The interval criterion checked on a three-element chain and a diamond

The interval partitions of $C$. By [F1] the intervals of $C$ are $\{0\},\{a\},\{1\},\{0,a\},\{a,1\}$ and $C$, so a partition of $C$ into intervals is a family of pairwise disjoint of these sets with union $C$. The possibilities are: the three singletons; $\{0,a\}$ with $\{1\}$; $\{0\}$ with $\{a,1\}$; and $C$ itself. There is no other: any such partition other than these would have to contain a two-element block $\{0,1\}$, but $\{0,1\}$ is not an interval because $0<1$ and $a$ lies strictly between, while $a\mathrm{\notin }\{0,1\}$. Hence there are exactly four interval partitions of $C$, the four listed in the Example.

The interval partitions of $D$. By [F2] the intervals of $D$ are the four singletons, the four two-element sets $\{0,a\},\{0,b\},\{a,1\},\{b,1\}$, and $D$. No three-element subset of $D$ is an interval: a block of the form $[d,u]$ contains its least element $d$ and its greatest element $u$, but $\{0,a,b\}$ has no greatest element and $\{a,b,1\}$ has no least element, while the two remaining three-element subsets $\{0,a,1\}$ and $\{0,b,1\}$ have least element $0$ and greatest element $1$ with $[0,1]=D$ strictly larger than either. A partition of $D$ into intervals therefore consists of the four singletons ($1$ way), or one two-element interval and two singletons ($4$ choices of the pair, the complement being two singletons), or two two-element intervals, of which the three pairings of $D$ contribute the admissible $\{0,a\}\mathrm{\mid }\{b,1\}$ and $\{0,b\}\mathrm{\mid }\{a,1\}$ but not $\{a,b\}\mathrm{\mid }\{0,1\}$ since neither $\{a,b\}$ nor $\{0,1\}$ is an interval, or $D$ itself ($1$ way): altogether $1+4+2+1=8$ interval partitions, the eight listed in the Example.

The four partitions of $C$ are congruences. For each of the four partitions of step 1.1 the endpoints $d(x)\le u(x)$ are the least and greatest members of the block of $x$: for the discrete partition $d=u=\mathrm{i}\mathrm{d}$, both maps order-preserving; for $\{0,a\}\mathrm{\mid }\{1\}$ one has $d(0)=d(a)=0$, $d(1)=1$ and $u(0)=u(a)=a$, $u(1)=1$, and both maps are nondecreasing along $0<a<1$; for $\{0\}\mathrm{\mid }\{a,1\}$ one has $d(0)=0$, $d(a)=d(1)=a$ and $u(0)=0$, $u(a)=u(1)=1$, again both nondecreasing; and for the all-one partition both maps are constant. By the criterion [F3] all four are lattice congruences. Their quotients, computed with [F4], are: $C$ for the discrete partition (the blocks are the elements and $[x]\vee [y]=[x\vee y]$ reproduces the order of $C$); the two-element chain $\{0,a\}<\{1\}$ for $\{0,a\}\mathrm{\mid }\{1\}$, since $0\vee 1=1$ makes $\{0,a\}\vee \{1\}=\{1\}$ and $\{0,a\}\le \{1\}$; the two-element chain $\{0\}<\{a,1\}$ for $\{0\}\mathrm{\mid }\{a,1\}$, since $0\vee 1=1$ makes $\{0\}\vee \{a,1\}=\{a,1\}$; and the one-element lattice for the all-one partition.

The criterion on the eight partitions of $D$. The endpoint maps of each partition of step 1.2 are read off from its blocks, and monotonicity is checked on the five comparable pairs $0\le a$, $0\le b$, $0\le 1$, $a\le 1$, $b\le 1$. The discrete partition has $d=u=\mathrm{i}\mathrm{d}$ and is monotone; the all-one partition has constant maps and is monotone; the partition $\{0,a\}\mathrm{\mid }\{b,1\}$ has $d(0)=d(a)=0$, $d(b)=d(1)=b$, $u(0)=u(a)=a$ and $u(b)=u(1)=1$, and on the five pairs: $0\le a$ gives $0\le 0$ and $a\le a$, $0\le b$ gives $0\le b$ and $a\le 1$, $0\le 1$ gives $0\le b$ and $a\le 1$, $a\le 1$ gives $0\le b$ and $a\le 1$, and $b\le 1$ gives $b\le b$ and $1\le 1$, so $d$ and $u$ are order-preserving; the partition $\{0,b\}\mathrm{\mid }\{a,1\}$ has $d(0)=d(b)=0$, $d(a)=d(1)=a$, $u(0)=u(b)=b$ and $u(a)=u(1)=1$, and on the five pairs: $0\le a$ gives $0\le a$ and $b\le 1$, $0\le b$ gives $0\le 0$ and $b\le b$, $0\le 1$ gives $0\le a$ and $b\le 1$, $a\le 1$ gives $a\le a$ and $1\le 1$, and $b\le 1$ gives $0\le a$ and $b\le 1$, so these maps are order-preserving too. These four partitions satisfy the criterion [F3]. The other four fail: $\{0,a\}\mathrm{\mid }\{b\}\mathrm{\mid }\{1\}$ has $u(0)=a$ and $u(b)=b$ with $0\le b$ but $a\le ̸b$; $\{0,b\}\mathrm{\mid }\{a\}\mathrm{\mid }\{1\}$ has $u(0)=b$ and $u(a)=a$ with $0\le a$ but $b\le ̸a$; $\{0\}\mathrm{\mid }\{a\}\mathrm{\mid }\{b,1\}$ has $d(a)=a$ and $d(1)=b$ with $a\le 1$ but $a\le ̸b$; and $\{0\}\mathrm{\mid }\{b\}\mathrm{\mid }\{a,1\}$ has $d(b)=b$ and $d(1)=a$ with $b\le 1$ but $b\le ̸a$. Every lattice congruence of $D$ has interval classes by [F4], so its partition occurs in step 1.2. Hence the four partitions satisfying [F3] are exactly the lattice congruences of $D$.

Quotients and the count. By [F4] the quotient of $\{0,a\}\mathrm{\mid }\{b,1\}$ is the two-element chain $\{0,a\}<\{b,1\}$: the blocks are distinct, and taking the representatives $0\in \{0,a\}$ and $b\in \{b,1\}$ the class operations give $\{0,a\}\vee \{b,1\}=[0\vee b]=[b]=\{b,1\}$ and $\{0,a\}\wedge \{b,1\}=[0\wedge b]=[0]=\{0,a\}$, so the order is total on the two classes. The quotient of $\{0,b\}\mathrm{\mid }\{a,1\}$ is likewise the two-element chain $\{0,b\}<\{a,1\}$, by the representatives $0$ and $a$. The quotient of the discrete partition is $D$ and the quotient of the all-one partition is the one-element lattice. Therefore $D$ has exactly four lattice congruences and the congruence lattice of $D$ has exactly four elements: order congruences by refinement, meaning that each class of the finer congruence is contained in a class of the coarser one. The discrete congruence is the least, the all-one congruence is the greatest, and the two 2+2 congruences are incomparable (one identifies $0$ with $a$ but not $b$, and the other identifies $0$ with $b$ but not $a$). Thus this order is a diamond: the two middle elements have meet the discrete congruence and join the all-one congruence; comparable pairs have meet the smaller and join the larger. This proves directly that the four congruences form a lattice. Together with steps 1.1, 1.2, 2.1 and 2.2 this verifies every claim of the Example: the four interval partitions of $C$ and their quotients, the eight interval partitions of $D$, the four surviving the criterion with their explicit failures, and the four-element congruence lattice of $D$.