Let $x$ and $y$ denote variables over the set $\{a, b, c\}=S$, and $t$ some constant of $S$. Let "X" denote an unknown binary operation on $S$. Suppose we have the following set of properties:
- For all x, xxX=t
- There exists a y, such that for all x, xyX=y=t (the same constant as in 1.)
- There exists an x, such that for all y, xyX=t
- There exists an x, such that for all y, xyX=y
- There exists an x, such that there exists a y not equal to x, and xyX=x.
There exist at least 6 structures (all automorphic to each other, which indicates how I obtained them in the first place, THEN I wrote the properties) which satisfy this set of properties as follows:
A a b c
a c c c
b b c c
c a b c
B a b c
a b b b
b a b c
c c b b
C a b c
a c a c
b c c c
c a b c
D a b c
a a b c
b a a a
c a c a
E a b c
a b b a
b a b c
c b b b
F a b c
a a b c
b a a b
c a a a
Do any other binary operations on {a, b, c} satisfy properties 1.-5. above? I would think "yes", since, if I've gotten things right, property 1. specifies something about the diagonal, property 2. specifies a particular column, for which all of its values equal that of the diagonal, 3. specifies a row that always equals the valued of the diagonal, 4. specifies a row which always equals the values of the second coordinate, and I think 5. specifies the only value left undetermined by 1.-4. Do any other structures exist here?