Timeline for Structure of $Hom(L_1,L_2)$, where $L_i$ are distributive lattices
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Aug 8, 2018 at 21:17 | comment | added | Todd Trimble | @DuchampGérardH.E. Yes, I assume $f, g$ preserve all distributive lattice structure: meets and joins (and also empty meets and joins if those are considered as part of the definition of lattice). The point is that the inequality $(a \wedge b) \vee (c \wedge d) \leq (a \vee c) \wedge (b \vee d)$ typically isn't an equality, as one can easily see by putting the right side in disjunctive normal form, or using Venn diagrams, etc. | |
| Aug 8, 2018 at 20:15 | comment | added | Duchamp Gérard H. E. | @ToddTrimble For your first equality, you suppose $f,g$ to preserve meets, am I right ? Otherwise, Dominic's lattice property is all right (it is a sublattice of the lattice product $L^P$). | |
| Feb 9, 2018 at 14:39 | comment | added | Todd Trimble | Yes, but I think OP is looking not just at poset maps $L_1 \to L_2$, but at maps that preserve finite meets and joins. It's not true that your pointwise definition for $f \vee g$ gives a distributive lattice map, because $(f \vee g)(x \wedge y) = (f(x) \wedge f(y)) \vee (g(x) \wedge g(y))$ does not match $(f(x) \vee g(x)) \wedge (f(y) \vee g(y))$. | |
| Feb 9, 2018 at 13:59 | history | answered | Dominic van der Zypen | CC BY-SA 3.0 |