The universal theory of ordered equidecomposability types semigroups
Résumé
We prove that a commutative preordered monoid $S$ embeds into the space of all equidecomposability types of subsets of some set equipped with a group action (in short, a full type space) if and only if for all $x$, $y$, $u$, $v$ in $S$, the following statements hold: $0\leq x$; $x\leq y$ and $y\leq x$ implies that $x=y$; $x+u\leq y+u$ and $u\leq v$ implies that $x+v\leq y+v$; $mx\leq my$ implies that $x\leq y$, for all positive integers $m$. Furthermore, such a structure can always be embedded into a reduced power of the space $T$ of all nonempty initial segments of $Q_+$ (the non-negative rationals) with rational (possibly infinite) endpoints, where $A+B$ is the set of all elements $a+b$, where $a\in A$ and $b\in B$, for all sets $A$, $B$ of rationals. As a corollary, we obtain that the set of all universal formulas of $(+,\leq)$ satisfied by all full type spaces is decidable.