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We say that $$S_i \\prec S_j$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \u227a<\/mml:mo>\n \n S<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if $$S_j$$<\/jats:tex-math>\n \n S<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has to be removed before we can remove $$S_i$$<\/jats:tex-math>\n \n S<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The relation $$\\prec $$<\/jats:tex-math>\n \u227a<\/mml:mo>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> defines a partial order $$P(S,\\prec )$$<\/jats:tex-math>\n \n P<\/mml:mi>\n (<\/mml:mo>\n S<\/mml:mi>\n ,<\/mml:mo>\n \u227a<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on S<\/jats:italic> which we call the separability order<\/jats:italic> of S<\/jats:italic> and $$ \\overrightarrow{v}$$<\/jats:tex-math>\n \n v<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A partial order $$P(X, \\prec ')$$<\/jats:tex-math>\n \n P<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n ,<\/mml:mo>\n \n \u227a<\/mml:mo>\n \u2032<\/mml:mo>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on $$X=\\{x_1, \\ldots , x_n\\}$$<\/jats:tex-math>\n \n X<\/mml:mi>\n =<\/mml:mo>\n {<\/mml:mo>\n \n x<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n x<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is called a separability order<\/jats:italic> if there is a collection of convex sets S<\/jats:italic> and a vector $$ \\overrightarrow{v}$$<\/jats:tex-math>\n \n v<\/mml:mi>\n \u2192<\/mml:mo>\n <\/mml:mover>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in some $$\\mathbb {R}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that $$x_i \\prec ' x_j$$<\/jats:tex-math>\n \n \n x<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \n \u227a<\/mml:mo>\n \u2032<\/mml:mo>\n <\/mml:msup>\n \n x<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$P(X, \\prec ')$$<\/jats:tex-math>\n \n P<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n ,<\/mml:mo>\n \n \u227a<\/mml:mo>\n \u2032<\/mml:mo>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if and only if $$S_i \\prec S_j$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \u227a<\/mml:mo>\n \n S<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$P(S,\\prec )$$<\/jats:tex-math>\n \n P<\/mml:mi>\n (<\/mml:mo>\n S<\/mml:mi>\n ,<\/mml:mo>\n \u227a<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We prove that every partial order is the separability order of a collection of convex sets in $$\\mathbb {R}^4$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 4<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and that any poset of dimension 2<\/jats:bold> is the separability order of a set of line segments in $$\\mathbb {R}^3$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We then study the case when the convex sets are restricted to be boxes in d<\/jats:italic>-dimensional spaces. We prove that any partial order is the separability order of a family of disjoint boxes in $$\\mathbb {R}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for some $$d \\le \\lfloor \\frac{n}{2} \\rfloor +1$$<\/jats:tex-math>\n \n d<\/mml:mi>\n \u2264<\/mml:mo>\n \u230a<\/mml:mo>\n \n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:mfrac>\n \u230b<\/mml:mo>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We prove that every poset of dimension 3<\/jats:bold> has a subdivision that is the separability order of boxes in $$\\mathbb {R}^3$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, that there are partial orders of dimension 2<\/jats:bold> that cannot be realized as box separability in $$\\mathbb {R}^3$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and that for any d<\/jats:italic> there are posets with dimension d<\/jats:italic> that are separability orders of boxes in $$\\mathbb {R}^3$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We also prove that for any d<\/jats:italic> there are partial orders with box separability dimension d<\/jats:italic>; that is, d<\/jats:italic> is the smallest dimension for which they are separable orders of sets of boxes in $$\\mathbb {R}^d$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s11083-023-09628-8","type":"journal-article","created":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T13:57:02Z","timestamp":1684159022000},"page":"699-712","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Separability, Boxicity, and Partial Orders"],"prefix":"10.1007","volume":"40","author":[{"given":"Jos\u00e9-Miguel","family":"D\u00edaz-B\u00e1\u00f1ez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paul","family":"Horn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mario A.","family":"Lopez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nestaly","family":"Mar\u00edn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adriana","family":"Ram\u00edrez-Vigueras","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8399-3321","authenticated-orcid":false,"given":"Oriol","family":"Sol\u00e9-Pi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Stevens","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jorge","family":"Urrutia","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,5,15]]},"reference":[{"issue":"9","key":"9628_CR1","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1002\/j.1538-7305.1966.tb01713.x","volume":"45","author":"FW Sinden","year":"1966","unstructured":"Sinden, F.W.: Topology of thin film rc circuits. 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