Examples of Commutative Semigroups Where the Cardinality of the Carrier Set is Greater Than $\mathfrak c$.

Given:

• 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 $$|\mathbb R |$$?

• Keywords: first-order theory, Löwenheim–Skolem, ultrapower – bof Oct 19 '18 at 7:48
• 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. – 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 $$+$$.