Congruence Lattices 101

This lecture—based on the author's book, General Lattice Theory , Birkhäuser Verlag, 1978—briefly introduces the basic concepts of lattice theory, as needed for the lecture " Some combinatorial aspects of congruence lattice representations ". 1.1. Posets. A partially ordered set A; ≤≤ consists of a nonvoid set A and a binary relation ≤ on A such that the relation ≤ satisfies properties (P1)–(P3) for all a, b, c ∈ A: (P1) Reflexivity: a ≤ a. (P2) Antisymmetry: a ≤ b and b ≤ a imply that a = b. (P3) Transitivity: a ≤ b and b ≤ c imply that a ≤ c. A poset A; ≤≤ that also satisfies: (P4) Linearity: a ≤ b or b ≤ a. is called a chain (also called fully ordered set, linearly ordered set, and so on). The length, l(C), of a finite chain C is |C| − 1. Let C n denote the set {0,. .. , n − 1} ordered by 0 < 1 < 2 < · · · < n−1; then C n is an n-element chain and l(C n) = n−1. Let H ⊆ P and a ∈ P. The element a is an upper bound of H iff h ≤ a for all h ∈ H. An upper bound a of H is the least upper bound (supremum) of H iff, for any upper bound b of H, we have a ≤ b. We shall write a = sup H or a = H. The concepts of lower bound and greatest lower bound (infimum) are similarly defined; the latter is denoted by inf H or H. 1.2. Lattices. A poset L; ≤≤ is a lattice iff inf{a, b} and sup{a, b} exist for all a, b ∈ L. We will use the notation a ∧ b = inf{a, b}, a ∨ b = sup{a, b}, and call ∧ the meet and ∨ the join. In lattices, they are both binary operations, which means that they can be applied to a pair of elements a, b of L to yield again an element of L.