A ring
(
A
;
+
,
⋅
)
(A; + , \cdot )
is constructed such that the congruence lattice
L
A
{L_A}
of the ring
(
A
;
+
,
⋅
)
(A; + , \cdot )
is distributive, the elements of
L
A
{L_A}
are pairwise permutable and there is no
L
A
{L_A}
-compatible function
p
p
on
A
A
such that
\[
p
(
a
,
b
,
b
)
=
p
(
a
,
b
,
a
)
=
p
(
b
,
b
,
a
)
=
a
for
all
a
,
b
∈
A
.
p(a,b,b) = p(a,b,a) = p(b,b,a) = a\quad {\text {for}}\;{\text {all}}\;a,b \in A.
\]
(1)