. Proof, We show ¬(2) ? ¬(1) Suppose that T 1 = {0} is a nonempty bounded unary relation in ?. By Lemma 13, there is a non-empty unary relation T 2 in ? that is not 0-valid. Therefore, for some c ? Q, the unary relation U = T 1 ? c · T 2 in ? is non-empty, p.0

?. ). Howeverm and . For-some-m-<-?, Both U and the equation k · y = x are pp-definable in ?, so the claim follows from Lemma 17 We show ¬(3) ? ¬(2) Assume that there exists a unary relation U in ? containing a positive point and If U is bounded, then ¬(2) follows immediately), there exists a non-empty bounded unary relation in {R + , U }} and, consequently, there exists such a relation in ?, Otherwise, by Lemma 2, there exists some M < ? such that, pp.69-76, 1975.

A. Bulatov, P. Jeavons, and A. Krokhin, Classifying the Complexity of Constraints Using Finite Algebras, SIAM Journal on Computing, vol.34, issue.3, pp.720-742, 2005.
DOI : 10.1137/S0097539700376676
URL : http://www.dur.ac.uk/andrei.krokhin/papers/SIAMclassifying.pdf

T. Feder and M. Y. Vardi, Monotone monadic SNP and constraint satisfaction, Proceedings of the twenty-fifth annual ACM symposium on Theory of computing , STOC '93, pp.612-622, 1993.
DOI : 10.1145/167088.167245

T. Feder and M. Y. Vardi, The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory, SIAM Journal on Computing, vol.28, issue.1, pp.57-104, 1998.
DOI : 10.1137/S0097539794266766
URL : http://theory.stanford.edu/~tomas/constraint.ps

M. Bodirsky and M. Grohe, Non-dichotomies in Constraint Satisfaction Complexity, Proceedings of the 35th International Colloquium on Automata , Languages and Programming, pp.184-196, 2008.
DOI : 10.1007/978-3-540-70583-3_16
URL : http://www.informatik.hu-berlin.de/~grohe/pub/bodgro08.pdf

M. Bodirsky, P. Jonsson, and T. Von-oertzen, Essential Convexity and Complexity of Semi-Algebraic Constraints, Logical Methods in Computer Science, vol.8, issue.4, 2012.
DOI : 10.2168/LMCS-8(4:5)2012
URL : https://hal.archives-ouvertes.fr/hal-00756926

M. Bodirsky, P. Jonsson, and T. Von-oertzen, Horn versus full first-order: Complexity dichotomies in algebraic constraint satisfaction, Journal of Logic and Computation, vol.22, issue.3, pp.643-660, 2012.
DOI : 10.1145/321864.321877
URL : https://hal.archives-ouvertes.fr/hal-00756925

D. Marker, Model Theory: An Introduction, 2002.

P. Jonsson and T. Lööw, Computational complexity of linear constraints over the integers, Artificial Intelligence, vol.195, pp.44-62, 2013.
DOI : 10.1016/j.artint.2012.10.001
URL : https://doi.org/10.1016/j.artint.2012.10.001

P. Jonsson and J. Thapper, Affine Consistency and the Complexity of Semilinear Constraints, Proceedings of 39th International Symposium on Mathematical Foundations of Computer Science, pp.420-431, 2014.
DOI : 10.1007/978-3-662-44465-8_36
URL : https://hal.archives-ouvertes.fr/hal-01762331

P. Jeavons, On the algebraic structure of combinatorial problems, Theoretical Computer Science, vol.200, issue.1-2, pp.185-204, 1998.
DOI : 10.1016/S0304-3975(97)00230-2
URL : https://doi.org/10.1016/s0304-3975(97)00230-2

W. Hodges, A Shorter Model Theory, 1997.

A. Schrijver, Theory of linear and integer programming, 1986.

G. Stengle, J. Mcenerney, and R. Robson, Convex polarities over ordered fields, Journal of Pure and Applied Algebra, vol.214, issue.4, pp.370-379, 2010.
DOI : 10.1016/j.jpaa.2009.06.006
URL : https://doi.org/10.1016/j.jpaa.2009.06.006

T. J. Schaefer, The complexity of satisfiability problems, Proceedings of the tenth annual ACM symposium on Theory of computing , STOC '78, pp.216-226, 1978.
DOI : 10.1145/800133.804350

M. Bodirsky, B. Martin, and A. Mottet, Constraint Satisfaction Problems over the Integers with Successor, Proceedings of the 42nd International Colloquium on Automata, Languages and Programming, p.2015
DOI : 10.1007/978-3-662-47672-7_21
URL : http://dro.dur.ac.uk/20040/1/20040.pdf

P. Hell and J. , Colouring, constraint satisfaction, and complexity, Computer Science Review, vol.2, issue.3, pp.143-163, 2008.
DOI : 10.1016/j.cosrev.2008.10.003
URL : http://kam.mff.cuni.cz/~kamserie/serie/clanky/2008/s892.pdf

M. Bodirsky and J. Kára, The complexity of temporal constraint satisfaction problems, J. ACM, vol.57, issue.2
DOI : 10.1145/1667053.1667058

M. Bodirsky, M. Pinsker, and T. Tsankov, Decidability of definability, Journal of Symbolic Logic, vol.78, issue.4, pp.1036-1054, 2013.
DOI : 10.2178/jsl.7804020
URL : http://arxiv.org/pdf/1012.2381

F. Dumortier, M. Gyssens, L. Vandeurzen, and D. V. Gucht, On the Decidability of Semilinearity for Semialgebraic Sets and Its Implications for Spatial Databases, Journal of Computer and System Sciences, vol.58, issue.3, pp.535-571, 1999.
DOI : 10.1006/jcss.1999.1625

F. Dumortier, M. Gyssens, L. Vandeurzen, and D. V. Gucht, On the decidability of semilinearity for semialgebraic sets and its implications for spatial databases ? CORRIGENDUM, J. Comput. System Sci, vol.59, issue.3, pp.557-562, 1999.