Modular, distributive and Boolean lattices

Abstract

In Chapter 2 we began an exploration of the algebraic theory of lattices, armed with enough axioms on ∨ and ∧ to ensure that each lattice 〈 L ; ∨, ∧〉 arose from a lattice 〈 L ;≤〉 and vice versa. Now we introduce identities linking join and meet which are not implied by the laws (L1)–(L4) and their duals (L1) ∂ –(L4) ∂ defining lattices (recall 2.9). These hold in many of our examples of lattices, in particular in powersets. In the second part of the chapter we abstract a different feature of powersets, namely the existence of complements. Lattices satisfying additional identities Before formally introducing modular and distributive lattices we prove three lemmas which will put the definitions in 4.4 into perspective. The import of these lemmas is discussed in 4.5. Lemma. Let L be a lattice and let a,b,c ∈ L. Then (i) a ∧ ( b ∨ c ) ≥ ( a ∧ b ) ∨ ( a ∧ c ), and dually , (ii) a ≥ c implies a ∧ ( b ∨ c ) ≥ ( a ∧ b ) ∨ c, and dually , (iii) ( a ∧ b ) ∨ ( c ) ∨ c ) ∧ ( a ) ≤ ( a ∨ b ) ∧ ( b ∨ c ) ∧ ( c ∨ a ). Proof. We leave (i) and (iii) as exercises. (Alternatively, see Exercise 2.9.) By the Connecting Lemma, (ii) is a special case of (i).