# Revision history [back]

I do not know whether there is such a builtin construction in Sage, so here is a possible construction.

Let S be a set, e.g.:

sage: S = {'a','b','c','d'}


First, we define the list of partitions over S, where each partition is a tuple and each atom of the partition is also a tuple. We do that because elements of a Poset must be hashable, so lists are not allowed:

sage: from sympy.utilities.iterables import multiset_partitions
sage: list(tuple([tuple(j) for j in i]) for i in multiset_partitions(S))
[(('a', 'b', 'c', 'd'),),
(('a', 'b', 'c'), ('d',)),
(('a', 'b', 'd'), ('c',)),
(('a', 'b'), ('c', 'd')),
(('a', 'b'), ('c',), ('d',)),
(('a', 'c', 'd'), ('b',)),
(('a', 'c'), ('b', 'd')),
(('a', 'c'), ('b',), ('d',)),
(('a', 'd'), ('b', 'c')),
(('a',), ('b', 'c', 'd')),
(('a',), ('b', 'c'), ('d',)),
(('a', 'd'), ('b',), ('c',)),
(('a',), ('b', 'd'), ('c',)),
(('a',), ('b',), ('c', 'd')),
(('a',), ('b',), ('c',), ('d',))]


Second, we define a function that decides whether one partition refines another one:

sage: refine = lambda p,q : all(any(set(i).issubset(set(j)) for j in q) for i in p)

sage: refine(((1,), (2, 3)), ((1,), (2,), (3,)))
False
sage: refine(((1,), (2,), (3,)), ((1,), (2, 3)))
True


With both the list of partitions and the refinment order, we can construct the poset:

sage: P = Poset((list(tuple([tuple(j) for j in i]) for i in multiset_partitions(S)), refine))

sage: P
Finite poset containing 15 elements
sage: P.plot()


I do not know whether there is such a builtin construction in Sage, so here is a possible construction.

Let S be a set, e.g.:

sage: S = {'a','b','c','d'}


First, we define the list of partitions over S, where each partition is a tuple and each atom of the partition is also a tuple. We do that because elements of a Poset must be hashable, so lists are not allowed:

sage: from sympy.utilities.iterables import multiset_partitions
sage: list(tuple([tuple(j) for j in i]) for i in multiset_partitions(S))
[(('a', : sage: list(SetPartitions(S))
[{{'a', 'b', 'c', 'd'),),
(('a', 'd'}},
{{'a', 'b', 'c'), ('d',)),
(('a', 'c'}, {'d'}},
{{'a', 'b', 'd'), ('c',)),
(('a', 'b'), ('c', 'd')),
(('a', 'b'), ('c',), ('d',)),
(('a', 'd'}, {'c'}},
{{'a', 'b'}, {'c', 'd'}},
{{'a', 'b'}, {'c'}, {'d'}},
{{'a', 'c', 'd'), ('b',)),
(('a', 'c'), ('b', 'd')),
(('a', 'c'), ('b',), ('d',)),
(('a', 'd'), ('b', 'c')),
(('a',), ('b', 'd'}, {'b'}},
{{'a', 'c'}, {'b', 'd'}},
{{'a', 'c'}, {'b'}, {'d'}},
{{'a', 'd'}, {'b', 'c'}},
{{'a'}, {'b', 'c', 'd')),
(('a',), ('b', 'c'), ('d',)),
(('a', 'd'), ('b',), ('c',)),
(('a',), ('b', 'd'), ('c',)),
(('a',), ('b',), ('c', 'd')),
(('a',), ('b',), ('c',), ('d',))]
'd'}},
{{'a'}, {'b', 'c'}, {'d'}},
{{'a', 'd'}, {'b'}, {'c'}},
{{'a'}, {'b', 'd'}, {'c'}},
{{'a'}, {'b'}, {'c', 'd'}},
{{'a'}, {'b'}, {'c'}, {'d'}}]
Second, we define a function that decides whether one partition refines another one: sage: refine = lambda p,q : all(any(set(i).issubset(set(j)) for j in q) for i in p)

sage: refine(((1,), (2, 3)), ((1,), (2,), (3,)))
False
sage: refine(((1,), (2,), (3,)), ((1,), (2, 3)))
True
With both the list of partitions and the refinment order, we can construct the poset: sage: P = Poset((list(tuple([tuple(j) for j in i]) for i in multiset_partitions(S)), Poset((list(SetPartitions(S)), refine))
sage: P

sage: P
Finite poset containing 15 elements
sage: P.plot()


 3 No.3 Revision updated 2021-05-24 15:14:00 +0200 I do not know whether there is such a builtin construction in Sage, so here is a possible construction. Let S be a set, e.g.: sage: S = {'a','b','c','d'} First, we define the list of partitions over S: sage: list(SetPartitions(S)) [{{'a', 'b', 'c', 'd'}}, {{'a', 'b', 'c'}, {'d'}}, {{'a', 'b', 'd'}, {'c'}}, {{'a', 'b'}, {'c', 'd'}}, {{'a', 'b'}, {'c'}, {'d'}}, {{'a', 'c', 'd'}, {'b'}}, {{'a', 'c'}, {'b', 'd'}}, {{'a', 'c'}, {'b'}, {'d'}}, {{'a', 'd'}, {'b', 'c'}}, {{'a'}, {'b', 'c', 'd'}}, {{'a'}, {'b', 'c'}, {'d'}}, {{'a', 'd'}, {'b'}, {'c'}}, {{'a'}, {'b', 'd'}, {'c'}}, {{'a'}, {'b'}, {'c', 'd'}}, {{'a'}, {'b'}, {'c'}, {'d'}}] Second, we define a function that decides whether one a partition refines another one: sage: refine = lambda p,q : all(any(set(i).issubset(set(j)) for j in q) for i in p) sage: refine(((1,), (2, 3)), ((1,), (2,), (3,))) False sage: refine(((1,), (2,), (3,)), ((1,), (2, 3))) True With both the list of partitions and the refinment order, we can construct the poset: sage: P = Poset((list(SetPartitions(S)), refine)) sage: P sage: P Finite poset containing 15 elements sage: P.plot() 4 No.4 Revision updated 2021-05-24 15:15:36 +0200 I do not know whether there is such a builtin construction in Sage, so here is a possible construction. Let S be a set, e.g.: sage: S = {'a','b','c','d'} First, we define the list of partitions over S: sage: list(SetPartitions(S)) [{{'a', 'b', 'c', 'd'}}, {{'a', 'b', 'c'}, {'d'}}, {{'a', 'b', 'd'}, {'c'}}, {{'a', 'b'}, {'c', 'd'}}, {{'a', 'b'}, {'c'}, {'d'}}, {{'a', 'c', 'd'}, {'b'}}, {{'a', 'c'}, {'b', 'd'}}, {{'a', 'c'}, {'b'}, {'d'}}, {{'a', 'd'}, {'b', 'c'}}, {{'a'}, {'b', 'c', 'd'}}, {{'a'}, {'b', 'c'}, {'d'}}, {{'a', 'd'}, {'b'}, {'c'}}, {{'a'}, {'b', 'd'}, {'c'}}, {{'a'}, {'b'}, {'c', 'd'}}, {{'a'}, {'b'}, {'c'}, {'d'}}] Second, we define a function that decides whether a partition refines another one: sage: refine = lambda p,q : all(any(set(i).issubset(set(j)) for j in q) for i in p) sage: refine(((1,), (2, 3)), ((1,), (2,), (3,))) False sage: refine(((1,), (2,), (3,)), ((1,), (2, 3))) True With both the list of partitions and the refinment order, we can construct the poset: sage: P = Poset((list(SetPartitions(S)), refine)) sage: P sage: P Finite poset containing 15 elements sage: P.is_lattice() True sage: P.plot() 5 No.5 Revision updated 2021-05-24 18:25:27 +0200 I do not know whether there is such a builtin construction in Sage, so here is a possible construction. Let S be a set, e.g.: sage: S = {'a','b','c','d'} First, we define the list of partitions over S: sage: list(SetPartitions(S)) [{{'a', 'b', 'c', 'd'}}, {{'a', 'b', 'c'}, {'d'}}, {{'a', 'b', 'd'}, {'c'}}, {{'a', 'b'}, {'c', 'd'}}, {{'a', 'b'}, {'c'}, {'d'}}, {{'a', 'c', 'd'}, {'b'}}, {{'a', 'c'}, {'b', 'd'}}, {{'a', 'c'}, {'b'}, {'d'}}, {{'a', 'd'}, {'b', 'c'}}, {{'a'}, {'b', 'c', 'd'}}, {{'a'}, {'b', 'c'}, {'d'}}, {{'a', 'd'}, {'b'}, {'c'}}, {{'a'}, {'b', 'd'}, {'c'}}, {{'a'}, {'b'}, {'c', 'd'}}, {{'a'}, {'b'}, {'c'}, {'d'}}] Second, we define a function that decides whether a partition refines another one: sage: refine = lambda p,q : all(any(set(i).issubset(set(j)) for j in q) for i in p) sage: refine(((1,), (2, 3)), ((1,), (2,), (3,))) False sage: refine(((1,), (2,), (3,)), ((1,), (2, 3))) True With both the list of partitions and the refinment order, we can construct the poset: sage: P = Poset((list(SetPartitions(S)), refine)) sage: P sage: P Finite poset containing 15 elements sage: P.is_lattice() True sage: P.plot() 


 Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license. about | faq | help | privacy policy | terms of service Powered by Askbot version 0.7.59 Please note: Askbot requires javascript to work properly, please enable javascript in your browser, here is how //IE fix to hide the red margin var noscript = document.getElementsByTagName('noscript')[0]; noscript.style.padding = '0px'; noscript.style.backgroundColor = 'transparent'; askbot['urls']['mark_read_message'] = '/s/messages/markread/'; askbot['urls']['get_tags_by_wildcard'] = '/s/get-tags-by-wildcard/'; askbot['urls']['get_tag_list'] = '/s/get-tag-list/'; askbot['urls']['follow_user'] = '/followit/follow/user/{{userId}}/'; askbot['urls']['unfollow_user'] = '/followit/unfollow/user/{{userId}}/'; askbot['urls']['user_signin'] = '/account/signin/'; askbot['urls']['getEditor'] = '/s/get-editor/'; askbot['urls']['apiGetQuestions'] = '/s/api/get_questions/'; askbot['urls']['ask'] = '/questions/ask/'; askbot['urls']['questions'] = '/questions/'; askbot['settings']['groupsEnabled'] = false; askbot['settings']['static_url'] = '/m/'; askbot['settings']['minSearchWordLength'] = 4; askbot['settings']['mathjaxEnabled'] = true; askbot['settings']['sharingSuffixText'] = ''; askbot['settings']['errorPlacement'] = 'after-label'; askbot['data']['maxCommentLength'] = 800; askbot['settings']['editorType'] = 'markdown'; askbot['settings']['commentsEditorType'] = 'rich\u002Dtext'; askbot['messages']['askYourQuestion'] = 'Ask Your Question'; askbot['messages']['acceptOwnAnswer'] = 'accept or unaccept your own answer'; askbot['messages']['followQuestions'] = 'follow questions'; askbot['settings']['allowedUploadFileTypes'] = [ "jpg", "jpeg", "gif", "bmp", "png", "tiff" ]; askbot['data']['haveFlashNotifications'] = true; askbot['data']['activeTab'] = 'questions'; askbot['settings']['csrfCookieName'] = 'asksage_csrf'; askbot['data']['searchUrl'] = ''; /*<![CDATA[*/ $('.mceStatusbar').remove();//a hack to remove the tinyMCE status bar$(document).ready(function(){ // focus input on the search bar endcomment var activeTab = askbot['data']['activeTab']; if (inArray(activeTab, ['users', 'questions', 'tags', 'badges'])) { var searchInput = $('#keywords'); } else if (activeTab === 'ask') { var searchInput =$('#id_title'); } else { var searchInput = undefined; animateHashes(); } var wasScrolled = $('#scroll-mem').val(); if (searchInput && !wasScrolled) { searchInput.focus(); putCursorAtEnd(searchInput); } var haveFullTextSearchTab = inArray(activeTab, ['questions', 'badges', 'ask']); var haveUserProfilePage =$('body').hasClass('user-profile-page'); if ((haveUserProfilePage || haveFullTextSearchTab) && searchInput && searchInput.length) { var search = new FullTextSearch(); askbot['controllers'] = askbot['controllers'] || {}; askbot['controllers']['fullTextSearch'] = search; search.setSearchUrl(askbot['data']['searchUrl']); if (activeTab === 'ask') { search.setAskButtonEnabled(false); } search.decorate(searchInput); } else if (activeTab === 'tags') { var search = new TagSearch(); search.decorate(searchInput); } if (askbot['data']['userIsAdminOrMod']) { $('body').addClass('admin'); } if (askbot['settings']['groupsEnabled']) { askbot['urls']['add_group'] = "/s/add-group/"; var group_dropdown = new GroupDropdown();$('.groups-dropdown').append(group_dropdown.getElement()); } var userRep = $('#userToolsNav .reputation'); if (userRep.length) { var showPermsTrigger = new ShowPermsTrigger(); showPermsTrigger.decorate(userRep); } }); if (askbot['data']['haveFlashNotifications']) {$('#validate_email_alert').click(function(){notify.close(true)}) notify.show(); } var langNav = $('.lang-nav'); if (langNav.length) { var nav = new LangNav(); nav.decorate(langNav); } /*]]>*/ if (typeof MathJax != 'undefined') { MathJax.Hub.Config({ extensions: ["tex2jax.js"], jax: ["input/TeX","output/HTML-CSS"], tex2jax: {inlineMath: [["$","$"],["\$","\$"]]} }); } else { console.log('Could not load MathJax'); } //todo - take this out into .js file$(document).ready(function(){ $('div.revision div[id^=rev-header-]').bind('click', function(){ var revId = this.id.substr(11); toggleRev(revId); }); lanai.highlightSyntax(); }); function toggleRev(id) { var arrow =$("#rev-arrow-" + id); var visible = arrow.attr("src").indexOf("hide") > -1; if (visible) { var image_path = '/m/default/media/images/expander-arrow-show.gif?v=19'; } else { var image_path = '/m/default/media/images/expander-arrow-hide.gif?v=19'; } image_path = image_path + "?v=19"; arrow.attr("src", image_path); \$("#rev-body-" + id).slideToggle("fast"); } for (url_name in askbot['urls']){ askbot['urls'][url_name] = cleanUrl(askbot['urls'][url_name]); }