Isotone maps as maps of congruences. I: Abstract maps
Let \(K\) and \(L\) be lattices and let \(\varphi:K\rightarrow L\) be a lattice homomorphism. The associated restriction map \(\text{rs } \varphi:\text{Con } L \rightarrow \text{Con } K\) is defined by setting \(x \equiv y\) \(((\text{rs } \varphi)\Theta)\) iff \(\varphi x \equiv \varphi y \;(\Theta)\), for each \( \Theta \in \text{Con }L\). The extension of \(\varphi\), \(\text{xt }\varphi:\text{Con }K\rightarrow \text{Con }L\) is defined by setting, for each \(\Theta\in \text{Con }K\), \((\text{xt }\varphi) \Theta\) to be the congruence relation of \(L\) generated by the subset \(\varphi^2(\Theta)\) of \(L^2\) (where \(\varphi^2:K^2\rightarrow L^2\) is the map induced by \(\varphi\)): \((\text{xt }\varphi) \Theta=\bigvee (\Theta_L(\varphi x,\varphi y)| x\equiv y\;(\Theta))\). \smallskip The authors prove several variants of the following main result: Let \(D_1\) and \(D_2\) be finite distributive lattices, and let \(\psi:D_1\rightarrow D_2\) be an isotone map. Then there are finite lattices \(L_1\), \(L_2\), \(L\), a lattice embedding \(\varphi_1:L_1\rightarrow L\), a lattice homomorphism \(\varphi_2:L_2\rightarrow L\), and isomorphisms \(\alpha_1:D_1\rightarrow \text{Con }L_1\), \(\alpha_2:D_2\rightarrow \text{Con }L_2\) such that \(\alpha_2\circ \psi= (\text{rs }\varphi_2)\circ (\text{xt }\varphi_1)\circ \alpha_1\) where \(\text{xt }\varphi_1:\text{Con }L_1\rightarrow \text{Con }L\) and \(\text{rs }\varphi_2:\text{Con }L\rightarrow \text{Con }L_2\). Furthermore \(\varphi_2\) is also an embedding iff \(\psi\) preserves \(0\).
scientific article; zbMATH DE number 1336399