Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 14] ORDERED SETS AND LATTICES 363

LATTICES

14.65. Consider the latticeLin Fig. 14-19(a).

(a) Find all sublattices with five elements. (c) Find complements ofaandb, if they exist.

(b) Find all join-irreducible elements and atoms. (d) IsLdistributive? Complemented?

Fig. 14-19

14.66. Consider the latticeMin Fig. 14-19(b).

(a) Find all join-irreducible elements.

(b) Find the atoms.

(c) Find complements ofaandb, if they exist.

(d) Express eachxinMas the join of irredundant join-irreducible elements.

(e) IsMdistributive? Complemented?

14.67. Consider the bounded latticeLin Fig. 14-20(a).

(a) Find the complements, if they exist, ofeandf.

(b) ExpressIin an irredundant join-irreducible decomposition in as many ways as possible.

(c) IsLdistributive?

(d) Describe the isomorphisms ofLwith itself.

Fig. 14-20

14.68. Consider the bounded latticeLin Fig. 14-20(b).

(a) Find the complements, if they exist, ofaandc.

(b) ExpressIin an irredundant join-irreducible decomposition in as many ways as possible.

(c) IsLdistributive?

(d) Describe the isomorphisms ofLwith itself.