Question

Problem 1 Let (A,) and (B, 3) be posets. Consider A x B as a poset under the product order - that is, (a, b) = (a',V) if and only if a < d' and b<W. (a) Suppose IE A is a minimal element and ye B is a minimal element. Prove that (2,4) is a minimal element of Ax B. (3 pts) (b) Suppose I E A is a maximal element and y E B is a maximal element. Prove that (2,y) is a maximal element of Ax B. (3 pts) (c) Let ai, 22 € A and bi, b2 € B. Suppose that a = inf{Q1, a2} is in A and b = inf{b1,b2} is in B. Prove that inf{(qı, bı), (az, b2)} = (a,b) E A x B. (5 pts) (d) Let aj, az E A and bi, b2 € B. Suppose that a = sup{ai, a2} is in A and b = sup{bı, b2} is in B. Prove that sup{(,61), (Q2, 62)} = (a,b) E A x B. (5 pts) (e) Given what you proved in the previous parts: if A is a lattice and B is a lattice, does it follow that Ax B is a lattice? (4 pts)

Answer 1

ut (1, 2) and (B 5) be poset AXR as a Poset defined as (a,b) (a',6') # asal and bab! a. lit X4A and YER is a minimal Element. 10 Prove: (ny) is a minimal element of AXB It caibo + AXB xca and ycb (my)< (2,6) (ny) is the minimal element of AXR b. Luxe A cd yе в камак imag Elean TO Prove (n.) is masinal element of AXB et a & (a, b) E AXB aca and b sy (9,6)< (8,4) (2,4) is the maximal euren the maximal eument of AXB

c. et a, , A, E A and bi, bet B a= Infşa , 027 an is in A and be inf&b, bez is in B to prove: infş (a,bi), Can, bz) } = (a,b) EAXO since a = infşa, A₂ } & ac a, and a la and b = Inf & bi, ba? & be 6, and b<b, from (1) and (2) * (a,b) < (a, b,) and Caib) L (a₂, ba) * a (a, b) = Inf{ (a, b), (92, bz) ? Int { (a, b) (a₂,6)} = (a,b) E AXB Hence Proved.

d. Suppose and To Prove: a = sup sa a, b is ma b = sup & B, ba? is in Sup { (arobi), Caz, 62 ) } = (a,b) E AXB since a= sup şa, A2} & aca and as a new b: sup & bi, be? & bish and by < b ② from equeation (1) and (2) & (a, b) 5 (a,b) and (a2, bz) < (a, b). Supę (a, b), (a, b)} = (a,b) € AXB e. Yes , It is a lattice and B is a lattice 6 then AXB is a lattice for every pair of (ab), ((x8 (916) (Cd) EAX BE Sup $19.6), (C, d)} and inf {(9,6) (Cd)} Exists