Ordered groups with greatest common divisors theory

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...

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Main Author:	Jiří Močkoř
Format:	Article
Language:	English
Published:	Hindawi Limited 2000-01-01
Series:	International Journal of Mathematics and Mathematical Sciences
Subjects:	po-groups l-groups GCD-theory complete GCD-theory.
Online Access:	http://dx.doi.org/10.1155/S0161171200004087

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spelling	doaj-4c7ef22b46f843a9b433f6c5742c20fe2020-11-24T22:27:31ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252000-01-0124746947910.1155/S0161171200004087Ordered groups with greatest common divisors theoryJiří Močkoř0Department of Mathematics, University of Ostrava, Ostrava CZ-702 00, Czech RepublicAn 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.http://dx.doi.org/10.1155/S0161171200004087po-groupsl-groupsGCD-theorycomplete GCD-theory.
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Jiří Močkoř
spellingShingle	Jiří Močkoř Ordered groups with greatest common divisors theory International Journal of Mathematics and Mathematical Sciences po-groups l-groups GCD-theory complete GCD-theory.
author_facet	Jiří Močkoř
author_sort	Jiří Močkoř
title	Ordered groups with greatest common divisors theory
title_short	Ordered groups with greatest common divisors theory
title_full	Ordered groups with greatest common divisors theory
title_fullStr	Ordered groups with greatest common divisors theory
title_full_unstemmed	Ordered groups with greatest common divisors theory
title_sort	ordered groups with greatest common divisors theory
publisher	Hindawi Limited
series	International Journal of Mathematics and Mathematical Sciences
issn	0161-1712 1687-0425
publishDate	2000-01-01
description	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.
topic	po-groups l-groups GCD-theory complete GCD-theory.
url	http://dx.doi.org/10.1155/S0161171200004087
work_keys_str_mv	AT jirimockor orderedgroupswithgreatestcommondivisorstheory
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Ordered groups with greatest common divisors theory

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