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awarded  Curious
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awarded  Nice Question
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Sep
24
comment Boolean algebras without atoms
You are right. I had missed that part. Nice proof.
Sep
24
comment Boolean algebras without atoms
I am not sure this proof works. The $y_{i,j}$ may not be all distinct, for different values of $i$. And similarly for the $w_{i,j}$. But there is no a priori guarantee that the equalities between the $y_{i,j}$ match exactly the equalities between the $w_{i,j}$. So it's not clear that $\pi$ is well defined.
Sep
6
comment nonisomorphic groups whose quotients are isomorphic
Like you say, subgroups can be embedded in all sorts of ways into other groups. So if $C$ and $C'$ are isomorphic but not exactly the same, it's a simple matter of modifying $A$ and $B$ and identifying elements in certain ways to make $C$ and $C'$ actually identical. See Ittay Weiss's answer for example.
Sep
6
answered nonisomorphic groups whose quotients are isomorphic
Sep
2
answered Non-isomorphic countable Boolean algebras
Jun
30
revised Show that $\langle G^+\rangle=G$ in a directed group
Subgroup was meant here
Jun
30
suggested suggested edit on Show that $\langle G^+\rangle=G$ in a directed group
May
15
answered Find a lattice with exactly three congruence relations
May
15
comment Find a lattice with exactly three congruence relations
Thanks for the example.
May
15
accepted Find a lattice with exactly three congruence relations
May
14
comment Find a lattice with exactly three congruence relations
You may be on to something. It seems that the lattice $M_3$ with 5 elements $a$, $b$, $c$ and $0$, $1$ that you mention has only the trivial congruences. So maybe a variation of this would have a single non-trivial congruence?
May
14
comment Find a lattice with exactly three congruence relations
In general, each congruence class of a congruence relation on a lattice is a sublattice, i.e., closed under $\lor$ and $\land$.
May
14
comment Find a lattice with exactly three congruence relations
@user69810 Because $a\sim b$ implies $a\lor a\sim b\lor a$, that is, $a\sim 1$.
May
14
awarded  Caucus
May
14
comment Find a lattice with exactly three congruence relations
@vadim123 If the diamond lattice has 4 elements $0$, $a$, $b$, $1$ with $0$ as minimum element, $1$ as maximum element, and $a$ and $b$ in between and not comparable, one congruence relation has the two blocks $\{0,a\}$ and $\{b,1\}$. The other congruence relation has the blocks $\{0,b\}$ and $\{a,1\}$.
May
13
asked Find a lattice with exactly three congruence relations