Answer to Question #280711 in Discrete Mathematics for Ashwini

Question

Draw the Hasse diagram for the lattice (D₂4, 1). Hence find. a) maximal & minimal element b) glb of 446 geb of 8 & 2 d) lub of 8 & 6

Answer 1

Solution:

Given, (D₂₄, /)

D is the set of all positive divisors of 24.

So, D={1, 2, 3, 4, 6, 8, 12, 24} 

poset A = {(1/2), (1/3), (1/4), (1/6), (1/8),(1/12), (1/24), (2/4), (2/6), (2/8), (2/12), (2/24), (3/6), (3/12), (3/24), (4/8), (4/12), (4/24), (6/12), (6/24)} 

So, now the Hasse diagram will be-

Maximal element is an element of a POSET which is not less than any other element of the POSET. Or we can say that it is an element which is not related to any other element. Top elements of the Hasse Diagram.

Minimal element is an element of a POSET which is not greater than any other element of the POSET. Or we can say that no other element is related to this element. Bottom elements of the Hasse Diagram.

Greatest element(glb) (if it exists) is the element succeeding all other elements.

Least element(lub) is the element that precedes all other elements.

Now, using these definitions & from the hasse diagram:

(a) maximal element = 24, minimal element = 1

(b) glb of 4 & 6 = not exists

(c) glb of 8 & 2 = 2

(d) lub of 8 & 6 = 8