Skip to main content
1 of 4
Salvo Tringali
  • 10.5k
  • 2
  • 29
  • 64

As a complement to my previous non-answer, let me show that the OP's conjecture holds for any order-$3$ semigroup $S$ that is not a group.

Up to the canonical anti-isomorphism between $S$ and its opposite semigroup, there are $17$ such semigroups (excluding the cyclic group of order $3$), of which only $11$ are commutative; see the Wikipedia article on order-$3$ semigroups here. Below, I'll list these $11$ commutative semigorups one by one and prove that they all satisfy the OP's conjecture. It may be a tediousv task, but is going to help with another reduction.

All semigroups will be defined on the set $\{x, y, z\}$ through their Cayley tables. Since $\lfloor (3-1)/2 \rfloor = 1$, we need to find, for each semigroup on the list, a permutation $(a,b,c)$ of the ordered triple $(x,y,z)$ such that $ac = bc$.

  1. $S$ is the cyclic semigroup (with $y = x^2$, $z = x^3$, and $x^2 = x^4$) defined by the following table:
x y z
x y z y
y z y z
z y z y

It satisfies the OP's conjecture by Proposition 2 here.

  1. $S$ is the cyclic semigroup (with $y = x^2$, $z = x^3$, and $x^3 = x^4$) defined by the following table:
x y z
x y z z
y z z z
z z z z

It satisfies the OP's conjecture by Proposition 2 here.

  1. $S$ is the monoid (with identity element $z$) defined by the following table:
x y z
x z y x
y y y y
z x y z

It is isomorphic to the (unital) submonoid $\{0, \pm 1\}$ of the integers under multiplication. In particular, it has a non-trivial unit (namely, $x$); hence, it satisfies the OP's conjecture by Proposition 1 here.

  1. $S$ is the monoid (with identity element $y$) defined by the following table:
x y z
x z x x
y x y z
z x z z

Its group of units is trivial, so we can't apply any of Propositions 1 to 3 here. However, $yx = zx = x$.

  1. $S$ is the nilsemigroup (with zero element $z$) defined by the following table:
x y z
x z x x
y x z z
z x z z

It satisfies the OP's conjecture by Proposition 3 here.

  1. $S$ is the null semigroup (with zero element $z$) defined by the following table (a null semigroup is a semigroup with zero in which any product of any two elements is zero):
x y z
x z z z
y z z z
z z z z

Every null semigroup is a nilsemigroup, so $S$ satisfies the OP's conjecture by Proposition 3 here.

  1. $S$ is the non-unital semigroup defined by the following table:
x y z
x z z z
y z y z
z z z z

It is neither a cyclic semigroup nor a nilsemigroup ($y$ and $z$ are both idempotents), so we can't apply any of Propositions 1 to 3 here. However, $yx = zx = x$.

  1. $S$ is the non-unital semigroup defined by the following table:
x y z
x z y z
y y y y
z z y z

It is neither a cyclic semigroup nor a nilsemigroup ($y$ and $z$ are both idempotents), so we can't apply any of Propositions 1 to 3 here. However, $xy = zy = y$.

  1. $S$ is the non-unital semigroup defined by the following table:
x y z
x z x z
y x y z
z z z z

It is neither a cyclic semigroup nor a nilsemigroup ($y$ and $z$ are both idempotents), so we can't apply any of Propositions 1 to 3 here. However, $xz = yz = z$.

  1. $S$ is the non-unital semilattice defined by the following table (a semilattice is a commutative semigroup in which every element is idempotent):
x y z
x x y z
y y y z
z z z z

It is neither a cyclic semigroup nor a nilsemigroup, so we can't apply any of Propositions 1 to 3 here. However, $xz = yz = z$.

  1. $S$ is the unital semilattice (with identity/bottom element $x$) defined by the following table (a semilattice is a commutative semigroup in which every element is idempotent):
x y z
x x y z
y y y z
z z z z

It is neither a cyclic semigroup nor a nilsemigroup, so we can't apply any of Propositions 1 to 3 here. However, $xz = yz = z$. []

I hope I haven't made too many typos.

Salvo Tringali
  • 10.5k
  • 2
  • 29
  • 64