
24 Chapter 1 First concepts
(a) If fM , g is a lattice, then fS, g is a sublattice.
(b) If fM , g is a complete lattice and
S :D S [fi.M/g,thenfS, g is a complete
sublattice.
Proof. (a) Let x, y 2 S.Thenx a and y a and therefore inf
M
fx, yg
sup
M
fx, yga, i.e., inf
M
fx, yg,sup
M
fx, yg2S, which proves the assertion
by Corollary 1.16.
(b) Let ¿ ¤ A S. Then for all x 2 S, x a, and therefore inf
M
S sup
M
S
a, i.e., inf
M
S,sup
M
S 2 S .But¿ S also. Then in order to apply Corol-
lary 1.16, inf
M
¿ D i.M/ and sup
M
¿ D o.M/ must be elements of S.Since
o.M/ a, it is automatically an element of S and therefore of
S .Thati.M/ 2 S,
however, has been assumed ...