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Miklós Schweitzer
1993 Miklós Schweitzer
4
4
Part of
1993 Miklós Schweitzer
Problems
(1)
iterated function
Source: miklos schweitzer 1993 q4
10/22/2021
Let f be a ternary operation on a set of at least four elements for which (1)
f
(
x
,
x
,
y
)
≡
f
(
x
,
y
,
x
)
≡
f
(
x
,
y
,
y
)
≡
x
f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x
f
(
x
,
x
,
y
)
≡
f
(
x
,
y
,
x
)
≡
f
(
x
,
y
,
y
)
≡
x
(2)
f
(
x
,
y
,
z
)
=
f
(
y
,
z
,
x
)
=
f
(
y
,
x
,
z
)
∈
{
x
,
y
,
z
}
f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}
f
(
x
,
y
,
z
)
=
f
(
y
,
z
,
x
)
=
f
(
y
,
x
,
z
)
∈
{
x
,
y
,
z
}
for pairwise distinct x,y,z. Prove that f is a nontrivial composition of g such that g is not a composition of f. (The n-variable operation g is trivial if
g
(
x
1
,
.
.
.
,
x
n
)
≡
x
i
g(x_1, ..., x_n) \equiv x_i
g
(
x
1
,
...
,
x
n
)
≡
x
i
for some i (
1
≤
i
≤
n
1 \leq i \leq n
1
≤
i
≤
n
) )
algebra
function