The Completion of a Lattice Ordered Group

Conrad, Paul; McAlister, D. B.

doi:10.1017/s1446788700005760

Cited by 67 publications

(57 citation statements)

References 19 publications

(36 reference statements)

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“…In case G = A 1 -© ^^ the following properties of projections />! and p 2 onto y*" 1 " and A ±J~ respectively are easily proved: To obtain these results it is sufficient to bear in mind that any /-group is a distributive lattice and that g y +g 2 = g x vgi+ffi A 9i-F°r other concepts used and not denned we refer to Birkhoff [1 ]. 70 T. P. Speed and E. Strzelecki [2] 2.…”

Section: Notations and Supplementary Resultsmentioning

confidence: 98%

“…Let a be discrete and let g ± \ig 2 . In this case b x = \a\A\gy\ and b 2 = \a\ A \g 2 \ are disjoint positive elements dominated by \a\.…”

Section: Notations and Supplementary Resultsmentioning

confidence: 99%

“…Conversely, suppose that 0 ^ g l ^ \a\, 0 ^ g 2 ^ \a\ and that g x ±g 2 . According to (ii), we may assume that e.g.…”

Section: Notations and Supplementary Resultsmentioning

confidence: 99%

“…Let p l and p 2 denote the projections on gf x and gf respectively. We have then a = p t (a)+ Pi( a ) with Pi( a ) -L Pi{ a ) a n d since a > 0, p^a), p 2 Repeating the reason from [7], we obtain…”

Section: An Element a Of A Stone L-group G Is An Atom If And Only If mentioning

confidence: 92%

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A Note on Commutative l-Groups

Speed¹,

Strzelecki²

1971

J. Aust. Math. Soc.

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Let G be a commutative lattice ordered group. Theorem 1 gives necessary and sufficient conditions under which a⊥ with a∈G is a maximal l-ideal. A wide family of, l-groups G having the property that the orthogonal complement of each atom is a maximal l-ideal is described. Conditionally σ-complete and hence conditionally complete vector lattices belong to the family.It follows immediately that if a is an atom in a conditionally complete vector lattice then a⊥ is a maximal vector lattice ideal. This theorem has been proved in [7] by Yamamuro. Theorem 2 generalizes another result contained in [7]. Namely we prove that if M is a closed maximal l-ideal of an archimedean l-group G then there exists an atom a ∈ G such that M = a⊥.

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Section: Notations and Supplementary Resultsmentioning

confidence: 98%

“…Let a be discrete and let g ± \ig 2 . In this case b x = \a\A\gy\ and b 2 = \a\ A \g 2 \ are disjoint positive elements dominated by \a\.…”

Section: Notations and Supplementary Resultsmentioning

confidence: 99%

“…Conversely, suppose that 0 ^ g l ^ \a\, 0 ^ g 2 ^ \a\ and that g x ±g 2 . According to (ii), we may assume that e.g.…”

Section: Notations and Supplementary Resultsmentioning

confidence: 99%

Section: An Element a Of A Stone L-group G Is An Atom If And Only If mentioning

confidence: 92%

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A Note on Commutative l-Groups

Speed¹,

Strzelecki²

1971

J. Aust. Math. Soc.

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“…It is well known (cf. [5]) that for any abelian Archimedean l-group Γ its completion Γ can be constructed such that…”

Section: Corollary 32 Let H : G / / γ Be a Complete Gcd Theory Thementioning

confidence: 99%

Ordered groups with greatest common divisors theory

Močkoř

2000

International Journal of Mathematics and Mathematical Sciences

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Abstract. An embedding (called a GCD theory) of partly ordered abelian group G into abelian l-group Γ is investigated such that any element of Γ is an infimum of a subset (possible non-finite) from G. It is proved that a GCD theory need not be unique. A complete GCD theory is introduced and it is proved that G admits a complete GCD theory if and only if it admits a GCD theory G / / Γ such that Γ is an Archimedean l-group. Finally, it is proved that a complete GCD theory is unique (up to o-isomorphisms) and that a po-group admits the complete GCD theory if and only if any v-ideal is v-invertible.

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Conditionally orthogonally complete l‐groups

Jakubík

1975

Mathematische Nachrichten

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A lattice ordered group G is called (conditionally) orthogonally complete if each (bounded) disjoint subset of G has the supremum. Orthogonally complete 2-groups were investigated in several papers; cf. e.g., [l], [3], [6]. G is said t o be complete (o-complete) if each bounded subset (each bounded countable subset) of G has the supremum and the infimum. VEKSLER and GEJLER [ill have found necessary and sufficient conditions under which a conditionally orthogonally complete vector lattice is complete. The aim of this note is to show that if an 1-group G is conditionally orthogonally complete and o-complete, then it is complete. A particular case of this result (concerning singular Z-groups) was proved in [7]. Further we show that if G is archimedean and conditionally orthogonally complete, then the largest singular l-ideal of G is a direct factor of G. This generalizes a result of CONRAD and MCALLISTER [4]. S 1. Preliminaries For the basic notions and denotations concerning lattice ordered groups (Z-groups) cf. BIRKHOFF [2] and FUCHS [ 5 ] . The lattice operations are denoted by A, v and the group operation is written additively. Let G be an 1-group. For any subset X G put Xa = {YE G: lyl A 1x1 = 0 for each x E X ) .The set X' is called a polar of G. Each polar of G is a convex closed 1-subgroul) of G . For g E G we denote {g)6a = [g]; the l-group [g] is said to be a principal polar of G. A subset 1M & G will be called disjoint if m > 0 for each m E M and m i A m2 = 0 for any pair of distinct elements ml, m2 E M . A convex 1-subgroup A of G is said to be a direct factor of G if there exists a convex l-subgroup B of G such that A n B = 0 and A + B = G. If such 1-group B does exist, then it is uniquelly determined; namely, B = A'. This situation is denoted by writting G = A x B and B = A*. Then each g E G can be written

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The Completion of a Lattice Ordered Group

Abstract: A lattice ordered group(‘l-group’) is called complete if each set of elements that is bounded above has a least upper bound (and dually). A complete l-group is archimedean and hence abelian, and each archimedean l-group has a completion in the sense of the following theorem.

Cited by 67 publications

References 19 publications

A Note on Commutative l-Groups

A Note on Commutative l-Groups

Ordered groups with greatest common divisors theory

Conditionally orthogonally complete l‐groups

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