Completely separating system(CSS) which is also an antichain (Sperner family)












0














Suppose we have a family $F$ of subsets of a set $S$ which is both an antichain and a CSS. That is, members of $F$ are pairwise incomparable w.r.t. subset relation and for every pair of members $s_1,s_2in S$ there exist $F_1,F_2in F$ such that $s_1in F_1, s_1notin F_2, s_2notin F_1 $ and $ s_2in F_2$.



Is this studied in the literature?



I know there is a vast literature on antichains and CSS. So I guess, something like this should be touched by someone. Interestingly, it seems to force the antichain to be 'nearly flat' (flat antichains are standard terminology. By 'nearly flat', I mean difference between levels of members in the Boolean lattice is small)










share|cite|improve this question
























  • Yes, you are right. I shall correct it.
    – Cyriac Antony
    Nov 28 '18 at 7:32
















0














Suppose we have a family $F$ of subsets of a set $S$ which is both an antichain and a CSS. That is, members of $F$ are pairwise incomparable w.r.t. subset relation and for every pair of members $s_1,s_2in S$ there exist $F_1,F_2in F$ such that $s_1in F_1, s_1notin F_2, s_2notin F_1 $ and $ s_2in F_2$.



Is this studied in the literature?



I know there is a vast literature on antichains and CSS. So I guess, something like this should be touched by someone. Interestingly, it seems to force the antichain to be 'nearly flat' (flat antichains are standard terminology. By 'nearly flat', I mean difference between levels of members in the Boolean lattice is small)










share|cite|improve this question
























  • Yes, you are right. I shall correct it.
    – Cyriac Antony
    Nov 28 '18 at 7:32














0












0








0







Suppose we have a family $F$ of subsets of a set $S$ which is both an antichain and a CSS. That is, members of $F$ are pairwise incomparable w.r.t. subset relation and for every pair of members $s_1,s_2in S$ there exist $F_1,F_2in F$ such that $s_1in F_1, s_1notin F_2, s_2notin F_1 $ and $ s_2in F_2$.



Is this studied in the literature?



I know there is a vast literature on antichains and CSS. So I guess, something like this should be touched by someone. Interestingly, it seems to force the antichain to be 'nearly flat' (flat antichains are standard terminology. By 'nearly flat', I mean difference between levels of members in the Boolean lattice is small)










share|cite|improve this question















Suppose we have a family $F$ of subsets of a set $S$ which is both an antichain and a CSS. That is, members of $F$ are pairwise incomparable w.r.t. subset relation and for every pair of members $s_1,s_2in S$ there exist $F_1,F_2in F$ such that $s_1in F_1, s_1notin F_2, s_2notin F_1 $ and $ s_2in F_2$.



Is this studied in the literature?



I know there is a vast literature on antichains and CSS. So I guess, something like this should be touched by someone. Interestingly, it seems to force the antichain to be 'nearly flat' (flat antichains are standard terminology. By 'nearly flat', I mean difference between levels of members in the Boolean lattice is small)







combinatorics order-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 28 '18 at 7:33







Cyriac Antony

















asked Nov 28 '18 at 5:32









Cyriac AntonyCyriac Antony

12910




12910












  • Yes, you are right. I shall correct it.
    – Cyriac Antony
    Nov 28 '18 at 7:32


















  • Yes, you are right. I shall correct it.
    – Cyriac Antony
    Nov 28 '18 at 7:32
















Yes, you are right. I shall correct it.
– Cyriac Antony
Nov 28 '18 at 7:32




Yes, you are right. I shall correct it.
– Cyriac Antony
Nov 28 '18 at 7:32










0






active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016752%2fcompletely-separating-systemcss-which-is-also-an-antichain-sperner-family%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























0






active

oldest

votes








0






active

oldest

votes









active

oldest

votes






active

oldest

votes
















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016752%2fcompletely-separating-systemcss-which-is-also-an-antichain-sperner-family%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Plaza Victoria

Puebla de Zaragoza

Musa