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Forcing extensions of partial lattices. (English) Zbl 1030.03039
Let $$K$$ be a lattice. Then $$\text{Con}_CK$$ denotes the $$\{\vee, 0\}$$-semilattice of all finitely generated congruences of $$K$$. The congruence lattice problem of Dilworth asks whether every distributive $$\{\vee, 0\}$$-semilattice is isomorphic to $$\text{Con}_CL$$ for some lattice $$L$$. This problem is still open. Here the author makes some contributions toward this problem. It is shown: Let $$K$$ be a lattice, $$D$$ a distributive lattice with $$0$$ and $$\varphi:\text{Con}_CK\to D$$ be a $$\{\vee,0\}$$-homomorphism. Then there are a lattice $$L$$, a lattice homomorphism $$f: K\to L$$ and an isomorphism $$\alpha: \text{Con}_CL\to D$$ with $$\alpha\circ \text{Con}_Cf= \varphi$$. It is shown that $$f$$ and $$L$$ satisfy some additional properties.
The author uses methods which come from forcing and Boolean-valued models. He generalizes some known results. So he shows: Every lattice $$K$$ such that $$\text{Con}_CK$$ is a lattice, admits a congruence preserving extension into a relatively complemented lattice.

##### MSC:
 03E40 Other aspects of forcing and Boolean-valued models 06B10 Lattice ideals, congruence relations
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