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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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