =1 p η (Φ ∧ h ) i ∧ h. Uniqueness follow s from t he ab o ve, the bijectivit y of η in Prop osition 3, and Prop osition 4.  THE D UAL GEOMETR Y OF BOOLEAN SEMIRINGS 13 The “Hairy Cub e” for n = 3 p 1 ∧ p 2 ∧ p 3          p 1 ∧ p 2 ∧ p 3 = = = = = = = = = p 1 ∧ p 2 ∧ p 3 ∧ h j j j j j j j j j j j j j j j j j j j T T T T T T T T T T T T T T T T T T T p 1 ∧ p 2 ∧ p 3          p 1 ∧ p 2 ∧ p 3 ∧ h N N N N N N N N N N N N N N N N N N N N N N N N p 1 ∧ p 2 ∧ p 3          p 1 ∧ p 2 ∧ p 3 ∧ h p p p p p p p p p p p p p p p p p p p p p p p p p 1 ∧ p 2 ∧ p 3 = = = = = = = = = p 1 ∧ p 2 ∧ p 3 = = = = = = = = = p 1 ∧ p 2 ∧ p 3 ∧ h j j j j j j j j j j j j j j j j j j j T T T T T T T T T T T T T T T T T T T p 1 ∧ p 2 ∧ p 3          p 1 ∧ p 2 ∧ p 3 ∧ h p 1 ∧ p 2 ∧ p 3 ∧ h N N N N N N N N N N N N N N N N N N N N N N N N p 1 ∧ p 2 ∧ p 3 ∧ h p p p p p p p p p p p p p p p p p p p p p p p p p 1 ∧ p 2 ∧ p 3 = = = = = = = = = p 1 ∧ p 2 ∧ p 3 ∧ h 4. Neither Full nor Strong, A Small Str ong Stru cture F rom the optimal duality established in Theorem 1, w e ha v e obtained geometric and p olynomial c haracterizations of X ( f S n , f S ) J . How ev er, this dualit y is neither full nor strong, as t he following result sho ws. Theorem 6. The duality yielde d by f S is neither ful l nor str ong. Pr o of. First w e will sho w that the duality yielded by f S on A is not strong. Consider the Second Strong Duality Theorem [1, 3.2 .9 ]. Since f S is a total structure, if it w ere to yield a strong dualit y on A , it would satisfy the Finite T erm Closure Condition: FTC: F or any n ∈ N , X 6 f S n and y ∈ 6 f S n \ X , ∃ term f unctions σ , τ : f S n → f S on S (that is morphisms) that a g ree on X but not at y . Consider X = { 0 , 1 } 6 f S a nd y = h . Up on viewing the diag ram in L emma 3 we see that ( f S , f S ) | X = { (0 , 0) , (0 , h ) , (0 , 1) , ( h, h ) , ( h, 1) , (1 , h ) , (1 , 1) } and hence no pair σ , τ ∈ X ( f S , f S ) can a gree o n { 0 , 1 } (and differ a t h ). T o sho w that the dualit y yielded on A by f S is not ev en a full dualit y , consider the First Strong Duality Theorem [1, 3 .2 .4], whic h states that f S yields a strong dualit y on A if and only if f S yields a full dualit y on A and f S is injectiv e in X . Since f S is injectiv e in X , b y Theorem 1, it follows that f S do es no t yield a full dualit y o n A .  14 D. J. CLOUSE AND F. GUZM ´ AN The Dual Adjunction Theorem [1, 1.5.3] establishes em b eddings of X into D E ( X ) for ev ery X ∈ X . The failure of the duality to b e full, a nd therefore strong, is in the failure of X to b e isomorphic to D E ( X ) = A ( X ( X , f S ) , S ) for ev ery X ∈ X . In o rder to obtain a strong dualit y , w e need to add structure to f S that will eliminate ob jec ts o f X that are a closed substructure of a p ow er of f S a nd a r e not term/hom-closed. The NU Strong Duality Theorem, [1, 3.3 .8 ] uses t he irreducibilit y index of S , defined b elo w, to giv e an exact recip e for constructing a generating structure f S nu that will yield a strong dualit y on A . One simply needs to add the all the algebraic n-ary op erations and partial op erations f or 1 ≤ n ≤ Irr( S ) to the structure on f S , to obtain f S nu . W e refer to this metho d w e refer to as the “Near Brute F orce” metho d. One can then apply the methods of [1], particularly the M-Shift Strong Duality Lemma [1, 3.2.3], to w o rk to w ards obtaining an optimal strong duality . Definition 3. L et A b e a finite algebr a. The irreducibilit y of A is the le ast n ∈ N such that the zer o c ong ruenc e o n A is a me et of n me et-irr e d ucib le c o n gruenc e s . T h e irreducibilit y index of A denote d Irr( A ) , is the maximum of the irr e ducibilities of sub algebr as of A . Lemma 7. Irr( S ) is 2. Pr o of. F ro m the lattice of subalgebras of S 2 in Lemma 2 it is easy to c hec k that Con( S ) = { ∆ , r 3 , r 2 ∩ r 2 − 1 , S 2 } , and it is isomorphic to the 2 dimensional cub e. S has no subalgebra other than itself.  Recall that a n-ary op eratio n g on S is algebraic o v er S if g ∈ A ( S n , S ); a n-ar y partial op eration h on S is algebraic o v er S if h ∈ A ( X , S ) fo r some X ≤ S n . F urthermore, these conditions are equiv alent to sa ying that the corresp onding graphs form subalgebras of S n +1 . As can b e seen through the pro ofs of Lemma 8 and Prop osition 6, the n umber of algebraic binary partial o p erations on S is to o large, for the br ute force metho d to yield a useful structure. W e wan t a structure f S s that yields a strong duality on A that is a s simple as p ossible. On the other hand, the only a lgebraic binar y tota l op eratio ns on S ar e the pro jections. Prop osition 5. L et n ∈ N and Λ : S n − → S . T hen Λ ∈ A ( S n , S ) if and only if it is a pr oj e ction map, Λ = Π n i . for some 1 ≤ i ≤ n . Pr o of. Λ − 1 (1) (resp.Λ − 1 (0)) is a prime filter (resp. ideal) of S n , hence there is x (resp. y ) join-irreducible ( resp. meet-irreducible) suc h that Λ − 1 (1) = ↑ x (resp. Λ − 1 (0) = ↓ y ). As x is j o in-irreducible, it follow s that Π j ( x ) 6 = 0 for at most one j , Since h is a constan t, Λ( x ∧ h ) = Λ( x ) ∧ h = 1 ∧ h = h , and w e cannot hav e x ∧ h = x and therefore Π j ( x ) = 1. Similarly , Π i ( y ) 6 = 1 fo r at most one i , and Π i ( y ) = 0. If i 6 = j , then x ≤ y and Λ( x ) ≤ Λ ( y ) yielding a contradiction, hence i = j . Now, let z ∈ X and consider the following cases: (1) Π i ( z ) = 0. In this case z ∈ ↓ y and hence Φ( z ) = 0. (2) Π i ( z ) = h . In this case z / ∈ ↑ x and z / ∈ ↓ y , hence Φ( z ) = h . (3) Π i ( z ) = 1. In this case z ∈ ↑ x and hence Φ( z ) = 1. So we see tha t Φ = Π i . The con v erse ho lds as pro j ections a re homomorphisms.  THE D UAL GEOMETR Y OF BOOLEAN SEMIRINGS 15 As a result of Prop o sition 5 the only total op era t io ns in f S nu are Π 1 1 , Π 2 1 and Π 2 2 . There are no prop er algebraic unary partia l op erations, since the only subalgebra o f S is S itself, but the set of algebraic binary pa rtial op erations is to o la rge to b e useful. In order to get a manageable structure f S s that yields a strong duality on A , w e will reduce the set of algebraic binary partia l op erations using the M-Shift Strong Dualit y Lemma. Any structure that strongly en t a ils f S nu will a lso yield a strong dualit y on A . T o get suc h structure f S s , w e may delete from f S nu those pa rtial o p erations that are restrictions of other total or partial op erations left in the structure. In particular, w e ma y delete any partial op eration whic h is a restriction of a pro jection. Unlik e total algebraic op erations, whic h b y Prop osition 5 ha v e t o b e pro jections, for the algebraic bina r y partial op erations there is a little more ro om as the follo wing lemma sho ws. Lemma 8. L et A ≤ S 2 . F or i = 1 , 2 , let L i = Π 2 i r estricte d to ↓ h , U i = Π 2 i r estricte d to ↑ h . L et Λ ∈ A ( A, S ) . (1) I f A ∩ ↑ h c ontains ( h, 1) o r (1 , h ) then ↑ h ⊆ A and Λ r estricte d to ↑ h e quals U 1 or U 2 . (2) Λ r estricte d to A ∩ ↓ h e quals L 1 or L 2 . Pr o of. If A ∩ ↑ h contains either ( h, 1) or (1 , h ), applying the complemen t op era t ion from Lemma 1 w e get the other and hence ↑ h ⊆ A . The r est follo ws from the f a cts that Λ is order preserving a nd eve ry elemen t of ∆ S is a constan t .  It is easy to chec k that the binary partia l op eratio n λ 1 with domain r 1 ha ving graph Γ( λ 1 ) = { (0 , 0 , 0) , ( h, h, h ) , (1 , 1 , 1) , ( 0 , h, h ) , (0 , 1 , h ) , ( h, 0 , 0) , ( h, 1 , h ) , (1 , h, 1) } , is in fact algebraic. It com bines U 1 and L 2 . Similarly , the com bination of U 2 and L 1 yields the a lg ebraic binary partia l op erat io n λ 2 with domain r − 1 1 ha ving graph Γ( λ 2 ) = { (0 , 0 , 0) , ( h, h, h ) , (1 , 1 , 1) , ( 0 , h, 0) , ( h, 0 , h ) , (1 , 0 , h ) , ( h, 1 , 1) , (1 , h, h ) } . These tw o partial op erations are examples o f algebraic binary partial op erations whic h are not restrictions o f pro jections. Lemma 8 places constrain ts o n suc h op erations to b e com binations of L i and U k with i 6 = k . In a sense, λ 1 and λ 2 are the only suc h examples, as it is more clearly stated in the pro of of the next pro p osition. Prop osition 6. L et n ∈ N and Π n i : f S n − → f S den o te the i -th pr oje ction map for any i ∈ N with 1 ≤ i ≤ n and T denote the discr ete top olo gy, then the structur e f S s = < S ; { r 1 , r 2 , r 3 } , { Π 2 1 , Π 2 2 } , { λ 1 , λ 2 } , T > yields a str ong duality on A . Pr o of. As Π 1 1 is the iden tity map on S , it has no effect on an y top olo gical category generated b y a structure with S as its carrier set. Therefore, w e do not need to include it in the list of total op erations. By the M-Shift Strong Duality Lemma w e only need to sho w that any algebraic binary partial opera t ion in the structure f S nu is a restriction of either a pr o jection or one of λ 1 , λ 2 . F rom Lemma 8, w e see that the only homomor phisms whic h are no t restrictions of pro jections m ust consist of either a combination of L 1 and U 2 or a com bination of L 2 and 16 D. J. CLOUSE AND F. GUZM ´ AN U 1 . Let us first conside r the subalgebra A = r 1 ≤ S 2 whic h contains the elemen t (0 , 1). Let λ : A → S b e a homomorphism whic h is no t a restriction of a pro jection. If w e ha d λ (0 , 1) = 0 this would force λ (0 , h ) = λ ( h, h ) ∧ λ (0 , 1 ) = h ∧ 0 = 0 and b y Lemma 8, λ restricted to A ∩ ↓ h w ould hav e to equal L 1 . Moreov er, w e would hav e λ ( h, 1) = λ ( h, h ) ∨ λ (0 , 1) = h ∨ 0 = h , and λ restricted to A ∩ ↑ h w ould ha ve to equal U 1 , making λ a restriction o f Π 2 1 . Similarly , if w e had λ (0 , 1) = 0 this w ould force λ to b e a restriction of Π 2 2 . Therefore, w e mus t ha ve λ (0 , 1) = h . As ab ov e, this forces λ ( 0 , h ) = λ ( h, h ) ∧ λ (0 , 1) = h ∧ h = h , and λ ( h, 1) = λ ( h, h ) ∨ λ (0 , 1) = h ∨ h = h , making λ a com bination of L 2 and U 1 , i.e. λ 1 . So, λ 1 is the o nly partial algebraic op eration with doma in r 1 whic h is not a restriction of a pro jection. A similar argument sho ws that λ 2 is t he only partial algebraic op eration with domain r − 1 1 , whic h is no t a restriction of a pro j ection. Note that the argumen t ab ov e do es not mak e use of the fa ct that ( h, 0 ) is in r 1 . Therefore, it also sho ws that the o nly partial algebraic op eration with domain r 2 whic h is not a r estriction of a pro j ection, mus t b e the restriction of λ 1 . Similarly , the only partial alg ebraic op eration with domain r − 1 2 whic h is not a restriction of a pro jection, mus t b e the r estriction of λ 2 . As sho wn in Lemma 2 an y other subalgebra A ≤ S 2 m ust b e a subalgebra of r 1 ∩ r − 1 1 . Hence, b y Lemma 8 , an y part ia l algebraic op eratio n with domain A , whic h is not a restriction of a pro jection, m ust b e a restriction of either λ 1 or λ 2 .  As an in termediate step tow a r ds an optima l strong dualit y , w e will sho w that w e can eliminate the total op erations and λ 2 from f S s and still ac hiev e a strong duality . First w e need the fo llo wing definitions: Definition 4. [1] A set of p artial op er ations on a set X is c al le d an enric hed partial clone if it is close d under c omp osition and c on tains { Π n i k 1 ≤ i ≤ n, n ∈ N } . T he enriche d p artial clone gener ate d by a set o f p artial op er ations H on a se t X is the smal lest enriche d p artial clone c on taining H and is denote d [ H ] . Given a structur e d top olo gic al sp a c e X = h X ; R, G, H , T i , the enriche d p artial clone of X is the sma l lest enriche d p artial cl o ne on X c o n taining G ∪ H and is deno te d [ G ∪ H ] . F urthermor e , let P den ote the set of al l finitary algebr aic, p artial or total op er ations on S . T h en P is an enriche d p artial clone on S and is r eferr e d to as the enric hed partial hom- clone of S . Definition 5. [1] L e t P ⊆ P and k ∈ P . We say that P hom-en tails k if, f or al l non-emp ty sets Ω , e a c h top olo gic al ly close d subset of S Ω which is close d under the p artial op er ations in P is also close d under k . De fine P = { k ∈ P | P hom -entails k } . Then P 7− → P is a closur e op e r ator on P a n d w e r efer to P as the hom-entailment closur e of P . Corollary 5. The structur e f S ′ s = h S ; { r 1 , r 2 , r 3 } , { λ 1 } , T i yields a str on g duality on A . Pr o of. Let G = { Π 2 1 , Π 2 2 } a nd H = { λ 1 , λ 2 } , then b y Lemma 6 and the Brute F orce Strong Dualit y Theorem [1, 3.2.2], G ∪ H = P , i.e. G ∪ H hom-en tails ev ery finita ry alg ebraic partial or to tal o p eration on S . No w consider Definition 4, and not e that [ H ] = [ G ∪ H ]. F urther note that λ 1 (Π 2 2 , Π 2 1 ) = λ 2 , and w e see that [ λ 1 ] = [ H ]. Let k : A − → S b e any elemen t of P and consider what w e refer to as the T est Algebra Lemma for maps [1, 9.4.1]. P arts (i.a) and (i.c) sho w { λ 1 } = P and hence the structure f S ′ s yields a strong dualit y on A .  THE D UAL GEOMETR Y OF BOOLEAN SEMIRINGS 17 No w we wis h to show that the partial op eration λ 1 en tails r 1 and r 3 , for that we need the follo wing L emma: Lemma 9. L et f S ′ = h S ; { λ 1 } , T i with λ 1 as the only p artial op e r ation. (1) L et X ∈ I S c P + ( f S ′ ) and g : X − → f S ′ b e a morphism. Then g pr eserve s r 1 and r 3 , and ther efor e { λ 1 } e n tails r 1 and r 3 . (2) Now, let Φ : f S ′ − → f S ′ b e gi v e n by Φ = ( h, 0 , 0) . T hen Φ p r eserv e s λ 1 but not r 2 , and ther efor e f S ′ do e s no t yield a duality on A . Pr o of. (1) Let ( Y , Z ) ∈ r X 1 = do m ( λ X 1 ). As g preserv es λ 1 , ( g ( Y ) , g ( Z )) ∈ dom( λ f S ′ 1 ) = r f S ′ 1 and g preserv es r 1 . Now let ( Y , Z ) ∈ r X 3 ⊆ dom( λ X 1 ), and consider ( Y , Z , λ 1 ( Y , Z )). F or an y i , ( Y i , Z i ) ∈ r f S ′ 3 hence Z i = λ 1 ( Y i , Z i ) and t herefore λ 1 ( Y , Z ) = Z ∈ X . Moreov er, g ( Z ) = g ( λ 1 ( Y , Z )) whic h implies that ( g ( Y ) , g ( Z )) ∈ r f S ′ 3 . The fact tha t { λ 1 } en tails r 1 and r 3 follo ws from the En tailmen t Lemma. (2) The fact that Φ preserv es λ 1 is easily seen b y insp ection of Φ( λ 1 ( x, y )) and λ 1 (Φ( x ) , Φ( y )) for eac h ( x, y ) ∈ dom( λ 1 ). Now note that (Φ(0) , Φ( h )) = ( h, 0) and henc e Φ fails to preserv e r 2 . F rom Lemmas 3 and 4 w e see that f S ′ do es not yield duality on A .  W e no w hav e enough results to prov e the ma jo r Theorem of this section. Theorem 7. L et T deno te the di s c r ete top olo gy and f S os = h S ; { r 2 } , { λ 1 } , T i , then f S os yields an optimal str on g duality on A . Pr o of. The fa ct that f S os yields a strong dualit y on A follows from Theorem 1, Corollary 5, Lemma 9.1 and the M-Shift Strong Dualit y Lemma. The fact tha t it is optimal follo ws from Lemma 9 .2 and Theorem 1.  Com bining this theorem with Lemma 4 we get the final result. Corollary 6. L et X os = I S c P + ( f S os ) , then for any n ∈ N , X os ( f S n os , f S ) J is the n- d imensional Hairy Cub e. Reference s References [1] David M. Clark and Bria n A. Dav ey, Natur al dualities for t he working algebr aist , Cambridge Stud- ies in Adv a nced Mathematics, v ol. 57 , Ca mb ridge Universit y Pres s, Ca mbridge, 1 998. MR 1663208 (2000d: 18 001) [2] Daniel J. Clouse , A Dual R epr esentation of Bo ole an Semirings in a Cate gory of St ructur e d T op olo gic al Sp ac es , Ph.D. dis sertation, Bingha m ton Univ er sit y, 2002 . [3] Samuel Eilenber g, Automata, languages, and machines. V ol. A , Academic Press [A s ubsidiary of Har c ourt Brace Jov anovich, Publishers ], New Y ork, 1974. Pure and Applied Ma thema tics, V o l. 58. MR 0530 382 (58 #2660 4a) [4] F erna ndo Guzm´ an, Th e variety of Bo ole an semirings , J. Pur e Appl. Algebra 78 (199 2), no. 3, 253– 2 70. MR 116327 8 (93d: 0800 7) [5] Peter T. Johnsto ne, Stone sp ac es , Ca m bridge Studies in Adv a nced Mathematics, vol. 3, Cambridge Univ ersity Press, Cambridge, 19 82. MR 698074 (85f: 5 4002) 18 D. J. CLOUSE AND F. GUZM ´ AN [6] W erner Kuich and Arto Sa lo maa, Semirings, automata, languages , EA TCS Monographs o n Theo retical Computer Science, vol. 5, Springer -V er lag, Berlin, 1986. MR 817983 (87h: 680 93) [7] M. H. Stone, Applic ations of the the ory of Bo ole an rings to gener al top olo gy , T rans. Amer. Math. So c. 41 (1937 ), no. 3 , 375–4 81. MR 1501905 Daniel J. Clouse Dep ar tment of Defense E-mail ad dr ess : beckc louse@ netzero.net Fernando Guzm ´ an Dept. of Ma thema tical Sciences, Binghamton U niversity, Binghamton N. Y., 13902-6000 E-mail ad dr ess : fer@m ath.bi nghamton.edu URL : http ://mat h.binghamton.edu/fer