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SEMILATTICES AND THE RAMSEY PROPERTY
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- Journal:
- The Journal of Symbolic Logic / Volume 80 / Issue 4 / December 2015
- Published online by Cambridge University Press:
- 22 December 2015, pp. 1236-1259
- Print publication:
- December 2015
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- Article
- Export citation
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We consider
${\cal S}$, the class of finite semilattices; 
${\cal T}$, the class of finite treeable semilattices; and 
${{\cal T}_m}$, the subclass of 
${\cal T}$ which contains trees with branching bounded by m. We prove that 
${\cal E}{\cal S}$, the class of finite lattices with linear extensions, is a Ramsey class. We calculate Ramsey degrees for structures in 
${\cal S}$, 
${\cal T}$, and 
${{\cal T}_m}$. In addition to this we give a topological interpretation of our results and we apply our result to canonization of linear orderings on finite semilattices. In particular, we give an example of a Fraïssé class 
${\cal K}$ which is not a Hrushovski class, and for which the automorphism group of the Fraïssé limit of 
${\cal K}$ is not extremely amenable (with the infinite universal minimal flow) but is uniquely ergodic.
