Geodesic
The Universal Theory of Ordered Equidecomposability Types Semigroups
Canadian journal of mathematics, Tome 46 (1994) no. 5, pp. 1093-1120

Voir la notice de l'article provenant de la source Cambridge University Press

We prove that a commutative preordered semigroup 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 it satisfies the following axioms: (i) (⩝x,y) (x ≤ x + y); (ii) (⩝x,y)((x ≤ y and y ≤ x) ⇒ x = y); (iii) (⩝x,y,u, v)((x + u ≤ y + u and u ≤ v) ⩝ x + v ≤ y ≤ v); (iv) (⩝x, u, V)((x + u = u and u ≤ v) ⇒ x + v = v); (v) (⩝x,y)(mx ≤ my ⇒ x ≤ y) (all m ∊ Ν \ {0}). Furthermore, such a structure can always be embedded into a reduced power of the space Τ of nonempty initial segments of + with rational (possibly infinite) endpoints, equipped with the addition defined by and the ordering defined by . As a corollary, the set of all universal formulas of (+, ≤) satisfied by all full type spaces is decidable.
DOI : 10.4153/CJM-1994-062-3
Mots-clés : 06F05, 20M14, 08C10, 06F20, 03C20, 03C10, equidecomposability, Cantor-Bernstein property, multiplicative cancellation property, initial segments of ordered groups, reduced powers 
Wehrung, Friedrich. The Universal Theory of Ordered Equidecomposability Types Semigroups. Canadian journal of mathematics, Tome 46 (1994) no. 5, pp. 1093-1120. doi: 10.4153/CJM-1994-062-3
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