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R E V I E W O F T H E M O N O G R A P H
STRUTURES PARAGRADUEES (groupes, anneaux, modules)
PARAGRADED STRUCTURES (Groups, Rings, Modules)
Queen's Papers in pure and applied Mathematics, No. 77, Queen's University,
Kingston, ON (1987), pp. VIII + 163.
OCLC No. 3610 4230; OL 625 989M; ISBN: QPPAM 77.
CLASSIFY OCLC Work Id: 22166894
VIAF Virtual International Authority File
Vuković, Mirjana
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WorldCat 870640200
VIAF ID: 40961814 (Personal)
Permalink: http://viaf.org/viaf/40961814
ISNI: 0000 0000 7401 2611
Id Ref: 176341250
MGP ID: 159055
Math-Net Ru: 75666 (person)
BNF: cb125985296 (data)
Bibliothèque nationale de France
Library of Congress Control Number LCCN
Library of Congress/NACO: LC-n88662504
Open Library ID: OL17925698M
100 1_ D
German National Library: DNB 177864095
100 1_
National Library of the Netherlands: NTA-133232034
100 1_
German National Library: DNB -1050443322
100 1_
SUDOC [ABES] SUDOC- 176341250
100 1_
Library of Congress/NACO: LC-n 88662504
100 1_
100 1_ ‡a Vukov
Library and Archives Canada: LAC-0060G9629
100 1_ ‡a Vuković
Math-Net Ru: personal 75666
WorldCat: 870640200
SUDOC-ABES.fr sudoc.fr/176341080
100 1_ ‡a Vuković, Mirjana AU ID: 00087077
CiNii-ID: DA01996367; AU ID: 087077
Libraries Australia ID: 57 13528
trove.ula.ua University Libraries Australia:
R E V I E W O F T H E M O N O G R A P H
Paragraded Structures (Groups, Rings, Modules), Queen' s Papers in pure and
applied Mathematics, No. 77, Queen's University, Kingston, ON (1987),
pp. VIII + 163.
In this monograph Marc Krasner and Mirjana Vukov developed a general theory of
algebraic graded structures (groups, rings, modules), which are at the same time a
generalization of the classical graduation, as defined by Bourbaki, as well as an
extension of the work Anneax graduées généraux (Colloque d' Albre, Rennes
(1980), pp. 209-308), done by M. Krasner in which is included also the notion of a
graduation introduced by M. Krasner in 1940’s in his paper Une généralisation de la
notion de corps-corpoide. Un corpoide remarquable de la theory des corps values
(Comptes Rendus, T. 219 (1944), 345-347). One of the most important properties of
the paragraded structures (groups, rings, modules) is that the homogeneous part of the
direct product and the direct sum is the direct product and the direct sum of the corres-
ponding homogeneous parts, respectively. This work extends many classical results
and opens a completely new branch of the research. (V. Perić)
The monograph "Structures paragraduées (groupes, anneaux, modules) consists of
detailed Introduction (pp. 1-54), where the historical developement of the graded
structures (groups, rings, modules) are presented and a motivation for the introduction
of more general paragraded and extragraded structures, and sketch of table of contents
is given. The main part of this monograph is divided into the following three chapters:
The Chapter I (pp. 55-98), called Extragraded and paragraded groups, the Chap-
ter II (pp. 99-127), where we observe structures derived from paragraded groups,
and the Chapter III (pp. 128-162) dedicated to paragraded rings and modules which
are defined from paragraduation of the additive groups similarly as in the case of the
graduation.
The graduation of a multiplicative group G is defined as a mapping γ : Δ Sg (G)
which joins an element Δ to some subgroup G Sg (G) of the group G, such that
the group G is the direct sum of the groups G ( Δ). The elements of a set Δ are
called grades (degrees) and they are essential if G {1}, and empty if G = {1}.
The set H = Δ G is the homogeneous part of the graded group G, and elements
of that set are homogeneous elements. If there are empty grades, then we usually
distinguish one of them and mark it as 0, that is grade of 1.
The theory of graduation by Marc Krasner has three aspects:
For the homogeneous element x 1, there is exactly = (x)Δ for which x G
and we call it the grade of the homogeneous element x.
1. non-homogeneous, where graded group G is observed as the group with
certain graduation;
2. semi-homogeneous, where the group (G, H) is observed as pair of the group G
and the so-called homogroupoid H G, so possiblly graduation of the group
G can be reconstructed (determined up to the equivalence) using the homo-
geneous part H;
3. homogeneous, where we observe the homogroupoid (H; ) and possibly we
can construct a so-called linearization (H; ) of this homogroupoid, which is
in fact the graded group for which (H; ) is the homogeneous part.
The homogroupoids are certain partial structures (given by axioms 1oo - 4oo, pp. 5-6 ).
The graduation of a ring (A,+, ) is a γ: Δ Sg(A) of its additive group (A, +),
such that for every pair of grades ξ, η Δ there exists ς Δ with AξAη Aς.
The homogeneous part (H,+, ) of a graded ring (A,+, ), as the partial structure of
the ring induced by the ring structure can be characterized with a certain system of
axioms (1o- 6o, pp. 15). Such partial structures are called aneids. The unitary graded
ring is the graded ring with the unit element which is homogeneous. If also
H* = H\{0} is the multiplicative group, then the graded ring (A,+, ) is called a
graded corp and the aneid (H,+, ) is called a corpoid.
Every aneid (H,+, ) determines a graded ring with homogeneous part H up to the
H-isomorphism which is called a linearization of the aneid H and is denoted by
(H,+, ).
The graded module is the graded Abelian group (M, +) which is a module over the
graded ring (A,+, ), such that for every grade Δ of the ring and for every grade
d D of the group there exists t D of the group for which we have A Md Mt. In the
homoneous part N of the group M we have apart from partial operation + and externel
partial operation on the aneid (H,+, ) which determine a partial structure (H, N) of
so-called H-moduloid.
The graded groups (rings, modules) are objects of a certain category whose morphi-
sms are the so-called homomorphisms. The category is not closed with respect to the
direct product in which the homogeneous part would be the direct product of the
homogeneous parts of the factors. Similar holds for the homogroupoids (aneids,
moduloids).
The chapter I is divided into three paragraphs. In the first one the theory of
paragraded and extagraded groups is presented from the non-homogeneous aspect.
First, an extragraduation of the group G is defined as a generalization of the
graduation. It is a certain mapping E: ΔSg(G) of a partially ordered set (Δ , <),
which is from bellow complete semi-lattice and from beyond inductively ordered set
Sg(G) of the subgroups of the group G, which fulfill a certain system of axioms
(I - VI, pp. 56- 60). One of them (axiom IV) requests that H has to generate the
group G. Four axioms result with certain commutative relations in H, which together
with a relation of the form xy = z in H, create a set of relations in H. If the mapping
E fulfills only first five axioms, and G is generated with H and the system of the
relations R, then E is called a paragraduation of the group G. The main result of this
paragraph is the Theorem 1 (p. 66) which states that each extragraduation is at the
same time a paragraduation. The homogeneous part is, of course, the set H =
ΔG, while the grade of the homogeneous element x is defined as
inf {Δ : x G }.
Each graduation is a special paragraduation. Namely, it is sufficient that the set of
empty grades is well ordered, taking the empty grade 0 as its first element, and after
all of them to put mutually incomparable essential grades of the given graduation.
In the second paragraph of the first chapter we observe extragraduation and para-
graduation from the semi-homogeneous aspect. There, the proof of the Theorem 4
(pp. 87-90) is given apart the other things, which gives sufficient and necessary
conditions for the certain subset H of the group G to be the homogeneous part of
that group for the paragraduation, resp. extragraduation.
The third paragraph has a homogeneous aspect. Some of the conditions (1o, 4o, and
partially of condition 5o) from the Theorem 4 are translated to the language of the
partial operation in H (conditions II* - IV*) and to these we add condition I*, so H
has a unit element 1. The partial structure (H, ) which fulfills those conditions is
called a quasigroupoid. The so-called morphisant function u(x, y) determines some
commutative relations which together with the relation of the form xy = z in H, in the
paragroupoid H, creates a set of the relations R.
If F is a free group generated by the set H, and Nu is its normal subgroup which
corresponds to the set of the relations R, then the mapping
u*: x xNu (x H)
is a homomorphism in the sense that for every x, y H with xy H,
u*(xy) = u*(x) u*(y).
In order that the paragraded group G exists with H as its homogeneous part, it is
necessary and sufficient that the following conditions are fulfilled:
1) x Nu H = { x }, (x H) and
2) xyNu H Ø x y H (p. 94).
In that case H is called an u-paragroupoid, and the corresponding paragraded group
(H, , u) is called its linearization, while the function u is called the linearizant
function. In the case that (H, , u) is an extragraded group, the partial structure
(H, , u) is called an u-extragroupoid. According to the Theorem 4, it is the case
when the transcription V* of the conditions 5o holds ( pp. 95-96).
In Chapter II we introduce structures derived from the paragraded groups. We start
with homogeneous groups and prove that from the paragraduation (extragraduation)
of the group G we get the corresponding graduation of its homogeneous subgroup
g
(Lemma 1.), show how the homogeneous part h of the homogeneous group
g
can be
characterised (Lemma 3.), in other words a subgroupoid h of the extragroupoid H
(Lemma 4.).
Further, we define a factor structure H/h and for that structure we proved The first
and The second theorem of isomorphism. With the help of the quasihomogeneous
isomorphism Φ of the paragraded group (G, , E) to the paragraded group (G', , E'),
(G, , E (Φ)) the so-called aglutinate of (G, , E ) is defined by Φ, which is a
paragraded group if ( G', , E') is.
For the family of paragraded groups the direct product, and the direct sum is defined,
in the natural way. Those are, again paragraded groups, and the direct sum is a
homogeneous subgroup of the direct product. The direct product (direct sum) of the
paragraded groups is an extragraded group, if and only if all of the factors are
extragraded groups. However, the direct product of the graded groups is only an
extragraded group, except if at most one of the factors is with non-trivial graduation,
i.e. Δα { 0 }.
Corresponding construction of the direct product of the quasi-groupoids gives a so-
called multigroupoid. In the set of Hom(H, H') of the quasi-homomorphisms of the
quasigroupoid (H, +) to the quasigroupoid (H',+), the additive operation is defined
which makes it into a commutative quasigroupoid.
If (H', +) is commutatively linearizable, that is if there exists 0 linearization
(H',+,0) such that the direct product (H',+,0)H is one commutative linearization of the
direct product (H',+) H . In this case (Hom(H, H'), +) is a paragroupoid.
Chapter III is dedicated to paragraded rings and modules, which are obtained from
the paragraduation of correspondings additive groups similarly as in the case of the
graduation. We will not be discussing in details the contents of this chapter, also very
rich with the results.
From what has been here said, it can be seen that large work "Paragraded
structures" contains many important results. They refer to first time introduced here,
paragraded structures (groups, rings, modules) that generalize earlier known graded
structures, with good justifications for that generalization.
In this work M.Krasner and M.Vuković developed a general theory of algebraic
graded structures which is at the same time a generalization of classical graduation, as
defined by Bourbaki, as well as an extension of the work done by M. Krasner (Anneax
gradués généraux, Colloque d'algèbre, Rennes (1980), pp. 209 308). So, this work
extends many classical results and opens a completely new branch of research.
(V. Perić)
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