• A set $M$.
  • A binary operation $+$ defined on $M$
    $+: M \times M \to M$
    $\text{ that is both associative and commutative.}$

satisfying the following properties:

P-1: $\text{For every } x,y,z \in M \text{, if } z + x = z + y \, \text{ then } \, x = y$.

P-2: $\text{For every } x,y,z \in M \text{, if } z = x + y \, \text{ then } \, z \ne x$.

P-3: $\text{For every } x,y \in M \text{, if } x \ne y \, \text{ then } \, [\exists u \; | \, x = y +u] \text{ or } [\exists u \; | \, y = x +u]$.

Example: The set of positive real numbers.

Are there examples where the the cardinality of $M$ is strictly greater that $|\mathbb R |$?

  • 2
    $\begingroup$ Keywords: first-order theory, Löwenheim–Skolem, ultrapower $\endgroup$ – bof Oct 19 '18 at 7:48
  • 1
    $\begingroup$ My first thought was this: Taking your example $(\mathbb{R}^+,+)$, let $M$ denote the set of all functions $f : \mathbb{R} \to \mathbb{R}^+$. An addition $+$ on $M$ is as usual defined by $(f + g)(x) = f(x) + g(x)$. Unfortunately axiom 3 is not satisfied. $\endgroup$ – Paul Frost Oct 19 '18 at 12:29

You can get such a semigroup by taking the set of positive elements in any totally ordered abelian group. Now, if $A$ is a totally ordered abelian group and $S$ is any totally ordered set, the direct sum $A^{\oplus S}$ is also a totally ordered group with respect to the lexicographic order. Explicitly, the semigroup of positive elements of $A^{\oplus S}$ is the set of functions $f:S\to A$ such that $f(s)=0$ for all but finitely many $s\in S$ and $f(s)>0$ for the least $s\in S$ for which $f(s)$ is nonzero. If $A$ is nontrivial then this semigroup has at least as many elements as $S$, so you can get an example of arbitrarily large cardinality by taking $S$ to be a totally ordered set of arbitrarily large cardinality.

Much more generally, any theory over a countable first-order language which has an infinite model has models of all infinite cardinalities, by the Löwenheim-Skolem theorem. Your semigroups are just models of a certain first-order theory over the language with a single binary operation $+$.


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