Schauder Hats for the 2-variable Fragment of BL

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1 Schauder Hats for the -variable Fragment of BL Stefano Aguzzoli D.S.I., Università di Milano Milano, Italy Simone Bova Department of Mathematics, Vanderbilt University Nashville TN, USA Abstract The theory of Schauder hats is a beautiful and powerful tool for investigating, under several respects, the algebraic semantics of Łukasiewicz infinite-valued logic [CDM99], [MMM7], [Mun94], [P95]. As a notably application of the theory, the elements of the free n-generated MV-algebra, that constitutes the algebraic semantics of the n-variate fragment of Łukasiewicz logic, are obtained as (t-conorm) monoidal combination of finitely many hats, which are in turn obtained through finitely many applications of an operation called starring, starting from a finite family of primitive hats. The aim of this paper is to extend this portion of the Schauder hats theory to the two-variable fragment of Hájek s Basic logic. This step represents a non-trivial generalization of the onevariable case studied in [AG5], [Mon], and provides sufficient insight to capture the behaviour of the n-variable case for n. I. INTRODUCTION For background notions and facts on Łukasiewicz and Basic logic (in short, BL), and their algebraic semantics, respectively the varieties of MV-algebras and BL-algebras, we refer the reader to [CDM99], [Háj98], [CEGT], [AM]. We only mention that the free BL-algebra over n-many generators, in symbols BL n, is the subalgebra of the BL-algebra of all functions from ((n + )[, ]) n to (n + )[, ] generated by the projections, where (n+)[, ] is the ordinal sum of n+ many copies of the generic MV-algebra [, ]. The generic MV-algebra [, ] is (term equivalent to) the algebra given by the interval [, ], equipped with the constant =, and the operations x y = max{, x + y } and x y = min{, x + y}. We define x = x, =, x y = x y, x y = x y, x y = x (x y), and x y = ((x y) y) ((y x) x). Moreover, for any integer m >, we denote mϕ and ϕ m the -disjunction and the -conjunction, respectively, of m occurrences of ϕ. We shall develop a notion of BL-Schauder hat (for short, BL-hat) such that the following two facts hold: (i) each element of BL n is a t-norm monoidal combination of finitely many BL-hats. (ii) each BL-hat in the above combination is constructed, as a BL-formula, via a refinement procedure consisting in a BL-combination of a finite set of primitive BL-hats. The key ingredients of the construction are presented in the general case n in [AB9], [Bov8], where the free n-generated BL-algebra is characterized as a BL-algebra of geometric-combinatorial objects called encodings. In this paper, in the interest of intuition and readability, we avoid the technicalities involved in the general case, and we study directly the two-variable case. Indeed, the two-variable case is complex enough to enlighten the construction in the general case, and allows for a neat geometrical intuition of the behaviour of BL-hats in the refinement procedure. II. FREE MV-ALGEBRAS AND FREE WAJSBERG HOOPS We collect from the literaure the following representations of the free n-generated MV-algebra, MV n, and the free n-generated Wajsberg hoop, WH n, in terms of n-ary McNaughton functions. Recall that a Wajsberg hoop is the {,, }-subreduct of an MV-algebra, and a continuous function f : [, ] n [, ] is a McNaughton function if and only if there are finitely many linear polynomials with integer coefficients, p,..., p k, such that, for every x [, ] n, there is j {,..., k} such that f(x) = p j (x). Theorem ([McN5], [AP]). MV n is (isomorphic to) the algebra of n-ary McNaughton functions, where is realized by the constant, and and are realized by the operations pointwise defined by the corresponding operations of the generic MV-algebra [, ]. WH n is (isomorphic to) the algebra of n-ary McNaughton functions f such that f(,,..., ) =, where, and are realized by the operations pointwise defined by the corresponding operations of the generic MV-algebra [, ]. III. THE FREE -GENERATED BL-ALGEBRA In this section, we introduce the notion of (binary) encoding, and we describe the free -generated BL-algebra, BL, in terms of (binary) encodings, as in [AB9]. Given a subset K = {j, j,..., j k } of {,..., n} we denote π K the projection over K, that is, π K (t,..., t n ) = (t j,..., t jk ). By a (rational) prism we mean a set P [, ] either of the form [, ] Q or of the form Q [, ] for Q [, ) being either (a singleton containing) a rational point or an open interval with rational endpoints. The set Q is called the base of P and is denoted B(P ). Definition. Let K {, }. A function f is essentially K-ary prismwise Wajsberg if the following holds. Case K = : In this case, f =, the empty function (the only function with empty domain). Case K = {} or K = {}: In this case, the following holds. (i) dom(f) is the union of as set of finitely many

2 prisms P [, ], of the first form if K ={}, or of the second if K = {}. (ii) For each P there is g WH such that f(x, x ) = g(π K (x, x )) for all (x, x ) P. Let Q = B(P ). We denote the restriction of f to P by g Q. If dom(f) = {P }, then we denote f simply by g Q. If P B(P ) = [, ) then we say f is total. Case K = {, }: In this case, f WH. We let PW denote the set of all essentially K-ary prismwise Wajsberg functions, for all K {,}. For each function f : [, ] [, ], each b {, } and each i {, }, we let b i (f) = π {,}\{i} (f (b) {(x, x ) π {i} (x, x ) = }) \ {}. Definition ((Binary) Encoding). A (binary) encoding is a 6-tuple, f = f, f, f, f, f, f, satisfying the following properties: ) f ij PW for all (i, j) {, } {,, }. ) f WH, and, either f WH or f =. ) Let b {, } such that b = if and only if f =, let i {, }, and let j {, }. Then, dom(f ij ) = {(x, x ) x j b j (f i )}. We let A denote the set of all binary encodings. It follows that f = implies f = f =. For any pair (f, g) where f is an encoding and g either an encoding or an encoding component, we set ν f (g) = g if f, ν f (g) = g if f =. Theorem. The free -generated BL-algebra, BL, is (isomorphic to) the BL-algebra, BL = A,,,, obtained by equipping the binary encodings with the following constant and operations. Let f, g A. Then, =,,,,,. f g = e, where e A is defined as follows. dom(e ij ) = dom(f ij ) dom(g ij ), for each (i, j) {, } {,, }; for all (x, x ) dom(e ij ), if (i, j) ({, } {,, }) \ {(, )} then e ij (x, x ) = f ij (x, x ) g ij (x, x ), while f g if f = and g =, g f if f = and g, e = f g if f and g =, f g if f and g. f g = e, where e A is defined as follows. g f if f = and g =, f g if f = and g, e = f g if f and g =, f g if f and g. If f then e = f g and for each (i, j) {, } {, }, dom(e ij ) = dom(g ij ) {(x, x ) ν f (f ij (y, y )) ν g (g ij (y, y )), y j =, y j = x j } and e ij (x, x ) = (f ij g ij )(x, x ) if (x, x ) dom(f ij ) dom(g ij ), e ij (x, x ) = otherwise. If f = then e j is defined as above for each j {, }, while e j is total and coinciding with for all j {,, }. The two generators are x BL = x, x,, x, x,, and x BL = x,, x, x,, x. The interpretation ϕ BL of a formula ϕ in the two-variable fragment of BL is the image ι(ϕ) of ϕ under the {,, }- homomorphism ι from the algebra of all two-variable formulas of BL to BL, uniquely determined by ι(x i ) = x BL i. IV. CONVEX GEOMETRY BACKGROUND To recall the notion of Schauder hat and define BL -hats we need to introduce some notions of convex geometry (see [Ewa96], for further background). An n-simplex S R m (for m n) is the convex hull of n + many affinely independent points of R m, called the vertices of S. That is, a -simplex is a (set containing exactly one) point, a -simplex is a line segment, a -simplex is a triangle, etc. By rational n-simplex in R m we mean an n- simplex S whose vertices v,..., v n+ are rational points in [, ] m, that is each component of each v i is a rational number δ, δ. In the following we shall consider only rational n-simplices, which we will call simply n-simplices, or even simplices when the dimension does not need to be specified. A k-dimensional face of a n-simplex S, for k n is the convex hull of k + vertices of S. An open simplex is the relative interior of a simplex (note that vertices, that is, -faces of simplices, are both -simplices and open -simplices; the empty set is the only ( )-dimensional face of any simplex). The denominator den(v) of a rational point v ([, ] Q) m is the least common denominator den(v) of the coordinates of v. The homogeneous expression of v is den(v)(v, ) Z m+. The Farey mediant of a finite set of rational points {v j } j J ([, ] Q) m is the point ( j J den(v j)v j )/( j J den(v j)). A rational m-simplex S R m is unimodular if is the absolute value of the determinant of the matrix whose rows are the homogeneous expressions of the vertices of S. A rational n-simplex F R m, with n m is unimodular if it is a face of a unimodular m-simplex. Note that a rational -simplex (a vertex) is always unimodular. A unimodular triangulation of [, ] m is a finite collection U of n-simplices, for all n m, such that {S U} = [, ] m, the intersection of any two members S, S of U is a common face of both S and S, and U is closed under taking faces. We say that an open simplex S belongs to U (in symbols, S U) if there is T U such that S is the relative interior of T. V. SCHAUDER HATS In this section we collect basic notions and results about Schauder hats that we shall be using in the paper (see [CDM99], [Mun94], [P95]).

3 Definition. Let U be a unimodular triangulation of [, ] n and let S be a k-simplex of U. Then the starring of U at S, in symbols U S, is the set of simplices obtained as follows. ) Put in U S all simplices of U not containing S. ) Display v,..., v k the vertices of S. Then, for each d {,..., k } and each d-dimensional face T of S, displaying {w,..., w d+ } the vertices of T, replace each simplex T R U with the collection {R,..., R d+ }, where R i is the simplex whose vertices are those of R with w i replaced by the Farey mediant v S of v,..., v k. Note that U S is again a unimodular triangulation of [, ] n. If S is a -simplex, the starring U S is called an edge starring. Definition 4. Given a unimodular triangulation U of [, ] n and a vertex (-simplex) v of U, the Schauder hat with apex v in U is the continuous function h v,u : [, ] n [, ] determined by the following conditions: ) h v,u (v) = /den(v). ) h v,u (u) = for all vertices u v of U. ) h v,u is linear over each simplex of U. The star of v in U is the set of all simplices of U having v among their vertices. The Schauder set H U associated with a unimodular triangulation U is the set of all hats of the form h v,u for v a vertex of U. Let v,... v k be the vertices of a simplex S U. Then the star refinement H U S of H U at S is obtained as follows: Let h i be the hat in H U with apex v i, and let h S = q i= h i. Then put in H U S the function h S together with all hats of H U distinct from any h i and replace h j by h j h S, for each j {,..., k}. Lemma. H U S is a Schauder set. In particular H U S = H U S and the apex of h S is the Farey mediant of the vertices of S. Definition 5. Let T be the n-simplex whose set of vertices is {v j } n j=, for π {i}(v j ) = if i + j n, π {i} (v j ) =, otherwise. Let Sym n be the group of all permutations of the set {,,..., n}. For each σ Sym n let T σ be the simplex whose ith vertex is such that its jth component is π {σ(j)} (v i ). Let F σ be the set of all faces of T σ. Then U n = σ Sym n F σ is a unimodular triangulation of [, ] n, called the fundamental partition of [, ] n. Example. U = {{}, [, ], {}}, and H U = {x, x }. The -simplices of U are {(t, t ) t t } and {(t, t ) t t }. Moreover, H U = {x x, x x, x x, x x }. Before stating the normal form theorem for MV n we collect for later use a fundamental technical result on unimodular triangulations. Lemma. Let S be either a rational -simplex or a - simplex lying on an edge of the hypercube [, ] n. Then there is a unimodular triangulation U of [, ] n such that S U. Moreover, U is obtained via finitely many edge starrings from U n. Lemma. For each McNaughton function f : [, ] n [, ] there is a unimodular triangulation U f of [, ] n such that f is linear over each simplex S U f. Moreover, U f is obtained via finitely many edge starrings from U n. Proof: This is one of the main arguments in Panti s geometric proof of the completeness of Łukasiewicz infinitevalued logic, see [P95, Lemma.]. Theorem. For each element f MV n there is a Schauder set H f = {h i } i I and nonnegative integers {m i } i I such that f = i I m i h i. Proof: One takes H f = H Uf, and m i = f(v i )den(v i ), for v i the apex of h i, for each i I. VI. BL -HATS As is well known, each MV-algebra A = A,,, is isomorphic to its order-dual A = A,,, via the map : a a. Note that (a b) = b a, and clearly, (a b) = a b and (a b) = a b. We call Schauder co-hat any function of the form h for h a Schauder hat. Let U be a unimodular triangulation of [, ] n, for some n. The apex and the star of a co-hat k in U are the apex and the star of the hat k in U, respectively. The co-schauder set associated with U is the set of all co-hats of the Schauder set of U. The star refinement of the co-schauder set H U at a simplex S U is defined as for Schauder sets, replacing by duality h i with h i and h j h S with h S h j. Definition 6. A Schauder co-hat k : [, ] n [, ] is virtual iff its apex is (,,..., ); k is actual iff it is not virtual. Note that a Schauder co-hat h: [, ] n [, ] is an element of WH n iff it is actual. Theorem 4. For each element f WH n there is a co- Schauder set H f = {h i } i I and nonnegative integers {m i } i I such that f = h mi i, i I where m i = if h i is virtual. Proof: Immediate from Theorem and Theorem. A primitive Schauder co-hat is a function h for h H U n. Example. The set of primitive Schauder co-hats for MV is H = {x, x }. The set of primitive Schauder co-hats for MV is H = {x x, x x, x x, x x }. Definition 7. A BL -hat is a 6-tuple of functions h = h, h, h, h, h, h belonging to one of the following kinds:

4 k: Either h = k, (k), (k),,, or h =,,, k, (k), (k), and k is a Schauder cohat. k: There is a pair (i, j) {, } {, }, a unimodular triangulation U of [, ] and an open unimodular simplex Q U such that h i j = for all (i, j ) ({, } {,, }) \ {(i, j)}, and h ij = Q for every open simplex Q Q in U, while h ij = k Q for k a Schauder co-hat in one variable. We say k is the Schauder co-hat associated with h. The star and the apex of a BL -hat h are the star and the apex of the associated Schauder co-hat. A BL -hat h is actual (resp. virtual) if so is its associated Schauder co-hat. A BL -hat h is total if it belongs to kind k or Q {{}, (, )}. Lemma 4. Let h be a BL -hat. Then h BL iff h is actual. VII. REFINEMENT PROCESS Let U be a unimodular triangulation of [, ]. Then a relevant face of U is an open k-simplex F of U, for k {, }, such that F {} [, ) or F [, ) {}. We denote FU the set of relevant faces of U of the first form, and F U the set of relevant faces of U of the second form. Definition 8. A BL -triangulation is a 6-tuple U, U, U, U, U, U such that U j is a unimodular triangulation of [, ] for each j {, }, and U ji is a map that associates with each relevant face in FU i j a unimodular triangulation of [, ], for each j {, } and each i {, }. We say that a k-simplex S is a simplex of U if either S U i for some i {, } or there is (i, j) {, } {, }, and a simplex R dom(u ij ) such that S U ij (R). The BL -fundamental partition is B = U, V, V, U, V, V, where V is the following map: {} U, (, ) U (recall from Definition 5 that U n is the fundamental partition of [, ] n ). The BL -set H U associated with a BL -triangulation U is a 6-tuple H, H, H, H, H, H such that, for each i {, }, H i is the set of k BL -hats such that their associated co-hats form the co-schauder set for U i ; for each j {, }, H ij is the map with the same domain as U ij defined as follows. For each S dom(h ij ), H ij (S) is the set of total k BL - hats such that their associated co-hats form the co-schauder set for U ij (S). Note that each hat of H U is linear over each simplex of U. Definition 9. Let p = x x,,,,,, p = x x,,,,,, p = x x,,,,,, ˆp = x x, {}, {},,,, p =,,, x x,,, p =,,, x x,,, p =,,, x x,,, ˆp =,,, x x, {}, {}, p =, x,,,,, ˆp =, x,,,,, p =,, x,,,, ˆp =,, x,,,, p =,,,, x,, ˆp =,,,, x,, p =,,,,, x, ˆp =,,,,, x. Let further, for j, p ij = (pj i (pj i pj i )) p ij, p ij = p ij p ij, and ˆp ij = (pj i (pj i pj i )) ˆp ij, ˆp ij = ˆp ij ˆp ij. Then the set P of primitive BL -hats is the 6-tuple P = P, P, P, P, P, P, where P i = {p i, p i, p i, ˆp i} for i {, }, P ij is the map {} {p ij, ˆp ij }, (, ) {p ij, ˆp ij }, for (i, j) {, } {, }. Note that the hats of the form ˆp ij, ˆp b ij are virtual, and all other hats are actual. Proposition. P is the BL -set associated with the BL - fundamental partition B. We now adapt the definition of starring of triangulations (Def. ) and star refinements of Schauder sets (Def. 4) to our current BL setting. Let U be a BL -triangulation, and let S be a -simplex of U. Let v S be the Farey mediant of the vertices v and v of S, and let S, S be the -simplices obtained by replacing v and v by v S, respectively. Let S = {v S }. Then the starring of U at S, in symbols U S is the 6-tuple U, U, U, U, U, U defined as follows: If S U i for some i {, } then: U i j = U i j for i = i and j {,, }; U i = U i S; If S F j U i, for one j {, }, then the map U ij has domain (dom(u ij ) \ {S}) {S, S, S }, and U ij (S k) = U ij (S) for each k {,, }; otherwise U ij = U ij. If there is (i, j) and R such that S U ij (R), then: U i j = U ij for all i {, } and j j {,, }; dom(u ij ) = dom(u ij) and U ij (R ) = U ij (R ) for all R R dom(u ij ), while U ij (R) = (U ij(r) \ {S}) {S, S, S }. Let U be a BL -triangulation, and let v, v, v S be the vertices of a -simplex S of U and their Farey mediant, respectively. Then the star refinement H U S of H U at S is the 6-tuple K obtained by one of the following processes: k -refinement: S U i for some i {, }. Then let h i be the BL -hat with apex v i and let h S = h h. Set K i = ((H U ) i \ {h, h }) {h S, h S h, h S h }; moreover, if S F j U i for one j {, }, then dom(k ij ) = (dom((h U ) ij ) \ {S}) {S, S, S }, for S, S being the -simplices obtained by starring S at v S, S = {v S }, and K ij (S k ) = (H U ) ij (S) for all k {,, }, while for all other R dom(k ij ), K ij (R) = (H U ) ij (R). If S F j U i F j U i, set K ij = (H U ) ij for all j {, }. Set K i j = (H U ) i j for i = i and j {,, }.

5 k -refinement: There is (i, j) and R such that S U ij (R). Then let h i be the BL -hat with apex v i and let h S = h h. Set K ij (R) = ((H U ) ij (R) \ {h, h }) {h S, h S h, h S h }. Set K ij (R ) = (H U ) ij (R ) for all R R dom((h U ) ij ). Set K i,j = (H U ) i j for all (i, j ) ({, } {,, }) \ {(i, j)}. Proposition. H U S is the BL -set associated with U S. We now single out some families of functions in BL. Each function in BL will turn out to be a combination of suitably chosen functions in these special families. In turn, we shall represent any function belonging to one of these families as a combination of BL -hats in the same family. Lemma 5. Let f BL be either of the form g, (g), (g),,,, or of the form,,, g, (g), (g). Then f is a finite - combination of actual BL -hats obtained by finitely many k-refinements from the set of primitive hats P, or the set P, respectively. Proof: Consider first f = g, (g), (g),,,. Since g W H, by Lemma, g = i I kmi i for suitable integers {m i } i I and a finite set of actual Schauder co-hats {k i } i I obtained from U by finitely many edge star refinements. That is {k i } i I = H Uu for a unimodular triangulation U u of [, ], and there exist -simplices S, S,..., S u [, ] such that U u = U S S S u. As each S i is either a simplex of the fundamental partition of [, ] or it is obtained by a finite sequence of edge starrings from the fundamental partition, then we can form the BL - triangulation B u = B S S u. By Proposition and Proposition, P u = P S S u is the BL -set of B u. In particular (P u ) = {h i } i I, where each h i is a BL -hat of kind k whose associated Schauder co-hat is k i. Since = = =, then for each (j, l) ({, } {,, })\{, }, it holds that ( i I hmi i ) jl is constantly over its domain. Then h mi i = g, (g), (g),,,. i I The case f =,,, g, (g), (g) is dealt with analogously. Lemma 6. Let f BL be of the form g, (g), (g),,,, Then f is the negation of a finite -combination of actual BL -hats obtained by finitely many k-refinements from the set of primitive hats P. Proof: By Theorem, f is such that f = f. Now, f = g, (g), (g),,,, and by Lemma 5, there is a finite set {h i } i I of actual BL -hats obtained by finitely many k-refinements from the set of primitive hats P and suitable positive integers {m i } i I such that f = i I hmi i. Hence f = f = i I hmi i. Lemma 7. Each total BL -hat h belonging to kind k is obtained by finitely many k-refinements from the set of primitive hats P ij ({}) or P ij ((, )), for some (i, j) {, } {, }. Proof: Each Schauder co-hat k in one variable is obtained by finitely many star refinements from the set of primitive cohats {x, x }. Since = =, we immediately conclude that h is obtained by finitely many k-refinements from the set P ij ({}) or P ij ((, )). There remains to deal with BL -hats that are not total. Definition. Let U be a BL -triangulation. Fix (i, j) {, } {, }, and let ĥ be the only virtual BL -hat in (H U ) i. Pick k (H U ) ij (S) (then k is a total k-hat). Let further h, h be actual k-hats in (H U ) i. Denote H = (H U ) i \ {h, h, ĥ}, H(h ) = H {h } and H(h ) = H {h }. Then the function (h, k) = ( h) k h H(h ) is the vertical refinement of the pair (h, k). The function (h, h, k) = ( (h, k) (h, k)) (( h H h) k) is the vertical refinement of the triple (h, h, k). Lemma 8. Each non-total BL -hat h belonging to kind k is obtained by vertical refinement from a set of hats obtained by finitely many steps of k-k-refinement from P. Proof: Consider a non-total k BL -hat h, and let h ij = k Q as in Definition 7, for some (i, j) {, } {, }. Let g be the total hat of kind k such that g ij = k (, ). By Lemma 7, g is obtained by k-refinement from P. Consider first the case (i, j) = (, ) and suppose Q = {v} for some v (, ) Q. Then let U be any BL -triangulation, obtained via finitely many starrings from B, such that w U for the point defined by π (w) = v and π (w) =. Such U exists by Lemma. Let K = H U. Then K contains an actual BL -hat f with apex w. Let K = K \ {f, ê}, for ê the unique virtual hat of K. Then e K e is an element of BL of the form f, {v},,,, for some f WH. Direct computation using the operations defined in Theorem shows the vertical refinement (f, g) is h. Now suppose Q = (v, v ) (, ) is an open unimodular segment with rational endpoints. Let U be any BL -triangulation, obtained via finitely many starrings from B, such that [w, w ] U for points w l defined by π (w l ) = v l and π (w l ) =, for l {, }. The existence of such U is granted by Lemma, again. Let f, f be BL -hats with apices w, w in K. Direct computation now shows e K \{f,f,ê} e is the function f, [v, v ],,,, for some f WH, and hence (f, f, g) = h. The cases (i, j) ({, } {, })\{(, )} are dealt with analogously. Theorem 5 (Normal Form). Each f BL can be expressed as f = ν f ( h m,j,j ) h m,j,j, j J j J where J and J are finite index sets, and for each i {, }, and j J i, the exponent m i,j is a nonnegative integer and

6 h i,j is an actual BL -hat obtained by a finite process of k, k, vertical refinements from the set of primitive hats P. Proof: If f = then we use Lemma 6 to obtain a finite family of BL -hats {h,j } j J and integers {m,j } j J such that, setting g = j J h m,j,j, we have g = f, (f ), (f ),,,. Let U be a BL - triangulation such that f is linear over each simplex of U. Such U exists by Definition 8 and Lemma. Note that for each open simplex S F j U, j {, }, either f j is not defined over [, ] π j (S) (if j = ) or π j (S) [, ] (if j = ), or, in the notation of Definition, f j = g π j (S) for some g WH. In the latter case, use Theorem 4 to express g as i J S k mi i for some finite set J S, and then use Lemma 8 to build i J S h(k i ) mi, where h(k i ) is the k non-total hat such that (h(k i )) j = k i π j (S). Then i J S h(k i ) mi is everywhere but on [, ] π j (S) (if j = ) or π j (S) [, ] (if j = ) where it coincides with f j. Let J be the disjoint union of all sets J S such that S FU F U and f = g π j (S) for some g WH. Then g i J h(k i ) mi is the desired normal form for f. In case f we reason analogously, using Lemma 5 instead of Lemma 6, to obtain functions g and g such that g = f and g = f. We then use Lemma 8 as before to obtain all non-total k BL -hats needed. We remark that Theorem 5 cannot be strengthened by omitting virtual hats from P : the minimal set of actual BL - hats allowing to express all elements of BL with a finite normal form is not finite. The refinement procedure provides an explicit construction of the BL-terms whose interpretation in BL correspond to BL -hats. First, we provide BL-terms whose interpretation in BL correspond to the actual primitive BL -hats. We define, x y = (x y) ((y x) x), x y = ((x y) y) ((y x) x), and we prepare (i =, ), x i = (( x i ) ( x i ) (x i x i )) x i, x i = (( x i ) (x i x i )) x i, x i = (( x i ) ( x i ) (x i x i )) x i, x i = (( x i ) ( x i ) (x i x i )) x i. Proposition. The following hold: (x x ) BL = p ; (x x ) BL = p ; (x x ) BL = p ; (x x ) BL = p ; (x x ) BL = p ; (x x ) BL = p ; (x ) BL = p ; (x ) BL = p ; (x ) BL = p ; (x ) BL = p. Proof: Direct computation. Given the BL-terms for primitive hats, it is possible to iterate through the refinement process to construct BL-terms for all the actual BL -hats. We provide an example of such construction (compare Lemma 7, see also [AG5] for the virtual-hat elimination algorithm for the one-variable case). x_ x_ (a) F (p ). (c) F (p ). x_ x_ x_ (b) F (p ). x_ (d) F (p ). Fig.. Sampling the functional representation of some primitive BL -hats. In [AB9] we define an isomorphism of BL-algebras F from the BL-algebra of encodings BL n to the BL-algebra of real functions from [, n + ] n to [, n+] generated by the projections x i (t,..., t n) = t i (see [AM]). As an example of this functional representation in the -variable case, we depict here the graph of the functions corresponding to some primitive BL -hats. Example. We construct the BL-term whose interpretation in BL corresponds to f =, x x,,,,. The encoding f is obtained in a single step of k-refinement from the set {p, ˆp }. The BL-term corresponding to f is obtained as follows: eliminate the negations from the Schauder cohat x x (maintaining equivalence, compare [Bov8] for details), obtaining the term x x. Then substitute x by x. We have (x x ) BL = f. The total BL -hat h such that h = (x x ) {} and h = (, ) is obtained from f by substituting x with (p (p p )) x. REFERENCES [AB9] S. Aguzzoli and S. Bova. The Free n-generated BL-Algebra. Submitted. [AG5] S. Aguzzoli and B. 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