Abstract
AbstractThe lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In an earlier paper, we characterize this lattice as a complete lattice. Let m be an uncountable regular cardinal. The latticeLof all m-complete congruence relations of an m-complete latticeKis an m-algebraic lattice; ifKis bounded, then the unit element ofLis m-compact. Our main result is the converse statement: For an m-algebraic latticeLwith an m-compact unit element, we construct a bounded m-complete latticeKsuch thatLis isomorphic to the lattice of m-complete congruence relations ofK. In addition, ifLhas more than one element, then we show how to constructKso that it will also have a prescribed automorphism group. On the way to the main result, we prove a technical theorem, the One Point Extension Theorem, which is also used to provide a new proof of the earlier result.
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