Minimal lattice-sub spaces

Polyrakis, IA

dc.contributor.author	Polyrakis, IA	en
dc.date.accessioned	2014-03-01T01:14:48Z
dc.date.available	2014-03-01T01:14:48Z
dc.date.issued	1999	en
dc.identifier.issn	0002-9947	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/13228
dc.relation.uri	http://www.scopus.com/inward/record.url?eid=2-s2.0-22844455889&partnerID=40&md5=54d7aac7ad15f5dd542cc8d0b075ea81	en
dc.subject.classification	Mathematics	en
dc.title	Minimal lattice-sub spaces	en
heal.type	journalArticle	en
heal.language	English	en
heal.publicationDate	1999	en
heal.abstract	In this paper the existence of minimal lattice-subspaces of a vector lattice E containing a subset B of E+ is studied (a lattice-subspace of E is a subspace of E which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology tau on E and E+ is tau-closed (especially if E is a Banach lattice with order continuous norm), then minimal lattice-subspaces with tau-closed positive cone exist (Theorem 2.5). In the sequel it is supposed that B = {x(1); x(2),...,x(n)} is a finite subset of C+(Omega), where Omega is a compact, Hausdorff topological space, the functions xi are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function beta(t) = r(t)/parallel to r(t)parallel to 1 where r(t) = (x(1)(t), x(2)(t),...,x(n)(t). If R(beta) is the range of beta and K the convex hull of the closure of R(beta), it is proved: (i) There exists an m-dimensional minimal lattice-subspace containing B if and only if K is a polytope of R-n with m vertices (Theorem 3.20). (ii) The sublattice generated by B is an m-dimensional subspace if and only if the set R(beta) contains exactly m points (Theorem 3.7). This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (sublattice) exists and also determines these subspaces.	en
heal.publisher	AMER MATHEMATICAL SOC	en
heal.journalName	Transactions of the American Mathematical Society	en
dc.identifier.isi	ISI:000081517100014	en
dc.identifier.volume	351	en
dc.identifier.issue	10	en
dc.identifier.spage	4183	en
dc.identifier.epage	4203	en