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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.

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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 Vuković 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' Algèbre, 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)