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is an unperforated partially ordered group with a property slightly weaker than being a lattice-ordered group. Namely, a Riesz group satisfies the
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Elliott, George A. (1976). "On the classification of inductive limits of sequences of semisimple finite-dimensional algebras".
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to be integrally closed and to be
Archimedean is equivalent. There is a theorem that every integrally closed
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The
Archimedean property of the real numbers can be generalized to partially ordered groups.
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This property is somewhat stronger than the fact that a partially ordered group is
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506:. The partially ordered groups, together with this notion of morphism, form a
1217:
1186:
30:"Ordered group" redirects here. For groups with a total or linear order, see
1047:. This has to do with the fact that a directed group is embeddable into a
1197:, Lecture Notes in Pure and Applied Mathematics 187, Marcel Dekker, 1995.
638:
1300:, Lecture Notes in Pure and Applied Mathematics 27, Marcel Dekker, 1977.
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557:, where the group operation is componentwise addition, and we write (
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is a partially orderable group if and only if there exists a subset
1063: โ Group with a cyclic order respected by the group operation
358:. Being unperforated means there is no "gap" in the positive cone
1266:, Siberian School of Algebra and Logic, Consultants Bureau, 1996.
377:, i.e. any two elements have a least upper bound, then it is a
1051:
lattice-ordered group if and only if it is integrally closed.
170:. So we can reduce the partial order to a monadic property:
645:
is a partially ordered group: it inherits the order from
204:, the existence of a positive cone specifies an order on
1255:
V. M. Kopytov and A. I. Kokorin (trans. by D. Louvish),
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Partially ordered groups are used in the definition of
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Pages displaying wikidata descriptions as a fallback
1075: โ Algebraic object with an ordered structure
975:
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914:
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810:
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745:
709:
551:A typical example of a partially ordered group is
1364:Transactions of the American Mathematical Society
1445:Creative Commons Attribution/Share-Alike License
629:is some set, then the set of all functions from
1259:, Halsted Press (John Wiley & Sons), 1974.
486:are two partially ordered groups, a map from
8:
1101: โ Ring with a compatible partial order
1107: โ Partially ordered topological space
664:is a stably finite unital C*-algebra, then
658:approximately finite-dimensional C*-algebra
1081: โ ring with a compatible total order
1375:
1305:Lattices and Ordered Algebraic Structures
1270:Kopytov, V. M.; Medvedev, N. Ya. (1994).
1095: โ Vector space with a partial order
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1155:Lattice Ordered Groups: an Introduction
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27:Group with a compatible partial order
7:
1272:The Theory of Lattice-Ordered Groups
1195:The Theory of Lattice-Ordered Groups
697:Property: A partially ordered group
496:morphism of partially ordered groups
1262:V. M. Kopytov and N. Ya. Medvedev,
1202:Partially Ordered Algebraic Systems
154:By translation invariance, we have
25:
1358:Everett, C. J.; Ulam, S. (1945).
943:{\displaystyle n\in \mathbb {Z} }
625:is a partially ordered group and
373:. If the order on the group is a
1088:Ordered topological vector space
385:, though usually typeset with a
365:If the order on the group is a
1443:, which is licensed under the
1:
1296:R. B. Mura and A. Rhemtulla,
778:{\displaystyle e\leq a\leq b}
1462:Ordered algebraic structures
1345:10.1016/0021-8693(76)90242-8
541:is a partially ordered group
401:Riesz interpolation property
1424:Encyclopedia of Mathematics
1406:Encyclopedia of Mathematics
811:{\displaystyle a^{n}\leq b}
1488:
1210:Ordered Permutation Groups
1161:Birkhoff, Garrett (1942).
992:A partially ordered group
976:{\displaystyle b<a^{n}}
548:is a lattice-ordered group
369:, then it is said to be a
346:for some positive integer
322:A partially ordered group
136:. The set of elements 0 โค
29:
1401:"Partially ordered group"
1280:10.1007/978-94-015-8304-6
1167:The Annals of Mathematics
1153:M. Anderson and T. Feil,
675:) is a partially ordered
1238:Partially Ordered Groups
1235:Glass, A. M. W. (1999).
1218:10.1017/CBO9780511721243
1208:Glass, A. M. W. (1982).
1163:"Lattice-Ordered Groups"
1061:Cyclically ordered group
746:{\displaystyle a,b\in G}
660:, or more generally, if
1437:partially ordered group
1419:"Lattice-ordered group"
1417:Kopytov, V.M. (2001) ,
1399:Kopytov, V.M. (2001) ,
1204:, Pergamon Press, 1963.
1105:Partially ordered space
889:{\displaystyle a\neq e}
837:{\displaystyle n\geq 1}
43:partially ordered group
1307:. Universitext. 2005.
1099:Partially ordered ring
1067:Linearly ordered group
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915:{\displaystyle b\in G}
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371:linearly ordered group
200:For the general group
140:is often denoted with
32:Linearly ordered group
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870:. Equivalently, when
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59:translation-invariant
53:, +) equipped with a
18:Lattice ordered group
1264:Right-ordered groups
1257:Fully Ordered Groups
1093:Ordered vector space
1000:if for all elements
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539:ordered vector space
449:, then there exists
162:if and only if 0 โค -
144:, and is called the
1360:"On Ordered Groups"
863:{\displaystyle a=e}
621:More generally, if
326:with positive cone
1333:Journal of Algebra
1157:, D. Reidel, 1988.
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504:monotonic function
500:group homomorphism
1289:978-90-481-4474-7
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998:integrally closed
988:Integrally closed
710:{\displaystyle G}
679:. (Elliott, 1976)
146:positive cone of
16:(Redirected from
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689:Archimedean
546:Riesz space
224:such that:
120:An element
1456:Categories
1441:PlanetMath
1327:, chap. 9.
1248:981449609X
1200:L. Fuchs,
1173:(2): 313.
1148:References
996:is called
950:such that
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684:Properties
515:valuations
457:such that
216:(which is
208:. A group
128:is called
1429:EMS Press
1411:EMS Press
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1187:0003-486X
933:∈
907:∈
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614:= 1,...,
381:(shortly
288:for each
1055:See also
1049:complete
1041:directed
818:for all
639:subgroup
532:integers
525:Examples
508:category
350:implies
130:positive
1386:1990202
1045:abelian
383:l-group
132:if 0 โค
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1012:, if
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306:and -
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1317:ISBN
1284:ISBN
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218:G
214:H
210:G
206:G
202:G
196:.
194:G
190:b
186:a
184:-
177:b
173:a
168:b
164:a
160:b
156:a
148:G
142:G
138:x
134:x
126:G
122:x
115:b
113:+
111:g
107:a
105:+
103:g
99:g
95:b
91:g
87:a
83:b
79:a
75:G
71:g
67:b
63:a
51:G
49:(
34:.
20:)
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