Reimer s Inequality on a Finite Distributive Lattice
|
|
- Antonia Patterson
- 5 years ago
- Views:
Transcription
1 Reimer s Inequality on a Finite Distributive Lattice Clifford Smyth Mathematics and Statistics Department University of North Carolina Greensboro Greensboro, NC USA cdsmyth@uncg.edu May 1, 2013 Abstract We generalize Reimer s Inequality [6] (a.k.a the BKR Inequality or the van den Berg Kesten Conjecture [1]) to the setting of finite distributive lattices. (MSC 60C05) 1 Introduction Let n be a positive integer. Let Ω 1,..., Ω n be finite sets and let Ω = n i=1 Ω i. For b Ω and S [n] (where [n] := {1,..., n}) we define the cylinder Ω(b, S) = {x Ω : i S x i = b i }. Note that b Ω(b, S), Ω(b, ) = Ω, and Ω(b, [n]) = {b}. Let b Ω and let A, B Ω. We say b A and b B hold disjointly if there exist S, T [n] with S T = such that Ω(b, S) A and Ω(b, T ) B. It does not change the definition to additionally require that S and T be a partition of [n], for if S and T satisfy the conditions of the definition, then so do S 0 = [n] \ T and T 0 = T since S S 0 implies Ω(b, S 0 ) Ω(b, S). The box product of A and B is defined (in [1]) to be A B = {b Ω : b A and b B hold disjointly}. (1) 1
2 Note A B A B. We say a probability measure µ on Ω = n i=1 Ω i is a product probability measure if µ = n i=1 µ i where µ i is a probability measure on Ω i for each i with 1 i n. We have the following theorem. Theorem 1.1 (Reimer s Inequality [6]) If Ω = n i=1 Ω i is a finite product of finite sets and µ is a product probability measure on Ω then µ(a B) µ(a)µ(b), for all A, B Ω. (2) Inequality (2) was originally conjectured by van den Berg and Kesten [1] who proved it for increasing sets on products of chains. Theorem 1.1 is sometimes known as the BKR Inequality. We generalize this theorem to finite distributive lattices, see Theorem 1.2 below. Let L be a finite bounded lattice. Let 0 L = min L and 1 L = max L. If x, y L, let [x, y] L = {z L : x z y}. Let b L and let A, B L. We say b A and b B hold disjointly if there exist x, y, z, w L with x y, z w, x z = 0 L, y w = 1 L, and b = x z = y w, such that b [x, y] L A and b [z, w] L B. Note that [x, y] L [z, w] L = {b}. (In general, if x y and z w, S = [x, y] L [z, w] L if and only if x z y w, in which case S = [x z, y w] L.) The L-box product of A and B is A L B = {b L : b A and b B hold disjointly}. (3) Note that A L B A B. If L is a lattice, a function µ : L [0, + ) is log-supermodular if and is log-modular if µ(a)µ(b) µ(a b)µ(a b), for all a, b L, (4) µ(a)µ(b) = µ(a b)µ(a b), for all a, b L. (5) Our main theorem is the following. Theorem 1.2 Let L be a finite distributive lattice and let µ be a log-modular probability measure on L. Then µ(a L B) µ(a)µ(b), for all A, B L. (6) 2
3 Distributivity is a necessary condition in Theorem 1.2. In the nondistributive lattice N 5 = ({, {1}, {2}, {2, 3}, {1, 2, 3}}, ), the sets A = {{2}, {2, 3}, {1, 2, 3}} and B = {, {2}, {2, 3}} furnish a counterexample when µ is uniform: µ(a L B) = µ(a B) = 2 > ( )2 = µ(a)µ(b). Essentially the same counterexample can be found in the non-distributive lattice M 3 = ({, {1}, {2}, {3}, {1, 2, 3}}, ). If P is a partially ordered set and X P, then X is increasing if for all x, y P, x X and x y imply y X. X is decreasing if for all x, y P, x X and x y imply y X. If f : P [0, + ), then f is increasing if for all x, y P, x y implies f(x) f(y). f is decreasing if for all x, y P, x y implies f(x) f(y). We state the following well-known theorem. Theorem 1.3 (The FKG Inequality [2]) If L is a finite distributive lattice and µ is a log-supermodular probability measure on L then g(x)µ(x), (7) x L f(x)g(x)µ(x) x L f(x)µ(x) x L for all increasing f, g : L [0, + ). Equivalently, µ(a B) µ(a)µ(b), for all increasing A, B L. (8) Let Ω = n i=1 Ω i be a finite product of finite sets. Let { n } L(Ω) = Ω i : Ω i is a total ordering of Ω i for 1 i n i=1 be the set of lattices obtained from Ω by viewing each factor Ω i as a chain. Theorem 1.4 (The Harris Kleitman Theorem [4, 5]) If Ω is a finite product of finite sets, µ is a product probability measure on Ω, and L L(Ω) then µ(a B) µ(a)µ(b), for all increasing A, B L. (9) The proofs in [4, 5] only handled the case where Ω = {0, 1} n. As a product measure is log-modular (see Proposition 1.6, statement 1), Theorem 1.3 implies Theorem 1.4. We make the following observations on our definitions, proved in Section 4. 3
4 Proposition 1.5 Let Ω = n i=1 Ω i be a finite product of finite sets and let L L(Ω). i. For all A, B Ω, A B A L B A B. ii. If A L is increasing and B L is decreasing then A L B = A B. iii. If Ω i 2 for all 1 i n then for all A, B Ω, A B = A L B. iv. If Ω i 3 for some i with 1 i n, then there are examples of A, B Ω for which = A B A L B = A B. Proposition 1.6 i. Suppose Ω is a finite product of finite sets, µ is a probability measure on Ω, and L L(Ω). Then the condition that µ is a product probability measure on Ω is equivalent to the condition that µ is a log-modular probability measure on L. ii. Suppose L and K are finite distributive lattices and L is a sublattice of K. If µ : L (0, + ) is log-modular, then there exists a log-modular ν : K (0, + ) such that µ = ν L. iii. If L is a finite distributive lattice and µ is a probability measure on L, then there is a product of chains, K, that contains L as a sublattice and a product probability measure ν on K such that µ( ) = ν( L). Given Propositions 1.5 and 1.6 it is easy to prove the following observations on how Theorem 1.2 relates to Theorems 1.1, 1.3, and 1.4. Proposition 1.7 We have the following implications. i. Theorem 1.2 implies Reimer s Inequality, Theorem 1.1, and is a stronger statement. ii. Theorem 1.2 implies the FKG Theorem, Theorem 1.3, in the case that µ is log-modular. iii. Theorem 1.2 implies the Harris Kleitman theorem, Theorem 1.4. We prove Theorem 1.2 in Section 3 after developing the preliminaries we need in Section 2. We ll prove Propositions 1.5, 1.6 and 1.7 in Section 4. This last section only requires the material from the introduction and the first paragraph of Section 2. 4
5 2 Butterflies Let L be a bounded lattice. An L-butterfly is a 6-tuple β = (x, y, z, w, b, L) where x, y, z, w, b L, x y, z w, x z = 0 L, y w = 1 L and b = x z = y w. R(β) = [x, y] L is the red wing of β, Y(β) = [z, w] L is the yellow wing of β, and b(β) = b is the body of β. If A, B L we say β is an (A, B)-butterfly if R(β) A and Y(β) B. Note that A L B = {b Ω : (A, B)-butterfly β with b(β) = b}. Butterflies have been defined in [6], but only for Ω = {0, 1} n. We ll soon see that the two definitions are equivalent on {0, 1} n. Lemma 2.1 Let L be a finite distributive lattice, let c, d L with c d. Consider the map f = f c,d : L [c, d] L defined by f(x) = (x c) d = (x d) c. Then f is a lattice homomorphism and hence order preserving. Thus if K L is a sublattice (respectively convex) then f 1 (K) is a sublattice (respectively convex). Note f(x) = x for all x [c, d] L, f 1 (c) [0 L, c] L and f 1 (d) [d, 1 L ] L. Proof: (Sketch.) Let x 1, x 2 L. f(x 1 ) f(x 2 ) = (x 1 c) (x 2 c) d = ((x 1 x 2 ) ((c x 1 x 2 ) c)) d = ((x 1 x 2 ) c) d = f(x 1 x 2 ). The proof that f preserves joins is similar. If K is a sublattice of L and f(x 1 ), f(x 2 ) K then f(x 1 x 2 ) = f(x 1 ) f(x 2 ) K. Similarly, f 1 (K) is also closed under joins and hence is a lattice. The other statements are just as straightforward. Butterflies on intervals restrict to subintervals in a nice way. Lemma 2.2 Let L be a finite distributive lattice. Let c, d, C, D L with C c d D. Let f(x) = f c,d (x). Let β = (x, y, z, w, b, [C, D]) be a [C, D]- butterfly. Then f(β) = (f(x), f(y), f(z), f(w), f(b), [c, d]) is a [c, d]-butterfly. If b [c, d] then b(β) = b = f(b) = b(f(β)), R(f(β)) = R(β) [c, d] and Y(f(β)) = Y(β) [c, d]. Proof: (Sketch.) We ll prove that if b [c, d] then R(f(β)) = R(β) [c, d]. The proof that Y(f(β)) = Y(β) [c, d] is similar and all the other statements 5
6 follow straightforwardly from Lemma 2.1: f(y) f(w) = f(y w) = f(d) = d, x b y implies f(x) f(b) = b f(y), etc. Since all the sets we now consider are intervals of L we ll drop the subscripts, R(f(β)) = [f(x), f(y)], R(β) = [x, y], etc. Since b [x, y] [c, d], [x, y] [c, d] = [x c, y d]. Since x c b d, f(x) = (x c) d = x c. Similarly, c b y d and f(y) = y d. Thus R(f(β)) = [f(x), f(y)] = [x, y] [c, d] = R(β) [c, d]. Let L be a bounded lattice. We say x, y L are L-complementary if x y = 0 L and x y = 1 L. In this case, we say x and y are complemented in L and that the complement of x in L is y, which is also denoted as (x) L. If L is distributive and x L is complemented, then there is a unique y L such that (x) L = y ( [3], p.62). If every element in L has a complement, we say L is complemented. Let C(L) = {x L : x is complemented in L}. Note that 0 L, 1 L C(L). By DeMorgan s identities, C(L) is a lattice ( [3], p.63). Since C(L) is distributive and complemented, it is by definition a Boolean lattice ( [3], p.63). By [3] (Corollary 21, p.85), C(L) is thus isomorphic to the lattice of subsets of some finite set, or, equivalently, to {0, 1} n for some n 1. In [6] Reimer defined a butterfly B b,a on {0, 1} n to be an ordered pair (b, a) with b, a {0, 1} n and also defined R(B b,a ) = [a, b], Y(B b,a ) = [a, b], and b(b b,a ) = b. Here [a, b] := {x {0, 1} n : x i {a i, b i }, for all 1 i n} and a is the complement of a in {0, 1} n. As can be easily checked, B b,a corresponds to the butterfly β = (a b, a b, a b, a b, b, {0, 1} n ) in the sense that R(B b,a ) = R(β), Y(B b,a ) = Y(β) and b(b b,a ) = b = b(β). In this same sense, the butterfly β = (x, y, z, w, b, {0, 1} n ) corresponds to the butterfly B b,a where a is the complement of b in [x, y]. Lemma 2.3 Let L be a finite distributive lattice and let c, d L with c d. Let β = (x, y, z, w, b, [c, d]) be an [c, d]-butterfly. If b C(Ω) then β 0 = (x, y, z, w, b, C([c, d])) is a C([c, d])-butterfly. Furthermore, x = b a, y = b a, z = b a, and w = b a where a = f x,y (b ), a = f z,w (b ), and b = (b) [c,d]. R(β 0) = R(β) C([c, d]) and Y(β 0 ) = Y(β) C([c, d]). Proof: First we show that a = (b) [x,y] and a = (b) [z,w]. This will mean that the formulas given for x, y, z, w hold. We then show that a = (a) [c,d]. This will mean a, a, b, b are all members of the lattice C([c, d]) and thus so 6
7 are x, y, z, w. The conditions that remain to be proved about x, y, z, w all immediately follow from the fact that β is a [c, d]-butterfly: y w = d = 1 C([c,d]), etc. Since b [x, y], b a = f x,y (b) f x,y (b ) = f x,y (b b ) = f x,y (c) = x. Similarly, b a = y. Thus a = (b) [x,y]. Similarly, a = (b) [z,w]. a a = ((b x) y) ((b z) w) = (b x) (b z) (y w) = (b x) (b z) b = (b b) (b z b) (x b b) (x z b) = c c c c = c. Similarly a a = d and thus a = (a) [c,d]. Clearly, R(β 0 ) = [x, y] C([c,d]) = [x, y] [c,d] C([c, d]) = R(β) C([c, d]). Similarly Y(β 0 ) = Y(β) C([c, d]). Let L be a bounded lattice and let B be a family of butterflies on L. Let R(B) = β B R(β) and Y(B) = β B Y(β). We say B has distinct bodies if for all β 1, β 2 B with β 1 β 2, b(β 1 ) b(β 2 ). If X L = {0, 1} n, let X = {(x) L : x X}. Reimer proved Lemma 2.4, below, in [6], where it was the crux of his proof of Theorem 1.1. We will use it in the next section to prove Theorem 1.2. Lemma 2.4 (Reimer s Butterfly Lemma [6]) If L = {0, 1} n is a finite Boolean lattice and B is a family of L-butterflies with distinct bodies, then B R(B) (Y(B)). 3 Proof of Theorem 1.2 Proof of Theorem 1.2. We write µ(a) = b L 1 A(b)µ(b) and µ(b) = b L 1 A(b )µ(b ), where 7
8 1 A (x) := 1 if x A, 0 otherwise. Thus µ(a)µ(b) = 1 A (b)1 B (b )µ(b)µ(b ) (b,b ) L 2 = 1 A (b)1 B (b )µ(b b )µ(b b ) (b,b ) L 2 = 1 A (b)1 B (b ) µ(c)µ(d) = (c,d) L 2 :c d (c,d) L 2 :c d (b,b ) L 2 :b b =c,b b =d (A C([c, d]) (B C([c, d])) µ(c)µ(d). The second equality follows from the log-modularity of µ. Similarly we multiply µ(a L B) = b L 1 A B(b)µ(b) and 1 = µ(l) = b L µ(b ) to get L µ(a L B)µ(L) = 1 A L B(b) µ(c)µ(d) or µ(a L B) = (c,d) L 2 :c d (c,d) L 2 :c d (b,b ) L 2 :b b =c,b b =d (A L B) C([c, d]) µ(c)µ(d). To complete the proof, we will show that (A L B) C([c, d]) (A C([c, d]) (B C([c, d])) for all c, d L with c d. Given b (A L B) C([c, d]), let β b = (x, y, z, w, b, L) be a butterfly witnessing the fact that b A L B, i.e. b(β b ) = b, R(β b ) A, and Y(β b ) B. Let f = f c,d. By Lemma 2.2, γ b = (f(x), f(y), f(z), f(w), b, [c, d]) is a [c, d]-butterfly with R(γ b ) = R(β b ) [c, d] A and Y(γ b ) = Y(β b ) [c, d] B. By Lemma 2.3, δ b = (f(x), f(y), f(z), f(w), b, C([c, d])) is a C([c, d])-butterfly with R(δ b ) = R(γ b ) C([c, d]) A C([c, d]). Similarly, Y(δ b ) B C([c, d]). By construction, B = {δ b : b (A L B) C([c, d])} is a family of butterflies of C([c, d]) with distinct bodies. Since C([c, d]) is isomorphic to {0, 1} n for some n 1, we apply Lemma 2.4 to finish the proof: (A L B) C([c, d]) = B R(B) (Y(B)) (A C([c, d])) (B C([c, d])). Note that the last inequality in the chain holds because of the inclusions R(B) A C([c, d]) and Y(B) B C([c, d]). Indeed, R(δ b ) A C([c, d]) for all δ b B, so R(B) = b R(δ b) A C([c, d]). 8
9 4 Proofs of Propositions 1.5, 1.6 and 1.7 Proof of Proposition 1.5. (i.) Suppose A, B Ω. A L B A B by definition. We now show A B A L B. Suppose b A B. Then there exist S and T, a partition of [n] such that Ω(b, S) A and Ω(b, T ) B. For U [n] we define b U, b U L as (b U ) i = { bi, if i U 0 Ω i, otherwise and (b U ) i = { bi, if i U 1 Ω i, otherwise. (10) Let x = b S, y = b S, z = b T, w = b T. It is easy to check that x z = b = y w, x z = 0 L, y w = 1 L, [x, y] L = Ω(b, S) A, and [z, w] L = Ω(b, T ) B. Thus b A L B as desired. (ii.) Suppose A L is increasing and B L is decreasing. By (i.), we only need to show that A B A L B. Suppose b A B. Then β = (b, 1 L, 0 L, b, b, L) is an (A, B)-butterfly with b(β) = b. Hence b A L B. (Note R(β) = [b, 1 L ] L A, since b A and A is increasing. A similar argument gives Y(β) B.) (iii.) We may assume Ω = L = {0, 1} n for some n 1 where x L y iff x i y i for all i. By (i.), we need only prove A L B A B. Suppose b A L B. Let β = (x, y, z, w, b, Ω ) be an (A, B) butterfly. To prove b A B, we exhibit a partition S,T of [n] so that Ω(b, S) = [x, y] L A and Ω(b, T ) = [z, w] L B. Let S = {i [n] : x i = y i } and T = {i [n] : z i = w i }. Since 1 L = y w, b = y w = x z, x z = 0 L, we have 1 = max(y i, w i ), b i = min(y i, w i ) = max(x i, z i ), and 0 = min(x i, z i ) for all i [n]. In particular, if b i = 1, y i = w i = 1 and {x i, z i } = {0, 1} while if b i = 0, x i = z i = 0 and {y i, w i } = {0, 1}. This means that S and T restricted to {i : b i = 1} form a partition. Similarly, S and T restricted to {i : b i = 0} form a partition. Thus S and T are a partition of [n]. It is not hard to prove that Ω(b, S) = [x, y] L. a [x, y] L is equivalent to x i a i y i for all i, which by definition of S is equivalent to a i = x i = y i for all i S. Since x i b i y i for all i, this last statement is equivalent to a i = b i for all i S or a Ω(b, S). (iv.) Suppose k 1,..., k n, i 0 are positive integers with 1 i 0 n and k i0 3. Let Ω = L = n i=1 [k i] where x L y iff x i y i for all i. Let A = {x Ω : 9
10 x i0 2} and let B = {x Ω : x i0 k i0 1}. Note that for any x Ω, Ω(x, S) is not contained in A or in B if i 0 S, thus A B =. Since A is increasing and B is decreasing, (ii.) implies A L B = A B = {x Ω : 1 x i0 k i0 1} A B. A similar example is given by Ω = L = {0, 1, 2} n, A = {x : x i 1, i}, B = {x : x i 1, i}. A L B = A B = {(1,..., 1)} while A B =. Proof of Proposition 1.6. (i.) If µ = µ i is a product measure on Ω = n i=1 Ω i then µ is a log-modular measure on the product of chains L = n i=1 Ω i: µ(a)µ(b) = n i=1 µ i(a i )µ i (b i ) = n i=1 µ i(a i b i )µ i (a i b i ) = µ(a b)µ(a b). The second equality follows because in the chain Ω i, {a i, b i } = {a i b i, a i b i }. Suppose now that µ is log-modular on L, then for any x 1, x 2,..., x k L, µ(x 1 )µ(x 2 ) µ(x k ) = µ(y 1 )µ(y 2 ) µ(y k ) where y 1,..., y k L have the property that for each i [n], ((y 1 ) i,... (y k ) i ) is a permutation of ((x 1 ) i,..., (x k ) i ). Indeed, both products are equal to µ(z 1 ) µ(z k ) where ((z 1 ) i,..., (z k ) i ) is a permutation of ((x 1 ) i,..., (x k ) i ) that is weakly increasing in Ω i, for all i. If i < j consider the products µ(a 1 ) µ(a k ) and µ(b 1 ) µ(b k ) where b i = a i a j, b j = a i a j, and b l = a l for l i, j. By log-modularity, the products are equal, but also for each l, (b i ) l = (a i a j ) l (a i a j ) l = (b j ) l. One can thus bring the product µ(x 1 ) µ(x k ) to the form µ(z 1 ) µ(z k ) by repeatedly using such two-factor replacements. If µ is additionally a probability measure, we claim that µ(x) = n i=1 µ i(x i ) is a product probability measure where µ i (x i ) = µ((z i,1,..., z i,i 1, x i, z i,i+1,..., z i,n )). z i,j Ω j,j i Clearly µ i is a probability distribution on Ω i, as x i Ω i µ i (x i ) = z L µ(z) = 1. Consider n µ i (x i ) = i=1 z i,j Ω,i j µ((x 1, z 1,2,..., z 1,n )) µ((z n,1,..., z n,n 1, x n )). 10
11 Let π S n such that π(i) is the unique number in [n] equivalent to i + 1 mod n. Since one can arbitrarily permute the ith coordinates of the products occurring in the summation, n µ i (x i ) = i=1 z i,j Ω,i j k 1 µ((x 1,..., x n )) µ((z π t (1),1,..., z π t (n),n)) t=1 = µ(x)(µ(l)) k 1 = µ(x). (ii.) It suffices to prove the result in the case that K = {0, 1} n. In the general case, one may use the fundamental theorem on finite distributive lattices ( [3], p.82) to embed K (and hence L) as a sublattice of some {0, 1} n. One can find ν log-modular on {0, 1} n restricting to µ on L and then restrict ν to K to get ν. It is completely straightforward to verify the following facts. If L is not trivial, pick some y L such that y < 1 L. Then L 1 = {x L : x y = 1 L } is a sublattice of L and furthermore L 1 is an interval L 1 = [a, 1 L ] L (where a L). L y = {x L : x y = y} is a sublattice of L and L 1 is isomorphic to the sublattice [a y, y] L y. The maps f : L 1 [a y, y] and g : [a y, a] L 1 given by f(x) = x y and g(x) = x a are inverse lattice isomorphisms. We assume now that K = {0, 1} n and proceed by induction on L. If L = 1 there is nothing to prove. If L = 2, let ν(x) = ar {i:x i=1} where a, r > 0. Note that ν is log-modular. Let w 0 = {i : (0 L ) i = 1} and w 1 = {i : (1 L ) i = 1}. Note w 0 < w 1. Pick a, r > 0 such that ν(0 l ) = ar w 0 = µ(0 L ) and ν(1 L ) = ar w 1 = µ(1 L ). Suppose now that L > 2. By induction, we get ν 1 log-modular on [0 K, y] K restricting to µ on L y. Let ν 2 be log-modular on [y, 1 K ] K restricting to the measure µ on {y, 1 L } given by µ (y) = 1 and µ(1 L ) = µ(1 L )/µ(y). We claim that ν(x) = ν 1 (x y)ν 2 (x y) is log-modular on K and restricts to µ on L. Indeed, since ν 1 is log-modular, ν 1 (x 1 y)ν 1 (x 2 y) = ν 1 ((x 1 y) (x 2 y))ν 1 ((x 1 y) (x 2 y)) = ν 1 ((x 1 x 2 ) y)ν 1 ((x 1 x 2 ) y). Thus the function ν 1 ( y) : K (0, + ) is log-modular. One can prove the same for ν 2 ( y) : K (0, + ). This means their product, ν, is also log-modular. If x L y then ν(x) = ν 1 (x)ν 2 (y) = µ(x) 1. If x L 1, ν(x) = ν 1 (x y)ν(1 L ) = µ(x y)µ(1 L )/µ(y) = µ(x). (This holds by the log-modularity of µ and the fact that x y = 1 L.) 11
12 Note that we do need that µ to be positive. If L = {x 1, x 2, x 3 } with x 1 < x 2 < x 3 is a sublattice of K = {0, 1} n and µ(x 1 ) = µ(x 2 ) = 1 while µ(x 1 ) = 0, then µ does not extend to any log-modular function ν on K. If x 1 is the complement of x 1 in [x 0, x 2 ] K, then we would have to have 0 = ν(x 1 )ν(x 1) = ν(x 0 )ν(x 2 ) = 1, an impossibility. (iii.) Suppose µ is a log-modular probability measure on L. Let K be a product of chains containing L as a sublattice. Let ν : K (0, + ) be log-modular restricting to µ on L. ν(x) = ν (x)/ν (K) is a log-modular probability measure on K. Note ν(x L) = ν({x} L)/ν(L) = ν ({x} L)/ν (L) = ν ({x} L)/µ(L) = ν ({x} L). Thus ν(x L) = µ(x) if x L and ν(x L) = 0 otherwise. Proof of Proposition 1.7. (i.) Let Ω be a finite product of finite sets. Let µ be a product measure on Ω. Let L L(Ω). By Proposition 1.6 (i.), µ is log-modular on L. Thus we may apply Theorem 1.2 and Proposition 1.5 (i.), to get µ(a B) µ(a L B) µ(a)µ(b). Proposition 1.5 (iv.) shows that one can have 0 = µ(a B) < µ(a L B) µ(a)µ(b). (ii.) Let L be a finite distributive lattice and let µ be a log-modular probability measure on L. Let A, B L be increasing. Note B 0 = L \ B is decreasing so by Proposition 1.5 (ii.), A L B 0 = A B 0. By Theorem 1.2, µ(a B) = µ(a) µ(a B 0 ) µ(a) µ(a)µ(b 0 ) = µ(a)µ(b). (iii.) By Proposition 1.6 (i.), Theorem 1.4 is a special case of the log-modular case of Theorem 1.3. References [1] J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability, J. Appl. Probab. 22 (1985), no. 3, MR MR (87b:60027) [2] C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), MR (46 #8607) 12
13 [3] George Grätzer, General lattice theory, Birkhäuser Verlag, Basel, 2003, With appendices by B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung and R. Wille, Reprint of the 1998 second edition [MR ]. MR [4] T. E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc. 56 (1960), MR MR (22 #6023) [5] Daniel J. Kleitman, Families of non-disjoint subsets, J. Combinatorial Theory 1 (1966), MR MR (33 #1242) [6] David Reimer, Proof of the van den Berg-Kesten conjecture, Combin. Probab. Comput. 9 (2000), no. 1, MR MR (2001g:60017) 13
Finite pseudocomplemented lattices: The spectra and the Glivenko congruence
Finite pseudocomplemented lattices: The spectra and the Glivenko congruence T. Katriňák and J. Guričan Abstract. Recently, Grätzer, Gunderson and Quackenbush have characterized the spectra of finite pseudocomplemented
More informationOn the variety generated by planar modular lattices
On the variety generated by planar modular lattices G. Grätzer and R. W. Quackenbush Abstract. We investigate the variety generated by the class of planar modular lattices. The main result is a structure
More informationUniquely complemented lattices The 45th Summer School on Algebra Sept. 2 7, 2007, Hotel Partizan Tale, Low Tatras Slovakia
Uniquely complemented lattices The 45th Summer School on Algebra Sept. 2 7, 2007, Hotel Partizan Tale, Low Tatras Slovakia G. Grätzer September 3, 2007 The Whitman Conditions P(X ) all polynomials over
More informationAlgebraic methods toward higher-order probability inequalities
Algebraic methods toward higher-orderprobability inequalities p. 1/3 Algebraic methods toward higher-order probability inequalities Donald Richards Penn State University and SAMSI Algebraic methods toward
More informationSUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS
SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS MARINA SEMENOVA AND FRIEDRICH WEHRUNG Abstract. For a partially ordered set P, let Co(P) denote the lattice of all order-convex
More informationarxiv:math/ v1 [math.gm] 21 Jan 2005
arxiv:math/0501340v1 [math.gm] 21 Jan 2005 SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, II. POSETS OF FINITE LENGTH MARINA SEMENOVA AND FRIEDRICH WEHRUNG Abstract. For a positive integer n, we denote
More informationProof of Reimer s Theorem
Proof of Reimer s Theorem Dan Crytser April 13, 2010 Abstract Two events in a product space A, B Ω = Ω 1... Ω n are said to occur disjointly if we can observe them occurring on disjoint sets of indices
More informationA CHARACTERIZATION OF LOCALLY FINITE VARIETIES THAT SATISFY A NONTRIVIAL CONGRUENCE IDENTITY
A CHARACTERIZATION OF LOCALLY FINITE VARIETIES THAT SATISFY A NONTRIVIAL CONGRUENCE IDENTITY KEITH A. KEARNES Abstract. We show that a locally finite variety satisfies a nontrivial congruence identity
More informationarxiv:math/ v1 [math.gm] 21 Jan 2005
arxiv:math/0501341v1 [math.gm] 21 Jan 2005 SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, I. THE MAIN REPRESENTATION THEOREM MARINA SEMENOVA AND FRIEDRICH WEHRUNG Abstract. For a partially ordered set P,
More informationNOTES ON PLANAR SEMIMODULAR LATTICES. IV. THE SIZE OF A MINIMAL CONGRUENCE LATTICE REPRESENTATION WITH RECTANGULAR LATTICES
NOTES ON PLANAR SEMIMODULAR LATTICES. IV. THE SIZE OF A MINIMAL CONGRUENCE LATTICE REPRESENTATION WITH RECTANGULAR LATTICES G. GRÄTZER AND E. KNAPP Abstract. Let D be a finite distributive lattice with
More information10. Finite Lattices and their Congruence Lattices. If memories are all I sing I d rather drive a truck. Ricky Nelson
10. Finite Lattices and their Congruence Lattices If memories are all I sing I d rather drive a truck. Ricky Nelson In this chapter we want to study the structure of finite lattices, and how it is reflected
More informationNOTES ON PLANAR SEMIMODULAR LATTICES. IV. THE SIZE OF A MINIMAL CONGRUENCE LATTICE REPRESENTATION WITH RECTANGULAR LATTICES
NOTES ON PLANAR SEMIMODULAR LATTICES. IV. THE SIZE OF A MINIMAL CONGRUENCE LATTICE REPRESENTATION WITH RECTANGULAR LATTICES G. GRÄTZER AND E. KNAPP Abstract. Let D be a finite distributive lattice with
More informationDistributive congruence lattices of congruence-permutable algebras
Distributive congruence lattices of congruence-permutable algebras Pavel Ruzicka, Jiri Tuma, Friedrich Wehrung To cite this version: Pavel Ruzicka, Jiri Tuma, Friedrich Wehrung. Distributive congruence
More informationJónsson posets and unary Jónsson algebras
Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal
More informationThe Strength of the Grätzer-Schmidt Theorem
The Strength of the Grätzer-Schmidt Theorem Katie Brodhead Mushfeq Khan Bjørn Kjos-Hanssen William A. Lampe Paul Kim Long V. Nguyen Richard A. Shore September 26, 2015 Abstract The Grätzer-Schmidt theorem
More informationSymmetric polynomials and symmetric mean inequalities
Symmetric polynomials and symmetric mean inequalities Karl Mahlburg Department of Mathematics Louisiana State University Baton Rouge, LA 70803, U.S.A. mahlburg@math.lsu.edu Clifford Smyth Department of
More informationFree trees and the optimal bound in Wehrung s theorem
F U N D A M E N T A MATHEMATICAE 198 (2008) Free trees and the optimal bound in Wehrung s theorem by Pavel Růžička (Praha) Abstract. We prove that there is a distributive (, 0, 1)-semilattice G of size
More informationMeasurable functions are approximately nice, even if look terrible.
Tel Aviv University, 2015 Functions of real variables 74 7 Approximation 7a A terrible integrable function........... 74 7b Approximation of sets................ 76 7c Approximation of functions............
More informationON POSITIVE, LINEAR AND QUADRATIC BOOLEAN FUNCTIONS
Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 177 187. ON POSITIVE, LINEAR AND QUADRATIC BOOLEAN FUNCTIONS SERGIU RUDEANU Abstract. In [1], [2] it was proved that a function f : {0,
More informationSTRICTLY ORDER PRIMAL ALGEBRAS
Acta Math. Univ. Comenianae Vol. LXIII, 2(1994), pp. 275 284 275 STRICTLY ORDER PRIMAL ALGEBRAS O. LÜDERS and D. SCHWEIGERT Partial orders and the clones of functions preserving them have been thoroughly
More informationAn embedding of ChuCors in L-ChuCors
Proceedings of the 10th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2010 27 30 June 2010. An embedding of ChuCors in L-ChuCors Ondrej Krídlo 1,
More informationCourse Notes. Part IV. Probabilistic Combinatorics. Algorithms
Course Notes Part IV Probabilistic Combinatorics and Algorithms J. A. Verstraete Department of Mathematics University of California San Diego 9500 Gilman Drive La Jolla California 92037-0112 jacques@ucsd.edu
More informationPOSITIVE AND NEGATIVE CORRELATIONS FOR CONDITIONAL ISING DISTRIBUTIONS
POSITIVE AND NEGATIVE CORRELATIONS FOR CONDITIONAL ISING DISTRIBUTIONS CAMILLO CAMMAROTA Abstract. In the Ising model at zero external field with ferromagnetic first neighbors interaction the Gibbs measure
More information8. Distributive Lattices. Every dog must have his day.
8. Distributive Lattices Every dog must have his day. In this chapter and the next we will look at the two most important lattice varieties: distributive and modular lattices. Let us set the context for
More informationThe category of linear modular lattices
Bull. Math. Soc. Sci. Math. Roumanie Tome 56(104) No. 1, 2013, 33 46 The category of linear modular lattices by Toma Albu and Mihai Iosif Dedicated to the memory of Nicolae Popescu (1937-2010) on the occasion
More informationMaximum union-free subfamilies
Maximum union-free subfamilies Jacob Fox Choongbum Lee Benny Sudakov Abstract An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called
More informationA loosely Bernoulli counterexample machine
A loosely Bernoulli counterexample machine Christopher Hoffman September 7, 00 Abstract In Rudolph s paper on minimal self joinings [7] he proves that a rank one mixing transformation constructed by Ornstein
More informationDISTINGUISHING PARTITIONS AND ASYMMETRIC UNIFORM HYPERGRAPHS
DISTINGUISHING PARTITIONS AND ASYMMETRIC UNIFORM HYPERGRAPHS M. N. ELLINGHAM AND JUSTIN Z. SCHROEDER In memory of Mike Albertson. Abstract. A distinguishing partition for an action of a group Γ on a set
More informationLecture 3: Boolean variables and the strong Rayleigh property
Lecture 3: Boolean variables and the strong Rayleigh property Robin Pemantle University of Pennsylvania pemantle@math.upenn.edu Minerva Lectures at Columbia University 09 November, 2016 Negative dependence
More informationConvex Optimization Notes
Convex Optimization Notes Jonathan Siegel January 2017 1 Convex Analysis This section is devoted to the study of convex functions f : B R {+ } and convex sets U B, for B a Banach space. The case of B =
More informationEquivalent Formulations of the Bunk Bed Conjecture
North Carolina Journal of Mathematics and Statistics Volume 2, Pages 23 28 (Accepted May 27, 2016, published May 31, 2016) ISSN 2380-7539 Equivalent Formulations of the Bunk Bed Conjecture James Rudzinski
More information1 Basic Combinatorics
1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set
More informationA dyadic endomorphism which is Bernoulli but not standard
A dyadic endomorphism which is Bernoulli but not standard Christopher Hoffman Daniel Rudolph November 4, 2005 Abstract Any measure preserving endomorphism generates both a decreasing sequence of σ-algebras
More informationZaslavsky s Theorem. As presented by Eric Samansky May 11, 2002
Zaslavsky s Theorem As presented by Eric Samansky May, 2002 Abstract This paper is a retelling of the proof of Zaslavsky s Theorem. For any arrangement of hyperplanes, there is a corresponding semi-lattice
More informationA Family of Finite De Morgan Algebras
A Family of Finite De Morgan Algebras Carol L Walker Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, USA Email: hardy@nmsuedu Elbert A Walker Department of Mathematical
More informationQuasi-invariant measures for continuous group actions
Contemporary Mathematics Quasi-invariant measures for continuous group actions Alexander S. Kechris Dedicated to Simon Thomas on his 60th birthday Abstract. The class of ergodic, invariant probability
More informationRemarks on categorical equivalence of finite unary algebras
Remarks on categorical equivalence of finite unary algebras 1. Background M. Krasner s original theorems from 1939 say that a finite algebra A (1) is an essentially multiunary algebra in which all operations
More informationTHE NUMBER OF POINTS IN A COMBINATORIAL GEOMETRY WITH NO 8-POINT-LINE MINORS
THE NUMBER OF POINTS IN A COMBINATORIAL GEOMETRY WITH NO 8-POINT-LINE MINORS JOSEPH E. BONIN AND JOSEPH P. S. KUNG ABSTRACT. We show that when n is greater than 3, the number of points in a combinatorial
More informationA GUIDE FOR MORTALS TO TAME CONGRUENCE THEORY
A GUIDE FOR MORTALS TO TAME CONGRUENCE THEORY Tame congruence theory is not an easy subject and it takes a considerable amount of effort to understand it. When I started this project, I believed that this
More informationAND SOME APPLICATIONS
A q-analogue OF THE FKG INEQUALITY AND SOME APPLICATIONS ANDERS BJÖRNER arxiv:0906.389v2 [math.co] 27 Jun 2009 Abstract. Let L be a finite distributive lattice and µ : L R + a logsupermodular function.
More informationTHE STRUCTURE OF RAINBOW-FREE COLORINGS FOR LINEAR EQUATIONS ON THREE VARIABLES IN Z p. Mario Huicochea CINNMA, Querétaro, México
#A8 INTEGERS 15A (2015) THE STRUCTURE OF RAINBOW-FREE COLORINGS FOR LINEAR EQUATIONS ON THREE VARIABLES IN Z p Mario Huicochea CINNMA, Querétaro, México dym@cimat.mx Amanda Montejano UNAM Facultad de Ciencias
More informationSpectra of Semidirect Products of Cyclic Groups
Spectra of Semidirect Products of Cyclic Groups Nathan Fox 1 University of Minnesota-Twin Cities Abstract The spectrum of a graph is the set of eigenvalues of its adjacency matrix A group, together with
More informationMATH5011 Real Analysis I. Exercise 1 Suggested Solution
MATH5011 Real Analysis I Exercise 1 Suggested Solution Notations in the notes are used. (1) Show that every open set in R can be written as a countable union of mutually disjoint open intervals. Hint:
More informationAN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES
AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov
More informationarxiv: v1 [math.co] 16 Feb 2018
CHAIN POSETS arxiv:1802.05813v1 [math.co] 16 Feb 2018 IAN T. JOHNSON Abstract. A chain poset, by definition, consists of chains of ordered elements in a poset. We study the chain posets associated to two
More informationLecture 4: Stabilization
Lecture 4: Stabilization There are many stabilization processes in topology, and often matters simplify in a stable limit. As a first example, consider the sequence of inclusions (4.1) S 0 S 1 S 2 S 3
More informationSum of dilates in vector spaces
North-Western European Journal of Mathematics Sum of dilates in vector spaces Antal Balog 1 George Shakan 2 Received: September 30, 2014/Accepted: April 23, 2015/Online: June 12, 2015 Abstract Let d 2,
More informationMULTIPLICITIES OF MONOMIAL IDEALS
MULTIPLICITIES OF MONOMIAL IDEALS JÜRGEN HERZOG AND HEMA SRINIVASAN Introduction Let S = K[x 1 x n ] be a polynomial ring over a field K with standard grading, I S a graded ideal. The multiplicity of S/I
More informationarxiv: v1 [math.fa] 14 Jul 2018
Construction of Regular Non-Atomic arxiv:180705437v1 [mathfa] 14 Jul 2018 Strictly-Positive Measures in Second-Countable Locally Compact Non-Atomic Hausdorff Spaces Abstract Jason Bentley Department of
More informationCONVERGENCE THEOREM FOR FINITE MARKOV CHAINS. Contents
CONVERGENCE THEOREM FOR FINITE MARKOV CHAINS ARI FREEDMAN Abstract. In this expository paper, I will give an overview of the necessary conditions for convergence in Markov chains on finite state spaces.
More informationTESTING FOR A SEMILATTICE TERM
TESTING FOR A SEMILATTICE TERM RALPH FREESE, J.B. NATION, AND MATT VALERIOTE Abstract. This paper investigates the computational complexity of deciding if a given finite algebra is an expansion of a semilattice.
More informationD-bounded Distance-Regular Graphs
D-bounded Distance-Regular Graphs CHIH-WEN WENG 53706 Abstract Let Γ = (X, R) denote a distance-regular graph with diameter D 3 and distance function δ. A (vertex) subgraph X is said to be weak-geodetically
More informationVC-DENSITY FOR TREES
VC-DENSITY FOR TREES ANTON BOBKOV Abstract. We show that for the theory of infinite trees we have vc(n) = n for all n. VC density was introduced in [1] by Aschenbrenner, Dolich, Haskell, MacPherson, and
More informationChapter 1. Measure Spaces. 1.1 Algebras and σ algebras of sets Notation and preliminaries
Chapter 1 Measure Spaces 1.1 Algebras and σ algebras of sets 1.1.1 Notation and preliminaries We shall denote by X a nonempty set, by P(X) the set of all parts (i.e., subsets) of X, and by the empty set.
More informationDISTRIBUTIVE LATTICES ON GRAPH ORIENTATIONS
DISTRIBUTIVE LATTICES ON GRAPH ORIENTATIONS KOLJA B. KNAUER ABSTRACT. Propp gave a construction method for distributive lattices on a class of orientations of a graph called c-orientations. Given a distributive
More information1 Take-home exam and final exam study guide
Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number
More informationG. de Cooman E. E. Kerre Universiteit Gent Vakgroep Toegepaste Wiskunde en Informatica
AMPLE FIELDS G. de Cooman E. E. Kerre Universiteit Gent Vakgroep Toegepaste Wiskunde en Informatica In this paper, we study the notion of an ample or complete field, a special case of the well-known fields
More informationFunctional BKR Inequalities, and their Duals, with Applications
Functional BKR Inequalities, and their Duals, with Applications Larry Goldstein and Yosef Rinott University of Southern California, Hebrew University of Jerusalem April 26, 2007 Abbreviated Title: Functional
More informationON k-subspaces OF L-VECTOR-SPACES. George M. Bergman
ON k-subspaces OF L-VECTOR-SPACES George M. Bergman Department of Mathematics University of California, Berkeley CA 94720-3840, USA gbergman@math.berkeley.edu ABSTRACT. Let k L be division rings, with
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More informationDefinition 2.3. We define addition and multiplication of matrices as follows.
14 Chapter 2 Matrices In this chapter, we review matrix algebra from Linear Algebra I, consider row and column operations on matrices, and define the rank of a matrix. Along the way prove that the row
More informationAn Investigation on an Extension of Mullineux Involution
An Investigation on an Extension of Mullineux Involution SPUR Final Paper, Summer 06 Arkadiy Frasinich Mentored by Augustus Lonergan Project Suggested By Roman Bezrukavnikov August 3, 06 Abstract In this
More informationNotes on ordinals and cardinals
Notes on ordinals and cardinals Reed Solomon 1 Background Terminology We will use the following notation for the common number systems: N = {0, 1, 2,...} = the natural numbers Z = {..., 2, 1, 0, 1, 2,...}
More informationarxiv: v2 [math.pr] 26 Jun 2017
Existence of an unbounded vacant set for subcritical continuum percolation arxiv:1706.03053v2 math.pr 26 Jun 2017 Daniel Ahlberg, Vincent Tassion and Augusto Teixeira Abstract We consider the Poisson Boolean
More informationOperators with numerical range in a closed halfplane
Operators with numerical range in a closed halfplane Wai-Shun Cheung 1 Department of Mathematics, University of Hong Kong, Hong Kong, P. R. China. wshun@graduate.hku.hk Chi-Kwong Li 2 Department of Mathematics,
More informationIDEMPOTENT n-permutable VARIETIES
IDEMPOTENT n-permutable VARIETIES M. VALERIOTE AND R. WILLARD Abstract. One of the important classes of varieties identified in tame congruence theory is the class of varieties which are n-permutable for
More informationLog-Convexity Properties of Schur Functions and Generalized Hypergeometric Functions of Matrix Argument. Donald St. P. Richards.
Log-Convexity Properties of Schur Functions and Generalized Hypergeometric Functions of Matrix Argument Donald St. P. Richards August 22, 2009 Abstract We establish a positivity property for the difference
More informationOpen Research Online The Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs Functions of genus zero for which the fast escaping set has Hausdorff dimension two Journal Item
More informationAccumulation constants of iterated function systems with Bloch target domains
Accumulation constants of iterated function systems with Bloch target domains September 29, 2005 1 Introduction Linda Keen and Nikola Lakic 1 Suppose that we are given a random sequence of holomorphic
More informationDecomposition of Riesz frames and wavelets into a finite union of linearly independent sets
Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets Ole Christensen, Alexander M. Lindner Abstract We characterize Riesz frames and prove that every Riesz frame
More informationREPRESENTING CONGRUENCE LATTICES OF LATTICES WITH PARTIAL UNARY OPERATIONS AS CONGRUENCE LATTICES OF LATTICES. I. INTERVAL EQUIVALENCE
REPRESENTING CONGRUENCE LATTICES OF LATTICES WITH PARTIAL UNARY OPERATIONS AS CONGRUENCE LATTICES OF LATTICES. I. INTERVAL EQUIVALENCE G. GRÄTZER AND E. T. SCHMIDT Abstract. Let L be a bounded lattice,
More informationDEPARTMENT OF ECONOMICS DISCUSSION PAPER SERIES
ISSN 1471-0498 DEPARTMENT OF ECONOMICS DISCUSSION PAPER SERIES TESTING FOR PRODUCTION WITH COMPLEMENTARITIES Pawel Dziewulski and John K.-H. Quah Number 722 September 2014 Manor Road Building, Manor Road,
More informationPosets, homomorphisms and homogeneity
Posets, homomorphisms and homogeneity Peter J. Cameron and D. Lockett School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS, U.K. Abstract Jarik Nešetřil suggested
More informationGraph coloring, perfect graphs
Lecture 5 (05.04.2013) Graph coloring, perfect graphs Scribe: Tomasz Kociumaka Lecturer: Marcin Pilipczuk 1 Introduction to graph coloring Definition 1. Let G be a simple undirected graph and k a positive
More informationCOUNTABLY COMPLEMENTABLE LINEAR ORDERINGS
COUNTABLY COMPLEMENTABLE LINEAR ORDERINGS July 4, 2006 ANTONIO MONTALBÁN Abstract. We say that a countable linear ordering L is countably complementable if there exists a linear ordering L, possibly uncountable,
More informationbe any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UFD. Therefore
More informationRoth s Theorem on 3-term Arithmetic Progressions
Roth s Theorem on 3-term Arithmetic Progressions Mustazee Rahman 1 Introduction This article is a discussion about the proof of a classical theorem of Roth s regarding the existence of three term arithmetic
More informationUNIVERSALITY OF THE LATTICE OF TRANSFORMATION MONOIDS
UNIVERSALITY OF THE LATTICE OF TRANSFORMATION MONOIDS MICHAEL PINSKER AND SAHARON SHELAH Abstract. The set of all transformation monoids on a fixed set of infinite cardinality λ, equipped with the order
More informationNegative Dependence via the FKG Inequality
Negative Dependence via the FKG Inequality Devdatt Dubhashi Department of Computer Science and Engineering Chalmers University SE 412 96, Göteborg, SWEDEN dubhashi@cs.chalmers.se Abstract We show how the
More informationLines, parabolas, distances and inequalities an enrichment class
Lines, parabolas, distances and inequalities an enrichment class Finbarr Holland 1. Lines in the plane A line is a particular kind of subset of the plane R 2 = R R, and can be described as the set of ordered
More informationOn the Sensitivity of Cyclically-Invariant Boolean Functions
On the Sensitivity of Cyclically-Invariant Boolean Functions Sourav Charaborty University of Chicago sourav@csuchicagoedu Abstract In this paper we construct a cyclically invariant Boolean function whose
More informationAlgebraic function fields
Algebraic function fields 1 Places Definition An algebraic function field F/K of one variable over K is an extension field F K such that F is a finite algebraic extension of K(x) for some element x F which
More informationNon-trivial intersecting uniform sub-families of hereditary families
Non-trivial intersecting uniform sub-families of hereditary families Peter Borg Department of Mathematics, University of Malta, Msida MSD 2080, Malta p.borg.02@cantab.net April 4, 2013 Abstract For a family
More informationCOUNTABLE CHAINS OF DISTRIBUTIVE LATTICES AS MAXIMAL SEMILATTICE QUOTIENTS OF POSITIVE CONES OF DIMENSION GROUPS
COUNTABLE CHAINS OF DISTRIBUTIVE LATTICES AS MAXIMAL SEMILATTICE QUOTIENTS OF POSITIVE CONES OF DIMENSION GROUPS PAVEL RŮŽIČKA Abstract. We construct a countable chain of Boolean semilattices, with all
More informationBoolean Inner-Product Spaces and Boolean Matrices
Boolean Inner-Product Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver
More informationMath 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together
More informationwhere c R and the content of f is one. 1
9. Gauss Lemma Obviously it would be nice to have some more general methods of proving that a given polynomial is irreducible. The first is rather beautiful and due to Gauss. The basic idea is as follows.
More informationPCF THEORY AND CARDINAL INVARIANTS OF THE REALS
PCF THEORY AND CARDINAL INVARIANTS OF THE REALS LAJOS SOUKUP Abstract. The additivity spectrum ADD(I) of an ideal I P(I) is the set of all regular cardinals κ such that there is an increasing chain {A
More informationAn FKG equality with applications to random environments
Statistics & Probability Letters 46 (2000) 203 209 An FKG equality with applications to random environments Wei-Shih Yang a, David Klein b; a Department of Mathematics, Temple University, Philadelphia,
More information4. Ergodicity and mixing
4. Ergodicity and mixing 4. Introduction In the previous lecture we defined what is meant by an invariant measure. In this lecture, we define what is meant by an ergodic measure. The primary motivation
More informationOvergroups of Intersections of Maximal Subgroups of the. Symmetric Group. Jeffrey Kuan
Overgroups of Intersections of Maximal Subgroups of the Symmetric Group Jeffrey Kuan Abstract The O Nan-Scott theorem weakly classifies the maximal subgroups of the symmetric group S, providing some information
More informationRINGS IN POST ALGEBRAS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVI, 2(2007), pp. 263 272 263 RINGS IN POST ALGEBRAS S. RUDEANU Abstract. Serfati [7] defined a ring structure on every Post algebra. In this paper we determine all the
More informationNOTES ON CONGRUENCE n-permutability AND SEMIDISTRIBUTIVITY
NOTES ON CONGRUENCE n-permutability AND SEMIDISTRIBUTIVITY RALPH FREESE Abstract. In [1] T. Dent, K. Kearnes and Á. Szendrei define the derivative, Σ, of a set of equations Σ and show, for idempotent Σ,
More informationA strongly rigid binary relation
A strongly rigid binary relation Anne Fearnley 8 November 1994 Abstract A binary relation ρ on a set U is strongly rigid if every universal algebra on U such that ρ is a subuniverse of its square is trivial.
More informationThe Caratheodory Construction of Measures
Chapter 5 The Caratheodory Construction of Measures Recall how our construction of Lebesgue measure in Chapter 2 proceeded from an initial notion of the size of a very restricted class of subsets of R,
More informationThe cocycle lattice of binary matroids
Published in: Europ. J. Comb. 14 (1993), 241 250. The cocycle lattice of binary matroids László Lovász Eötvös University, Budapest, Hungary, H-1088 Princeton University, Princeton, NJ 08544 Ákos Seress*
More information5 Set Operations, Functions, and Counting
5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,
More informationAN ALGEBRAIC APPROACH TO GENERALIZED MEASURES OF INFORMATION
AN ALGEBRAIC APPROACH TO GENERALIZED MEASURES OF INFORMATION Daniel Halpern-Leistner 6/20/08 Abstract. I propose an algebraic framework in which to study measures of information. One immediate consequence
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More information