Talk:Ideal (order theory)

Definition of Ideal

All the papers I've read in this area define an ideal in a poset just using the first condition (downward closure). Unless these are anomalous I feel the main page should be changed to avoid confusion. —Preceding unsigned comment added by 128.16.14.220 (talk) 18:24, 27 October 2009 (UTC)[reply]

I have the same experience - and I have been teaching the 'first condition only' definition, in courses based on Stanley's work. When the forerunner of this article was created a decade ago, it only contained the 'both conditions' definition, and did not even mention that there exists another usage (which actually I think is the only one I have encountered). This may be due to the original article creator at that time not having encountered more than one usage.

However, as the article now appears, it seems to contain flagrant breach of one of our first principles: Not to include new terminology in wp, but to report on the existing one.

There is some sense in deviating from our general policies in math articles, to the extent that we choose one common usage of a term to be employed in other math articles referring to the corresponding concept; after all, confused definitions probably are a greater source of trouble in advanced mathemathics than in most other fields, and it is not reasonable to define each term locally in each article where we employ it. However, the main articles definitely should present contradicting but common senses of the term directly in the opening paragraphs, at least when a reference to a dab is not feasible.

In this particular case, I'd like to know more about the sources for the usage of "ideal" and "filter". If User:Markus Krötzsch has the time (and still can find his sources with reasonably small effort -- ten years is a long time), I'd appreciate if he explained the reasons for his choices of terminology in terms of actual usage (apart from the immediate spread of the wp choice on the net), and if he or someone else provided an adequate reference. The motivations he wrote originally concerned the history of the terms (which is not uninteresting in itself, but not sufficient); and the motivations of the present article is just expressing the wish to have a bijection between terms and concepts in the wp (which as I stated above never should inhibit the information about actual usage to our readers).

Right now,the article Ideal (order theory) only contains two sources. They seem to be added only to prove that the term sometimes is not used in the sense chosen for this article. (One of the references is Stanley; and he explicitly uses "order ideal" as synonymous with "downset", i. e., employs the 'one condition definition', and he does not in any way indicate that he should be aware of any other usage of the term. I do not have immediate acces to the other reference given for the 'one condition definition', Lawson.) This of course is non-optimal; if there indeed are no independent references to the 'two conditions definition', then the article should be merged with downset. However, I hope and believe that there indeed were some references; it would be rather embarrassing if it should turn out that the 'two condistion definition' was originated at wp.

For filter (mathematics), there are more sources. I have not tried to evaluate them, but hopefully at least one of them does contain a reference to the 'two conditions definition' of filters of posets. However, since the article seems to cover several not quite coinciding usages of the term "filter", thiere is a risk that the references only cover other or specialised senses.

One possible source for confusion is the analogy with ideals in rings, and descending filtrations of modules. In these cases, the objects have algebraich structure, which is to be preserved by the ideals or filter components. Since a lattice also has a kind of algebraic structure, it may seem reasonable to include the condition that ideals and filters in lattices be sublattices (in addition to being downsets or upsets). If we consider lattices as special cases of posets, adding the sublattice condition is equivant to adding "the second condition" in our article. However, posets in general do not have any 'algebraich structure' to be respected by the ideals and filters.JoergenB (talk) 16:08, 16 February 2014 (UTC)[reply]

I have also not encountered the "second condition" anywhere outside Wikipedia. It is a strange condition to add, because if P is a poset with two distinct maximal elements, then the second condition means that the set P would not be considered to be one its own order ideals. — Preceding unsigned comment added by 216.249.49.43 (talk) 17:03, 10 March 2014 (UTC)[reply]

Apparently ideals in posets are often defined without the requirement of being directed, and ideals in lattices are always defined with this requirement? Quoting Davey & Priestley, Introduction to Lattices and Order (Second Edition), 2002:

On page 20: "Let P be an ordered set and Q ⊆ P. Q is a down-set (aternative terms include decreasing set and order ideal) if, whenever x ∈ Q, y ∈ P and y ≤ x, we have y ∈ Q."
On page 44: "Let L be a lattice. A non-empty subset J of L is called an ideal if (i) a, b ∈ J implies a ∨ b ∈ J, (ii) a ∈ L, b ∈ J and a ≤ b imply a ∈ J."

Anyway, here is one reference I found:

"If A is a poset, we say a subset of A is an ideal if it is (upwards) directed and a lower set." (Johnstone, Peter (1982), Stone Spaces, Cambridge studies in advanced mathematics, vol. 3, Cambridge University Press, p. 286, ISBN 978-0-521-33779-3)

The article itself says "While this is the most general way to define an ideal for arbitrary posets, it was originally defined for lattices only." Henri Cartan's definition of filters already required that they be directed.[1]:— Tobias Bergemann (talk) 19:55, 10 March 2014 (UTC)[reply]

D & P is an influential book at least among applied math practitioners. So their def, which is reflected in quite a few other texts, should be mentioned. 86.127.138.67 (talk) 17:22, 22 April 2015 (UTC)[reply]

Prime Ideals

"An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order."

Is it _really_ the /complement/ or the /dual/ that's meant here? Zero sharp 02:30, 29 September 2006 (UTC)[reply]