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m →A first application of Kuratowski's Free Set Theorem: add link to 'congruence permutable'. |
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The semilattice Con<sub>c</sub> F<sub>'''V'''</sub>(Ω) does not satisfy WURP, for any set Ω of cardinality at least ℵ<sub>2</sub> and any non-distributive variety '''V''' of lattices. Consequently, Con<sub>c</sub> F<sub>'''V'''</sub>(Ω) does not satisfy Schmidt's Condition.
It is proved by Tůma and Wehrung in 2001 that Con<sub>c</sub> F<sub>'''V'''</sub>(Ω) is not isomorphic to Con<sub>c</sub> ''L'', for any lattice ''L'' with [[congruence permutable|permutable congruences]]. By using a slight weakening of WURP, this result is extended to arbitrary [[Universal algebra|algebras]] with permutable congruences by Růžička, Tůma, and Wehrung in 2006. Hence, for example, if Ω has at least ℵ<sub>2</sub> elements, then Con<sub>c</sub> F<sub>'''V'''</sub>(Ω) is not isomorphic to the normal subgroup lattice of any group, or the submodule lattice of any module.
==Solving CLP: the Erosion Lemma==
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