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 Mar 21 asked Adding metric to matroids in order to describe graphs whose vertices are points in Euclidean space Feb 29 accepted How to deal with Ideals generation from a Poset of sets including the empty set? Feb 29 comment How to deal with Ideals generation from a Poset of sets including the empty set? In your example, subset $\{\{1\}\}$ when considered generates as ideal element $\{\emptyset,\{1\}\}$ right? So ideals for all subsets in poset are not distinct, some of them are the same and for this reason I do not have $2^{|S|}$ ideals but a less number of them. Feb 29 comment How to deal with Ideals generation from a Poset of sets including the empty set? Ah, ok some of ideals are equal... is it correct??? Feb 29 revised How to deal with Ideals generation from a Poset of sets including the empty set? added 145 characters in body Feb 29 awarded Critic Feb 29 comment How to deal with Ideals generation from a Poset of sets including the empty set? OK, wait a moment Brian, sorry for disturbing again, but actually I cannot understand how you are building the lattice. Shouldnt I consider all possible subsets in $S$? Feb 29 comment How to deal with Ideals generation from a Poset of sets including the empty set? Yes, thanks Patrick. Ah Patrick could you please tell me what you think about the two definitions I found of Ideal? In the linked question there is also a community wiki I started about this little debate of before, remember? What do you think about the two definitions? Feb 29 revised How to deal with Ideals generation from a Poset of sets including the empty set? added 624 characters in body Feb 29 comment How to deal with Ideals generation from a Poset of sets including the empty set? In the link to the question.... there you will find the definition. OK, I will write the definition also here... wait a min, sorry Feb 29 asked How to deal with Ideals generation from a Poset of sets including the empty set? Feb 27 revised Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset added 26 characters in body Feb 27 answered Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset Feb 27 comment Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset Thankyou very much for your answer Brian, it helped me but azarels one was a little more explicit for the construction of the set using an inductive cardinal oriented approach. Feb 27 accepted Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset Feb 27 comment Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset I see, but its kinda strange. I admit that these books are quite old, some of them preceding 1990... maybe conventions about ideals changed in time... thanks Brian, Ill investigate more about this. Feb 27 comment Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset OK think I got it... quite long procedure... Thanks a lot azarel, I will read better and try to understand it fully. Feb 27 revised Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset added 881 characters in body Feb 27 revised Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset added 674 characters in body Feb 27 comment Getting a Distributive-Lattice from Poset or, equivalently, a Greedoid from a Poset But to build the set of all ideals, what should I do? The relation is the following: $I \subseteq S$ is Ideal if $t \ in I, s \leq t \implies s \in I$. The condition $s \leq t$ is not $s \subseteq t$. So I should not take necessarly all subsets in $t$, but should look in $S$ for all those $s$ being $\leq t$ and include those in set $I$?