7. Congruence relations on lattices

7.1 Principles For $\theta \in$   Con$(A)$:

(1) If $a \equiv b \!\! \pmod \theta$, then $a \wedge b \equiv a \vee b \!\! \pmod \theta$.

(2) If $a \leq t \leq b$ and $a \equiv b \!\! \pmod \theta$, then $t \equiv a \equiv b\! \pmod \theta$.

(3) If $a \wedge b \equiv a \!\! \pmod \theta$, then $b \equiv a \vee b \!\! \pmod \theta$, and dually.

(4) If $a \equiv b \!\! \pmod \theta\;$ and $b \equiv c \!\! \pmod \theta$, then $a \equiv c\!\! \pmod \theta$.

7.2 Theorems

(A) A nonempty relation $\theta$ on a lattice is a congruence relation if and only if $\theta$ satisfies (1) through (4).

(B) For elements $a _ 0, b _ 0$ of a lattice $L$,    con$(a _ 0, b _ 0)$, the smallest congruence relation on $L$ that identifies $a _ 0$ and $b _ 0$, can be constructed by applying (1) (unless $a _ 0 \leq b _ 0$ already), then (2) and (3) repeatedly, and then (4) repeatedly. This is the principal congruence relation con$(a,b)$ (lower-case c).

For examples to try, see Figure . Congruence relations can be indicated by darkening each covering between two elements in the same block.

Figure: Some lattices for which to find congruence lattices