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The Universal Theory of Ordered Equidecomposability Types Semigroups

  • Friedrich Wehrung (a1)

Abstract

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) (xx + y); (ii) (⩝x,y)((xy and yx) ⇒ x = y); (iii) (⩝x,y,u, v)((x + uy + u and uv) ⩝ x + vyv); (iv) (⩝x, u, V)((x + u = u and uv) ⇒ x + v = v); (v) (⩝x,y)(mxmyxy) (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.

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References

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1. Banach, S. and Tarski, A., Sur la décomposition des ensembles de points en parties respectivement congruentes, Fund. Math. 6(1924), 244277.
2. Birkhoff, G., Lattice theory, Amer. Math. Soc, Providence, Rhode Island, 1967.
3. Chang, C. C. and Keisler, H. J., Model Theory, North Holland, 1973.
4. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Math. Surveys 7, Amer. Math. Soc, Providence, Rhode Island 1, 1961 and 2, 1967.
5. Goodearl, K. R., Partially ordered abelian groups with the interpolation property, Math. Surveys and Monographs 20, Amer. Math. Soc, 1986.
6. Grätzer, G., Universal Algebra, 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, 1979.
7. Hall Jr., M., Distinct representatives of subsets, Bull. Amer. Math. Soc. 54(1948), 922926.
8. Howie, J. M., An introduction to semigroup theory, Academic Press, London, New York, San Francisco, 1976.
9. Ketonen, J., The structure of countable Boolean algebras, Ann. of Math. (1) 108(1978), 4189.
10. Kiss, E. W., Marki, L., Prohle, P. and Tholen, W., Categorical algebraic properties. A compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity, Studia Sci. Math. Hungar. 18(1983), 79141.
11. Laczkovich, M., Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem, J. Reine Angew. Math. 404(1990), 77117.
12. Laczkovich, M., Invariant signed measures and the cancellation law, Proc Amer. Math. Soc, Vol. III, (2), February, 1991, 421431.
13. Laczkovich, M., private communication, May 25, 1991.
14. Laczkovich, M., private communication, October 23, 1992.
15. Bhaskara Rao, K. P. S. and Shortt, R. M., Weak cardinal algebras, Ann. New York Acad. Sci. 659(1992), 156162.
16. Shortt, R. M. and Wehrung, F., Common extensions of semigroup-valued charges, J. Math. Anal. Appl., to appear.
17. Tarski, A., Algebraische Fassung des Maβproblems, Fund. Math. 31(1938), 4766.
18. Tarski, A., Cardinal Algebras, New York, Oxford, 1949.
19. Wagon, S., The Banach Tarski-paradox, Cambridge University Press, New York, 1985.
20. Wehrung, F., Théoréme de Hahn-Banach et paradoxes continus et discrets, C. R. Acad. Sci. Paris Sér. I 310(1990), 303306.
21. Wehrung, F., Injective positively ordered monoids I, J. Pure Appl. Algebra 83(1992), 4382.
22. Wehrung, F., Injective positively ordered monoids II, J. Pure Appl. Algebra 83(1992), 83100.
23. Wehrung, F., Restricted injectivity, transfer property and decompositions of separative positively ordered monoids, Comm. Algebra (5) 22(1994), 17471781.
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The Universal Theory of Ordered Equidecomposability Types Semigroups

  • Friedrich Wehrung (a1)

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