## Notes on lattice theory by Nation J.B.

By Nation J.B.

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Hence, if we can show that R is a congruence relation, it will follow that R = i∈I con(ai , bi ). Note that (x, y) ∈ R if and only if (x ∧ y, x ∨ y) ∈ R. It also helps to observe that if x R y and x ≤ u ≤ v ≤ y, then u R v. To see this, replace the weaving polynomials w(t) witnessing x R y by new polynomials w (t) = (w(t) ∨ u) ∧ v. First we must show R ∈ Eq L. Reflexivity and symmetry are obvious, so let x R y R z with x ∨ y = r0 ≥ r1 ≥ · · · ≥ rk = x ∧ y using polynomials wj ∈ W , and y ∨ z = s0 ≥ s1 ≥ · · · ≥ sm = y ∧ z via polynomials vj ∈ W , as in the statement of the theorem.

Schmidt [8]), or (iii) D has at most ℵ1 compact elements (A. Huhn [4]). 3 In Chapter 10 we will prove (i), which includes the fact that every finite distributive lattice is isomorphic to the congruence lattice of a (finite) lattice. We need to understand the congruence operator con Q, where Q is a set of pairs, a little better. A weaving polynomial on a lattice L is a member of the set W of unary functions defined recursively by (1) w(x) = x ∈ W , (2) if w(x) ∈ W and a ∈ L, then u(x) = w(x) ∧ a and v(x) = w(x) ∨ a are in W , (3) only these functions are in W .

The proof uses three basic principles of universal algebra. 5 respectively. However, the proofs of these theorems involved nothing special to lattices except the operation symbols ∧ and ∨; these can easily be changed to arbitrary operation symbols. Thus, with only minor modification, the proof of this theorem can be adapted to show the existence of free algebras in any nontrivial equational class of algebras. Basic Principle 1. If h : A B is a surjective homomorphism, then B ∼ = A/ ker h. Basic Principle 2.