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What is a lattice? Which of the following graphs are lattice and why?
Ans: Let (L, ≤) be a poset. If each subset {x, y} consisting of any two elements of L, comprises a glb (Infimum) and a lub (Supremum), then the poset (L, ≤) is known as a lattice. A glb ({x, y}) is denoted by x∧y and it is called meet of x and y. Likewise, lub ({x, y}) is denoted by x∨y and it is called join of x and y. Hence, lattice is a mathematical structure equipped along with two binary operations meet and join.
In the specified examples, (a) is a lattice as each pair of elements has a meet and join in the set within the relation denoted by the graph. Graph in (b) does not denote a lattice as bottom two elements comprise no meet and top two elements have no join.
In the case of (c), the relation denoted is not even anti symmetric as two bottoms and two top level elements are at similar level and denoted as related to each other (symmetric) with no being similar element (equal).
Let R be the relation on S = {1, 2, 3, 4, 5} defined by R = {(1,3); (1, 1); (3, 1); (1, 2); (3, 3); (4, 4)}. (b) Write down the matrix of R. (c) Draw the digraph of R.
what is the answer
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