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The Universal Theory of Ordered Equidecomposability Types Semigroups

Published online by Cambridge University Press:  20 November 2018

Friedrich Wehrung
Friedrich Wehrung*
Affiliation:
Université de Caen, Département de Mathématiques 14032 Caen Cedex, France
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Abstract

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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.

Keywords

06F0520M1408C1006F2003C2003C10equidecomposabilityCantor-Bernstein propertymultiplicative cancellation propertyinitial segments of ordered groupsreduced powers
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 5 , 01 October 1994 , pp. 1093 - 1120
Copyright
Copyright © Canadian Mathematical Society 1994

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