Discrete Structures - CB0246 Lattices

Transcription

1 Discrete Structures - CB0246 Lattices Andrés Sicard-Ramírez EAFIT University Semester

2 Lattices from the Partial Orders Theory Definition (Lattice) A lattice (retículo) is a poset where every pair of elements has both a supremum and an infimum. Discrete Structures - CB0246. Lattices 2/31

3 Lattices from the Partial Orders Theory Definition (Lattice) A lattice (retículo) is a poset where every pair of elements has both a supremum and an infimum. Example The following poset is a lattice. e b d c f a Discrete Structures - CB0246. Lattices 3/31

4 Lattices from the Partial Orders Theory Example (counter-example) The following poset is not a lattice because the upper bounds of the pair {b, c} are d, e and f, but this set has not a least upper bound. f d e b c a Discrete Structures - CB0246. Lattices 4/31

5 Lattices from the Partial Orders Theory Example (counter-example) The following poset is not a lattice because for example, the pair {1, 2} has not supremum n Discrete Structures - CB0246. Lattices 5/31

6 Lattices from the Partial Orders Theory Examples (Z +, ) is a lattice where the supremum is the least common multiple and the infimum is the greatest common divisor. Discrete Structures - CB0246. Lattices 6/31

7 Lattices from the Partial Orders Theory Examples (Z +, ) is a lattice where the supremum is the least common multiple and the infimum is the greatest common divisor. Let A be a set. Is (P (A), ) a lattice? Discrete Structures - CB0246. Lattices 7/31

8 Algebraic Structures Definition (Algebraic structure) An algebraic structure on a set A 0 is essentially a collection of n-ary operations on A. 1 Example (Semigroup) A semigroup (S, ) is a set S with an associative binary operation S S S. Example (Monoid) A monoid (M,, ε) is a semigroup (M, ) with an element ε M which is an unit for, i.e. ( x)(x ε = ε x = x). 1 Cohn, P. M. (1981). Universal Algebra, p. 41. Discrete Structures - CB0246. Lattices 8/31

9 Lattices from the Algebraic Structures Theory Definition (Lattice) Let and be two binaries operations, called meet and join, respectively. A lattice (retículo) is an algebraic structure (L,, ), which satisfy the following axioms for all x, y and z in L: 2 x y = y x x y = y x (x y) z = x (y z) (x y) z = x (y z) x (x y) = x x (x y) = x (Commutative laws) (Associative laws) (Absortion laws) 2 Lipschutz, Seymour and Lipson, Marc Lars (2007). Discrete Mathematics. Schaum s Outline Series. 3rd ed. Discrete Structures - CB0246. Lattices 9/31

10 Lattices from the Algebraic Structures Theory Example Let A be a set. (P (A),, ) is a lattice. Discrete Structures - CB0246. Lattices 10/31

11 Lattices from the Algebraic Structures Theory Definition (Dual) The dual of any statement in a lattice (L,, ) is the statement obtained by interchanging and. 3 Lipschutz, Seymour and Lipson, Marc Lars (2007). Discrete Mathematics. Schaum s Outline Series. 3rd ed. Discrete Structures - CB0246. Lattices 11/31

12 Lattices from the Algebraic Structures Theory Definition (Dual) The dual of any statement in a lattice (L,, ) is the statement obtained by interchanging and. Example The dual of x (y x) = x x is x (y x) = x x. 3 Lipschutz, Seymour and Lipson, Marc Lars (2007). Discrete Mathematics. Schaum s Outline Series. 3rd ed. Discrete Structures - CB0246. Lattices 12/31

13 Lattices from the Algebraic Structures Theory Definition (Dual) The dual of any statement in a lattice (L,, ) is the statement obtained by interchanging and. Example The dual of x (y x) = x x is x (y x) = x x. Theorem (Principle of duality) The dual of any theorem in a lattice is also an theorem. 3 3 Lipschutz, Seymour and Lipson, Marc Lars (2007). Discrete Mathematics. Schaum s Outline Series. 3rd ed. Discrete Structures - CB0246. Lattices 13/31

14 Lattices from the Algebraic Structures Theory Definition (Dual) The dual of any statement in a lattice (L,, ) is the statement obtained by interchanging and. Example The dual of x (y x) = x x is x (y x) = x x. Theorem (Principle of duality) The dual of any theorem in a lattice is also an theorem. 3 Proof. The dual of every axiom in a lattice is also an axiom. Hence, the dual theorem can be proved by using the dual of each step of the proof of the original theorem. 3 Lipschutz, Seymour and Lipson, Marc Lars (2007). Discrete Mathematics. Schaum s Outline Series. 3rd ed. Discrete Structures - CB0246. Lattices 14/31

15 Lattices from the Algebraic Structures Theory Example Let (L,, ) be a lattice. Prove the idempotent laws x x = x, (1) x x = x. (2) Discrete Structures - CB0246. Lattices 15/31

16 Lattices from the Algebraic Structures Theory Example Let (L,, ) be a lattice. Prove the idempotent laws x x = x, (1) x x = x. (2) Proof of (1). x x = x (x (x y)) (second absortion law) = x (first absortion law) Discrete Structures - CB0246. Lattices 16/31

17 Lattices from the Algebraic Structures Theory Example Let (L,, ) be a lattice. Prove the idempotent laws x x = x, (1) x x = x. (2) Proof of (1). x x = x (x (x y)) (second absortion law) = x (first absortion law) Proof of (2). By principle of duality on (1). Discrete Structures - CB0246. Lattices 17/31

18 Lattices from the Algebraic Structures Theory Exercise (Rosen (2004), Problem 40, p. 500) Prove that if x and y are elements of a lattice (L,, ) then x y = y, if and only if, x y = x. Discrete Structures - CB0246. Lattices 18/31

19 Lattices from the Algebraic Structures Theory Exercise (Rosen (2004), Problem 40, p. 500) Prove that if x and y are elements of a lattice (L,, ) then x y = y, if and only if, x y = x. Proof. Let s suppose x y = y. Then x = x (x y) (first absortion law) = x y (hypothesis) Discrete Structures - CB0246. Lattices 19/31

20 Lattices from the Algebraic Structures Theory Exercise (cont.) Proof. Let s suppose x y = x. Then y = y (y x) (second absortion law) = y (x y) (commutative law) = y x (hypothesis) = x y (commutative law) Discrete Structures - CB0246. Lattices 20/31

21 Equivalence of the Definitions Theorem Let (L,, ) be a lattice. Then (L, ) is a partial order, where the relation is defined by 4 x y def = x y = x. 4 Lipschutz, Seymour and Lipson, Marc Lars (2007). Discrete Mathematics. Schaum s Outline Series. 3rd ed. Discrete Structures - CB0246. Lattices 21/31

22 Equivalence of the Definitions Theorem Let (L,, ) be a lattice. Then (L, ) is a partial order, where the relation is defined by 4 Proof. 1. The relation is reflexive x y def = x y = x. x x = x (idempotency), for all x L. Therefore x x, for all x L. 4 Lipschutz, Seymour and Lipson, Marc Lars (2007). Discrete Mathematics. Schaum s Outline Series. 3rd ed. Discrete Structures - CB0246. Lattices 22/31

23 Equivalence of the Definitions Proof (cont.) 2. The relation is antisymmetric Suppose x y and y x, then x y = x and y x = y. Therefore x = x y (hypothesis) = y x (commutative law) = y (hypothesis) That is, is antisymmetric. Discrete Structures - CB0246. Lattices 23/31

24 Equivalence of the Definitions Proof (cont.) 3. The relation is transitive Suppose x y and y z, then x y = x and y z = y. Therefore x z = (x y) z (hypothesis) = x (y z) (associativity law) = x y (hypothesis) = x (hypothesis) That is, x z. Discrete Structures - CB0246. Lattices 24/31

25 Equivalence of the Definitions Remark Let (L,, ) be a lattice and let be (L, ) the order partial induced by (L,, ). It is possible prove that (L, ) is a lattice. Discrete Structures - CB0246. Lattices 25/31

26 Equivalence of the Definitions Theorem (Rosen (2004), Problem 39, p. 500) Let (L, ) be a lattice. Then (L,, ) is a lattice, where x y def = inf(x, y), x y def = sup(x, y), Discrete Structures - CB0246. Lattices 26/31

27 Equivalence of the Definitions Theorem (Rosen (2004), Problem 39, p. 500) Let (L, ) be a lattice. Then (L,, ) is a lattice, where Proof. x y def = inf(x, y), x y def = sup(x, y), 1. Commutative laws for and (Rosen s solution). Because inf(x, y) = inf(y, x) and sup(x, y) = sup(y, x), it follows that x y = y x and x y = y x. Discrete Structures - CB0246. Lattices 27/31

28 Equivalence of the Definitions Proof (cont.) 2. Associative laws for and (Rosen s solution). Using the definition, (x y) z is a lower bound of x, y and z that is greater than every other lower bound. Because x, y and z play interchangeable roles, x (y z) is the same element. Discrete Structures - CB0246. Lattices 28/31

29 Equivalence of the Definitions Proof (cont.) 2. Associative laws for and (Rosen s solution). Using the definition, (x y) z is a lower bound of x, y and z that is greater than every other lower bound. Because x, y and z play interchangeable roles, x (y z) is the same element. Similarly, (x y) z is an upper bound of x, y and z that is less than every other upper bound. Because x, y and z play interchangeable roles, x (y z) is the same element. Discrete Structures - CB0246. Lattices 29/31

30 Equivalence of the Definitions Proof (cont.) 3. Absortion laws for and (Rosen s solution). To show that x (x y) = x it is sufficient to show that x is the greatest lower bound of x, and x y. Note that x is a lower bound of x, and because x y is by definition greater than x, x is a lower bound for it as well. Therefore, x is a lower bound. But any lower bound of x has to be less than x, so x is the greatest lower bound. The second statement is the dual of the first; we omit its proof. Discrete Structures - CB0246. Lattices 30/31

31 References Cohn, P. M. (1981). Universal Algebra. Revised edition. D. Reidel Publishing Company. Lipschutz, Seymour and Lipson, Marc Lars (2007). Discrete Mathematics. 3rd ed. Schaum s Outline Series. McGraw-Hill. Rosen, Kenneth H. (2004). Matemática Discreta y sus Aplicaciones. 5th ed. McGraw-Hill. Discrete Structures - CB0246. Lattices 31/31