Category:Naturally Ordered Semigroup is Unique

This category contains pages concerning Naturally Ordered Semigroup is Unique:

Let $\struct {S, \circ, \preceq}$ and $\struct {S', \circ', \preceq'}$ be naturally ordered semigroups.

Let:

$0'$ be the smallest element of $S'$

$1'$ be the smallest element of $S' \setminus \set {0'} = S'^*$.

Then the mapping $g: S \to S'$ defined as:

$\forall a \in S: \map g a = \circ'^a 1'$

is an isomorphism from $\struct {S, \circ, \preceq}$ to $\struct {S', \circ', \preceq'}$.

This isomorphism is unique.

Thus, up to isomorphism, there is only one naturally ordered semigroup.