(1)STUDIA NIVERSITATIS BABES-B0LYA1 MATHEMATICA 3 1980 CLUJ-NAPOCA (2)REDACTOR Ş E F : Prof

STUDIA

NIVERSITATIS BABES-B0LYA1

MATHEMATICA

3 1980

CLUJ-NAPOCA

REDACTOR Ş E F : Prof. I. VLAD

R E D A C T O R I Ş E F I A D JU N C Ţ I: Pro/. I. HAIDUC, prof. L KOVACS, prof. L A. RUS

COM ITETUD D E R ED A C Ţ IE MATEMATICA: Prof. C. KALIK, prof. I. MARUŞCIAC, prof. P . MOCANU, prof. I. MUNTEAN, prof. A. PAL (redactor responsabil), prof.

D. D. STANCU, con/. M. RĂDVLESCV (secretar de redac(le)

STUDIA

UNIVERSITATE BABEŞ BOLYAI

MATHEM ATICA

3 * * •

R e d a c ţ i a : 3400 CLUJ-NAPOCA, str. M, Kogălnlceanu, 1 • Telefon 1 34 50

S U M A R - C O N T E N T S - S O M M A I R E

GR. CĂEUGĂREANU, On an Enriched theory of Modules (II) # Asupra unei teorii îmbo­

găţite a modulelor (II) ... 3 N. I/UNGU, Calculul pulsaţiilor neliniare ale unei sfere de gaz în rotaţie • The Calculation

of Nonlinear Pulsations of the Gas Sphere in R o tatio n ... 18 D. ACU, Noi formule de cuadratură cu elemente fixe • Nouvelles formules de quadrature

à éléments f i x e s ... 25 E . SCHECHTER, Remarks on the numerical solution of a nonlinear parabolic equation #

• Observaţii asupra rezolvării numerice a unei ecuaţii parabolice neliniare... 32 GH. COMAN, On some practical quadrature and cubature formulas • Asupra unor formule

practice de cuadratură şi c u b a tu ră ... 40 P. ENGHIŞ, |P. SANDOVICl], M. ŢARINĂ, Surla récurrence de la métrique de Schwartz-

schild • Asupra recurenţei metricii lui Schwartzschild... 48 P . ENGHIŞ, Quelques remarques sur la métrisabllité des espaces An T-récurrents et T-ré-

currents • Cîteva observaţii asupra metrizabilităţii spaţiilor .¿„T-recurente sau T- recurente ... 50 U. BRĂDEANU, Un schéma implicite aux différences finies pour le problème de la couche

limite hydrodynamique # O schemă implicită cu diferenţe finite pentru problema stratului limită hidrodinamic ... 53 A. B. NEMETH, The comparison of the Michal-Bastiani and of the Clarke subdifferential

• Comparare între noţiunile de subdiferenţiale a lui Michal-Bastiani şi Clarke . . . . 60 C. K A EIK , Génération d'éléments spline à l'aide des applications monotones • Generarea

elementelor spline cu ajutorul aplicaţiilor monotone . ... 66 h . BITAY, Equations à quatre variables représentables par un nomogramme composé avec

deux échelles projectives sur des coniques 0 Ecuaţii cu patru variabile reprezentabile cu o nomogramă compusă cu două scări proiective situate pe co n ice ... 72

2

R e c e n z i i - B o o k s - L i v r e s p a r u s

C a r l d e B o o r , A Praetieal Guide to Spline (GH. M IC U L A )...

C h r u s t o p h e r T. H. B a k e r , The Numerical Treatment ot Integral Equations (GH.

M I C U L A ) ... * p S i n g e r , Programmierung m it COBOL (Z. RASA) ...

M. M. R i c h t e r , Logikkalkule' (N. BOTH) ...

W . J . P a u l , Komplexitätstheorie (C. TARŢIA) ...

79 80 80 79

. i. ’ ' 0

c%. ;

STUDIA U N IV . B A BE$-BO LYA I, M ATHEM ATICA, X X V , 3, 1980

ON AN E N R IC H E D TH EO R Y OF MODULES (II) ' : * GRIGORE CALUGAREANU

Introduction. Stimulated by the excelent monography [4], the author of the preseut paper works out the closed and monoidal closed part of the theory of modules over a fixed monoid, a theory for which, in [5], M a c l a n e only worked out the monoidal part.

The reader needs only the first section from [3]-where the basic notions : closed and monoidal monoids and the corresponding morphisms, left and right modules over monoids, are defined, and the basic situations studied, — in order to recover our main definitions.

From section two all the definitions and results following the corrolary 2.7 are needed.

In this way, we shall start this second part of this paper with section three. In what follows, we suppose th at V0 is a symmetric monoidal closed cate­

gory with equalizers.

3 . The closed and monoidal closed structure of PMV. Demma 3.1. — The morphism zA = n(yA • cAJt) : A -► (RA), considered for a left R-module (A, aA, yA) over V, factors through {E A }. Moreover, denoting by : A —*■ {E A } the factori­

zation morphism, this is an isomorphism in V 0.

Proof. First, we have to check (3) (aA, 1) • Rar • zA = (n, 1) • Las • zA.

Using [II, (3.1), (3.19), (3.22)] we have

(», 1) • LrA • zA = (*(»»), 1) • Lra • zA = P • K 1) • n(yA • cAR) =

= P * Mya • Car * 1 ®»») = * * (ya ‘ car • 1 ® m • a) = mz(yA • m ® 1 • ca,r9r -a).

The following commutative diagram

(A8R) 8R - a-* AWeRl-MRSRlGAÎ^â^RÔA

c®1 a'1

(R® A)®R —®—>R®(A8R)^_* R8(R8A)-^-^R®A enables us to continue the equalities above

= nn{yA . 1 ® (Y^ • cAR) • aRAR • ^ R® 1) = A < Ya‘ 1®(Yx ; car) • a) ‘ car) using again [II, (3,1)]. A t the same time, we have

(<xA, 1) • Efj* • zA = u(Mar- ^c(rauaa)• za®cca) = u{Mra • .«a® *a ■ car)- Thus, the equality (3) is equivalent to the following

4 GR. CALUGAREANU

m

*

• *A®*A = "ÍYx • 1®(Y

* c AR)I . aRAR) or to the one derived t

appljang

, which is proved as follows ea from

= P • (Yx • C^ar

, 1) • «x = P • "(Yx • * ® (Y

using [II, (3.1), (3.19), (3.22)].

Now, let us prove that ix 1: (-R4) —> {IA ) — > A is a

inverse for iA, that is, let us check the following two equalities (e, 1) . Zjl =

= *x» ^za

’ »

(e, 1) • 7t(Yx • Car) = A y a ' cAR • 1 ® « ) = * (Y x • * ® 1 • cAI) = tc (lA . cAl) =

= A *a) = *x-

As for the second, we first derive from [II, (7.4)] the following commutative diagram

(0.1) .-1

‘X

lA-M llAlS--- »(R/IA.10R1

'ilAi

(III A )- M/1A)

(lR>' -»(10R;(IA,^miK^í

Using i(jA) = (1, iA) and the following commutative diagram (by naturality of K)

(IA )-

1ÍA.

n,u)

*(I® R,A® R)

| (1, iA81)

(I (IA)--- iltUAl---, f J 0R,n A)® R)

above as follows ^

now prove

second

^ 1 ? , * ^ = n ^ A ‘ °AR ’ * cqUgA =

’ Yx * ^x* • zA '® 1) . U(/A)iR . (e, 1) . equ^x =

= (l~l ; yAr R AR ' iA 0 1 ) * {1r1> *A @ 1 ) • K *A • k 1) =

( * » Yx • • K

. (*, 1) . equju = (/;», . eAR) . (<®1,1) •

= .,0.1 ^ ’ Yx • cx*) * (c**, C

) • i r L • equ*x = „

= l l

*

!) • equS/1 = equRA, using also [II, (3.4)]

lem®3 y =I/

^RMA ~ Itft R-module (A, <xA, yA) over V, the morph ^

A RA ' C(RRH**) • w(w • cRR) ® e q u ^ : 2?® {R A } - (2?^) / « ^ fS

ON A N ENRICHED THEORY OF MODULBS (II)

5 {R A}. The unique factorization morphism ts>RA< R®{RA} - (RA) provides a left

R-module structure over V for {.R d}. J

Proof. Using the F-functoriality of RA and L*, the naturalitv of M a com m utative diagram derived from [III, (4.4)] and (3) from the previous lemma we have

(cla , 1) • Rar • Mra • C(R*), (^jm) • ^tc(m • Cjejj)®equjix =

= (n, 1) • L RA • MrA • c{RR)i {RA). 7t(m • cRR)®equRA, which proves the existence of the required morphism <oRA.

N ext, let us check th at uRA) actually is a left ^-module over V, th at is, the com m utativity of the following diagrams

(R8R )® [R A }-

m«1 |

r« [r a)

►R«(R®

ÎRA]

RA]) ,06jRA R®(RA]

B y left composition with equiX, the first one is equivalent with yRA • e ® l =

= equju • 1{RA) denoting by y RA = eq u ^ • <aRA, or, applying tc, n(yRA) • e =

= (1, e q u ^ ) • j {RA]. Using n(yRA) = (equ, 1) • Rrr • n(m- cRR) and n(m ■ cRR) • e=

— j R (from proposition 1.10) it is sufficient to verify Rrr • j A = j{RAy But this follows easily from [CC2], i.e., (jR, 1) • L RA = ifRA) applying tc-1 and using

[1 1 1 ,(4 .4 )] for [11 1 ,6 .4 ],

The com m utativity of the second diagram is equivalent with yRA • m<g>l =

= yXA ■ • a. We first mention that again from proposition 1.10 we have P • m = Mrr ■ p o p • cRR = Mrr • c(RR),(RR) • P®P- which implies, applying tc, (1, p) • tc(ot) = (p, 1) • Rrr • p. Next, using the F-functoriality of RA and applying 7c, we derive an analogous of [CC3]

(R R ) _

(r a)(r a»

m .

i(RA>

(((RAXRA I,(RA)(RA3)-

♦ÜRRXRR#

( I ^ ) ►IRRXIRAXRAM

Using all these, the proof is the following

Terr(y RA • W ® 1) = n (n (y RA) • m ) = (1, n ( y RA)) • tc(m) =

= (1, (equjM, 1)) • (1. Rrr) • ( ! . P) • =

= (l, (equjM, l)) • i1» R

) ' (P* ^ T

= (1, (equjM, 1)) • (P* 1) • {RrR> *) • * R * * * P =

6 GR. CALUGARBANU

= (P, 1) • (R

, l) • ((equj

, 1), 1) • ^ __

= (*, ( yR*)> 1) *

•

• P = p • ( y RA,

• . p ^

=

p

cRR)

p

n(yRA .

= nn{yRA • l®co • a).

Le m m a

3.3. — The construction described in lemma 2.8. defines '

metric monoidal closed category V with equalizers a bifunctor { — _ f J ? Wr

XrM Z ^ Zo- i - K m ? x

Proof. I f f : (A’, <x.A’) -* (A, ocA) and g : ( B , <x.B) -*• (B\

b.) are momlii««, of left ^-modules over V, then {/, g} : {A B } -+ { A 'B '} is the unique moroS

of factorization through the equalizers ^

equ.AB

tABi—

e,»uxe

.[A 'e 'j

The functoriality is derived from the uniqueness. Thus, in order to prove the existence of the factorization morphism on the above diagram we must check («s'» 1) • R

’A' • (/» g) • equ^js = (a^<, 1) • L

'B’ • (/» g) • equ^a- Using the follo­

wing commutative diagram

l(BBHAB)) — ilila i— ((BB)(ABD—i2LLiB— ((b bIAB’))

{AB}- equ

*BBA

- (AB) l£A_

(Ug).1)

-((BBXABl) (f.g)

(AB) ‘‘BA

((9,1).(f.D)

• dBEi (A'B)) ■■ (0Cb’1L.(R|a'b)) ( V 1

the first member of the required equality is = (1,(/, g)) * (aa> ^ l Bat this is = (1, (/, e )) • (a* 1) • • equxs. Finally, a jm ilal *£% e q * » (like the above one) leads us to the second member ot tne *s

T

3.4. — For each left R-module {A, *

> fAj oveh jfu R-tnodvh natural isomorphism in DM V. iA: A -+ {R A }, where {2^4} has t

structure given by lemma 3.2. * ieft R-&o^e

Proof. We must show that \A actually is a morphism l;;.

over V, i.e., that the following diagram commutes

ON A N ENRICHED THEORY O F MODULES (II)

7 B y left composition with equ** it is sufficient to check zA . yA = y . i ® j We derive this from the following equivalent equalities -Z i-y, ■ c = M*

• za®zr> v1' za) ’ z'a = (zr> 1; • Lra • zA, this last equality being checked ana- logously, like the one in proposition 1.6.

The naturality (in RMVl) of the family i = iA-A - {7L4} reduces to the naturality of the family z = zA which is readily checked.

Th eo r em 3.5. — iR: R —*■ {RR} is an isomorphism of monoids from the opposite monoid of R to the monoid {RR} of the R-endomorphisms of the left R-module R over itself.

Proof. Straightforward, using equalities from the proof of proposition 1.10.

Remark. I t can be shown, in the usual subjacent Way, th at if V preserves equalizers and V • W is an epifunctor, then the left .R-mo.dule (R, n, m) over R is a projective object in PMV. Analogously one could now define Quasi- Frobenius monoids over V in the usual way.

— L e t us point out the second bifunctor corresponding to the monoidal structure of PM V . W e assume t h a t . F 0 has coequalizers and, for a monoid (jR, e,n,m), th a t {A, yA) is a object in PMV and (B, SB) is an object of MV„.

1 W e define the tensor product B ® RA as an object in V0, namely coequ (((8B®1A) • ¿bra, 1*® Ya) : B® [R ®A ) -+ B®A).

• This will be a quotient object of B® A. We shall denote by co e q u a l : B® A - + B ® RA the canonic epimorphism to the coequalizer.

Propo sitio n 3.6. In a monoidal category with coequalizers V, the above construction defines a bifunctor ® R : M Ve X PMV -» VQ.

Proof. Evidently, ® R((B, 8B), {A, y j ) = B ® RA. If / : (B, 8B) (B', 8B') is a morphism in M VP and g: (A, yA) (A*, y^') is a morphism in RMV then f ® Rg : B ® r A B '®rA' is the unique morphism of factorization through the

coequalizers

.B0A • B9A

ccequ I coequBA

B0r A ---- B0rA'

The functionality is derived from the uniqueness. In order to prove the exis­

tence of the factorization morphism on the above diagram we must check coequ^^ • f ® g • • ¿bra = coequb*a' • f ®S : ®ut this easily

follows using the following commutative diagram ( ,

B©(r©a) ^-ir-^B0R )0A -:V . U m

ffln«gj| , (f0l)0gl -

8® (R8A'/——^ — >(88R)gA' :

8 GR. CALUGARBANU

D

3.1. A monoid {R, e, m) over V is called commutative iftn-c In what follows we suppose the monoid (R,e,n,m) commutative.

T , q n if ( R e m ) is a commutative monoid over V and (A „ ^ ■ Ufi < M ) f i * " * " « » M lik‘

mutative diagram

Proof. <xA being morphism of monoids over V we have

M Îa

• *A®ctA = aA * m = <x.A • m • cRR =

• • cRR =

=

•

• aA®ccA. Applying

we get (ocA,

. L AA . aA =

= (a^ , 1) • Raa • <*-a-

P

3.8. I f (A, aA) and ( B , <xB) are left R-modules, there is a morphism Y{

) : -*• {A B } which gives {A B } a structure of left R-module over V.

Proof. We consider the morphism

A

=

• aB(S>equ,iB :

From the previous lemma aB = equBB • xB, so that xAB factors through equ^

using lemma 2.9. Hence a morphism Y{

) exists and makes the following diagram commutative

r«(ab}—

m

As u sS^e^e^eauhralen^ actually is a left B-module over t x . n . e& i — ^ . en* conditions by left composition with equ^s. namely ing it and [ n T m ' " * 1 * " ' ” ® 1 = *•» ‘ >®Twn) • «■ For the

«H*r

• Us T± (iu]\r aVe ^ ^ 1) • ^ :

rahty of j and the fact^thit"» •* ' J (AB}> also usmg ^CCI-*' thC For the second we have 18 * ^ * - module over £ '

= il feau r T ^ = w(*<**») * m> = ^ = , *

’ n a* ))• (!, L BB) .( \ - afl) >7t(m) = (equ^fl> 1))-(1, L Ab b) - ( *bA ) ‘1 b B ‘H

— / * ! )} ^a’B> ^ ‘ ^ BB> 1) • ^\aB), {AB) * LbB * ^

_ , , ^ *) • ((e v w i), i ) . .

• «a =

= £ • n (M * AB^ ^ L(A^ (AB) ' L * B ' * B = P • (*AB’ !) ' LbB- * 8 - AB • ccB<S)xAB) = . n {x AB . 1 ® r{AB}) = im (xAB • 1 ® Y {^ *

ON A N ENRICHED THEORY O F MODULES (II)

9 R em ark. W e m ust show th at in the commutative case the left f?-module structures defined on in the lemma 3.2 and in the previous proposition are identical. Using m • cRR = m and applying 7i—1 to the definition of equ^, one can show th a t M ARA • <xA ® e q u ^ = M RA

• c(RR)t(RA)

Th e o r e m 3 .9 .

Lemma

defines a bifunctor

Proof.

(f,

({AB},y{ABi)

{{A'B'},

RiiÀ'B1)---{alg}

follows from the equivalent equality M%B> • *B ® (f,g) • 1 ® equ^B = (f,g)

• Mab• aB® 1 • 1 ®equ^fl, which is true using the following commutative diagram ..b

R81AB1- « B *

Ulf.1

HBB)0(AB)- Iab_

(1.g)0(U)

IBB)8(AS) •

-CAB)

B (Ad)

R0IAB)- * B » -IBB)0(A0) -Itjg 8(Àb)

P

-*Ens

lizers then the following diagram of functors is commutative

Proof‘ Straightforward from the remark following lemma 2.8.

P

a^),

which appears in lemma 3.7 is a morphism of left R-modules. Further, the family j = ) {A,<xA) = xA is natural in RMV.

Proof, The following commutative diagram shows th at xA is a morphism of left ^-modules

RiR- m

-R0{AA} * * * * ? & —iAAWAA) V l . . : <

eQ°AA

-IAA)

10 G R . C À L U G À R B A N U

The n atu rality of j follows fro m ( 1 ,/ ) • * A = (/, 1) • oc^, tru e for a .

of left jR-modules / .( A , aA) -*• (A ', <xA-), eq uAA being monom orphism 01011’1“8111 P^roposition 3.12.

RV )

Proof.

show a t th e subjacent le v e l^ th a t V{z(AA]){\A) = etA. F in a lly , one reduc *°

to F (w m a* ) ( 1 A = J * ® 1a ‘ 1 r ’ using also th e q u a l i t y z(AA) = ( l s , T (. .. . ^

• U[AA),it and axiom [C C 5]. '

*

For each left R-modules (A, <xA), (B, ;aJ and ic \

there is a transformation " ' ,ac)

L = (Ci0(c) : {B C } -* {{ A B }, {A C }} natural in nMV.

Proof. Let us, first, mention the following generalization of lemma 2.9

(ecc, 1) • Rca • Mac •

eqU|C®equ^B

equBC®equ^. Hence there is a morphism

{BC}(g>{.4B} -*• {A C} which closes the commutative diagram ‘

{BCj8fA8>

(BC181AB)-

5L

MAC

equAC (AC)

Using again n there is a morphism Lie- {B C }-* ({.<4 B},J^4C}) for which (equ^a, 1)'

• L

• equBc = (1, equ^c) • Lac is true. If we want L ie to give us by facton zation a morphism h i c ' {BC} -*■ {{A B}, {A C }} .we have to check the fo owing equality («{¿cp 1) • R\

),{

\ : L ie = («MB), 1) • L\

%,{A

) • L

-

may use the definitions of the left B-module structure on {A B} ana t ^ (l,equ^B) . a {AB) = (equ„B, 1) . L A BB - aB and the analogous c h a n g e » * ^ Again, using the F-functonality of

composing to the right wi

and applying n, we get another analogous of [CC3] : faci-

BC ((AB (ACM

^AC)

wac)(ac))!«a b](ac)))

tec»

P.Lgc)

(LCC-1) (ICO.ftABHACM) ,))

----- ^ ( L i b b e d

We shall prove the required equality

prove rne required equality

to the ^|'T1(^[uQctors)- ^

(which still is a monomorphism, the functors ( X , —) being m

= W t q . l ) - ((1, equ^c), 1) * * $ % {* * • * Lac "

ON AM ENRICHED THEORY O F MODULES (II)

11

= («C. 1) • (L ie , 1) • ((equ^c, 1), 1) • R\a%i{AB) • L i c =

= («c* !) • (Lee, 1) • R\ac),{ab) • (1, equxe) • L ie —

= (ac> 1) ‘ (Lee, 1) • i?(^c},{AB) • (equ^B, 1) • L ie • equBC =

= (ac> 1) ’ (Lee, 1) • (1, (equ^B, 1)) • E $ q . (i<q • L ie • equBc =

= (ac> (equ/g, 1)) • (1, Lie) ■ Rcb • equBC = (1» (cqtr,iB, 1) • Lbc) ■ (fltB, 1) • Lbc * cquBc —

= (as> (equ^B, 1)) • (Lis, 1) • Ljib}_ (AB) ■ Lie • equBC =

•— ((equ.iB, 1) • L i s • aB, 1) • L(abI,(ac) • L i e • equBC = . .

= (<*{ab), 1) • ((1, equ^B), 1) • L(ab), (^ac) • L ie • equBc =

= (<X(ab}> 1) • (1, (1, equxe)) • L\abi{aq • Lbc =

= equ^c)) • (<*{ab), 1) • L\i^h ^AC) • L i c .

Thus, there is a morphism Lb c: {BC} -* {{AB}, {AC}} in V0. The proof of the naturality in „MV of the corresponding family is left to the reader.

Th eo r e m 3.14. I f V is a symmetric monoidal closed category with equali­

zers, (R,e,n,m) is a commutative monoid over V and the subjacency functor V :V 0-*

-♦E n s preserves equalizers, then „MV, the category of the left R-modules over V, is a closed category.

Proof Using theorem 3.4, the remark following proposition 3.8, theorem 3.9, propositions 3.10, 3.11, 3.12, 3.13, all data for the closed structure of „MV are constructed, the „unit” object being obviously (R,n,m) as a E-module over (R ,e,n,m). Proposition 3.12 is axiom [CC5] for „MV. Hence, one has only to verify the remaining axioms [CC1—4] for RMV. In what follows we shall prove, for instance, axiom [CC2]. We have to check the commutativity of the follo­

wing diagram

B y left composition with equB<{ylq this is reduced to the following commuta­

tivity

[ac]---- AC--- ►ÇAAlfACj}

IR/aCI)

12 GR. C À LU G À R EA N U

A new composition with (1, equ^c) gives us th e required proof : (1, equ^c) • z{ac) = (1. equ^c) • ^ (ï ^ c } • C{ac},r) =

= Ttfequ^c • Y{AC} * C{a q,s) = n(%AC • cIAC) ,r) = n ( M AC • ac ®equ<c . c{ilC} fi) _ . = n (M Ac • C(AC),(cc) • equ^c ® ac) — (ac> 1) * Rca.• equ^c =

= («X» !) • • equ^ = (j* 1) • (equ^, 1) • L AC • equAC =

= (j^» 1) ' * ^-ac — (1» equ*c) • (j^, 1) • L Aç.

Remark.

— <g)X : V0 -* V0, preserve limits (equalizers) and m onom orphism s.

— L et us return now to the monoidal stru ctu re of „M V .

Theorem 3.15.

For a commutative monoid (R, e, m), if the functor R ® — :

—

preserves coequalizers, proposition 3.6 defines a bifunctor

®r ' °MV X pMV — „M V .

Proof

(A,

(A,

cRA)

: B®(R®A)

B®A).

B ® RA

R®(B<g)A

t(R ® B )

B ® A ^ ^ B ® RA

Re(Bg(R8A))-^ -* R 8 ((B 8 R )B A )-16(!',,B'CBS^ 11 R£©SAi 1 0 (1 8 ^ )

The functor R ® — preserving coequalizers, this will p rove th e existence of a morphism ygtsA : R ® (B ®rA) -* B ® RA in V 0 which will provide th e left R~jn°~

dule structure on B ® RA. W e shall avoid this verification which only uses defi­

nitions and coherence. So, our y Ri BA closes th e following com m u tative diagram

V

^R.BA (RBBJBA— — ► B 0A - - .q- B A - » B « R A

“l

R 8 IB 0 A ) ■•■9 COeqUBA--- * R 0 ( B 0 R A|

-Ne? ' we ^ave to Prove th a t (B ®rA, y R ba) a ctu a lly is a left R-module, is, e com m utativity of th e following diagram s

R e i s e r - t f l _

(R0R m«1

0{B0rA)-

R 0 (B 0rA)

R.BA"

j 1®*R,BA R 0 (B®oA)

' B 0r A

WR

^R.BA

ON A N ENRICHED THEORY OF MODULES (II)

13 As for th e first, we can check the equivalent one obtained by

tion with l / ®coequBil(this being epimorphism) right composi-

Yr.ba • e ® 1 • 1 ®coequflil — '{r.ba • 1 ®coequBX . e ® I = .

= coeqUiM • Y^b® 1 • «-1 • ¿ 0 1 = coequ^.Yfl(g)l-(e® l)® l . a ~ 1 =

= coequBX • /B® 1 . a~i = coequfl/1 . lA9B = 1B9rA . 1 ®coequ3 UBA‘

As for the second, a right composition with lj?®/e®coequBi4 gives us the requi­

red proof

Yr.ba • • 1 ®coequJM = coequ^ • yB0 l • o r1 • m ® l =

= coequ^ • ya ® l • (1®^yb)®1 • « 0 1 • «-1 =

= coequ^ • Ya®l * «~1 • 1 ® (Yb® 1 ) • 1®«—1 • « =

= Yr.ba • 1 ®coequflil • 1®(y/,® 1) • l® « -1 • a =

= YR.ba • 1 ® Y « ,ba • l® ( l® c o e q u flJ ■ a = y s . BA • 1 ®yjz.ba • «• 1 ®coequail.

i - '

Finally, one easily checks that, using notations from proposition 3.6, f ® Rg actually is a morphism of left i?-modules over V, i.e., the following diagram commutes

R 8 (B8^rA )-- ^biba»98^r A

»(f9 Rg)

R»(BfcRA’)— M

Remark. If we suppose th at V 0 is abelian, ® preserves cokernels in both variables and we take cokernels instead of coequalizers we recover the similar Maclvane's result.

— Moreover, the following result is true

T^heorem 3 .1 6 . I f V is a symmetric monoidal category with coequalizers, (R, e, m) is a commutative monoid over V_ and R® — preserves coequalizers, then RMV_

is a symmetric monoidal category.

Proof, Simple generalization of Macl^ane s result. F or instance, if (A, yA) is an object in pM V, we have from the definition 1.9 yx • m® \ = yA • l®Yi<’a » th at is, yA coequalizes the pair ((m • * aRR*> J*® Y )- ^ ^ s , factors through co eq u ^ g iv in g one of our natural isomorphisms iA. K ® RA A.

We shall end our paper with the principal result which uses all the results obtained above

Theorem 3.17. I / V i s a symmetric monoidal closed category, V 0 has equa­

lizers and coequalizers, V preserves equalizers, (R, e, n, m) is a commutative monoid over V and R ® — preserves coequalizers, then RMV, the category of the left R- -modules over V, is a symmetric monoidal closed category.

14

GR. c a l u g a r e a n u

Proof. First we must find in V 0 m orp h ism s psac : { B ^

and prove th at this is a n atu ral fam ily of isom orp h ism s in rm^ p £ W } } we prove the existence of m orphism s Pb a c w hich close com m u tatively the foil

wing diagram ' ~ °'

{B 8r a,c}. Pmc B8r AP

♦IB,{AC}).

(B 80 AC) (C° eqUBAll l >(B 8A,C)-fBAC_»|B ,( A C ))

Because (B, —) preserves equalizers it will suffice to ch eck

(1, («c* !)) * i1»

*

• (coequBii, 1) • equB®s^>c =

=

!)) ‘ i1-

•

• (coequBil, 1) • equB® ^,c

This verification needs th e following f a c t s :

(i) denoting by X = B ® RA, and applying 7t to a co n v en ien t diagram which ex­

presses the F-functoriality of R c, one has (1, Rca) • L x x = {Rxa, 1) • R{xc\,(ccyRcx- (ii) from [III, (4.4)] applying n, we ca n fin d th e eq u ality

{Rxa, 1) • R(xc), (xx) • L x c = { Rxa, 1) •L\ax), (ac) • Lxc

(iii) the definition of coequBil gives b y a double ap p licatio n of 7t, the equality ( 1 ,7t(coequflil)) • Tz{yB-cBR) = (a*, 1 )-LAAX • 7t(coequB/1)

(iv) we have {ccx, 1) - R$ A. 7t(coequBil) = (1, Tc(coequB^)) • 'rr(Yn-cBr) = (yii)

— it(n(coequBil) - yB-cBR) ; this follows also using th e re su lt of th e forthcom g Having this in mind th e proof goes like t h i s :

(M«c> !)) • (1 .J & ) 'Pb a c'(coequBi^, 1) -equxc =

= (Mac» 1)) • (1,

• (ir(coequB/4),l) • Lxc • equxc =

= (^(coequ^), 1) . (l,(a c, 1)) . (R%A, 1) • R $% Ac c > - R e x 888

— fa(coequBil), 1) • ( Rx a> 1) • R {x c),r* (ac* ^ c x ’ ec^Xxc (ji)

““ 1) • [ Rx a, 1) • -R jjrc),«* ( a A> ^ x c #

= (w(coequBil), 1) . (1, (« ^ i)) . ( J & , 1) . I ® < x o * L Axc • {.y)

= (*(ooeqUjM), 1) . ( R L , 1) - ((«*,. 1), 1) - L?ax)MC)- LxC • e<lu* c " (iii)

= (w(Yb • cBR), 1) . (( l, 7i(coequB^)), 1 )-L(ax),{ac) ‘Lxc ' &lnxC ^

= W coequ^), 1) .

{Lix,

* M c o e q u ^ ) , 1) . ( i , ( a ^ 1 }) . ( L ^ , i ) . l } $ ) M c V L x C ' &^ * C '

~ (Ma.i> 1)) • (1, L

) • (7c(coequB^), 1) • Lxc * eilnxC

= (1* {aA> 1)) • (1, L

) * P

• (coequBil, 1) • e<^UxC

15

ON AN ENRICHED THEORY OF MODULES (II)

Now, we must show the existence of morphisms

close the diagram ^Pbac which commutatively

[B8RA,C)-f - BAC BAC

Mac» equB,(AC}

>(b,{ac\)

and these will be the required isomorphisms. We have to check the following equality ( « {ac} ,1) • f?{Ac|,B • ft bac = (<xB, 1) • L%, (ac) - Pbac-Again, we need some

preliminary results ■ _

(v) from proposition 3.13 we take the equality

(l.eq u ^ ) • <X(ac} = (eqnAC ,1) . Lee- ac (vi) the following equality holds

(Lee ,1) • R [ac{ b • Pbac = (l,Pbac) * Rc.b q a',

indeed, this follows applying 7t to axiom MCC3, composing to the left, with C(b®a,o, (CC) and applying rc- 1 .

(vii) by a double application of ^tz to the definition of yx = y*. ^ba we get p • (coequB/(, 1) • a * = (l,ir(coequBJ ) • aB

So, the following „enriched diagram chasing” proves the required equality (1,(1, equ^c))- (a ^ q , 1) • R\ac] ,b • Pbac=

= («{ac), 1)-((1, equ«),l) • R\ac) ,b • Pbac — (v)

= (ac, 1) • (Lee, 1) • ((equ«,l),l) • R\ac}, b • Pbac =

= (ac,l) • (Lee A ) • R\ac\.b • (l,equac) - Pbac =

= (ac,l ) • (Lee, 1) • R\ac),b • Pbac • (coequal 1) • e q u « = (vi)

= (ac,l) • (1, pbac) • Rc.boa • (coequ^ ,1) • e q u « =

= (« cl) • (1 , P ^bac) ' (l,(coeqUflA, 1)) • Rex • equ« = . = (1, Pbac)• (l,(coequBil)) • ( a x . l ) X « • e q u « =

= (ocjf, 1) • ((coequ*, ,1),1) • (1, Pbac) • L*x%A » e q u « = MCC3

= (a * ,l) . ((coequB<4,1 ),1 ) • (Pbac- *) • L{ax),mo • L xc • e q u « = (vii)

== (aB,l) • ((1, it(coequBil)),l) • L (® C), (ac) • L x c • e q u « = ,

= (aB,l) • ! # ,( « ) • («(coequal), 1) • L « • e q u « = - (aB,l) -.Xg. (^ac) • Pbac • (coequjM ,1) • e q u « =

= M * Lb,(AC) • (1,equ^c) • Pbac = («^b> 1) * ( U ^ ^ac)) • Lbb, {ac) • ^ ^ac =

= (1.(1, equ^c)) • (afl,l) • L%.{ac\ ' Pbac

16

GR. CALUGAREANU

We now have, for each three left 2?-modules (A, a.. Y ) ( n

a morphism ¡>BAC: { 5 ® A ,C } - * ^ ' ' aa< Ya). (C, # ,

J o iamiUr nf __ — C*

OrpillSiii i’BAC • l~ v ---- ^

Analogously, we can determine a family of morphism« X-i

the commutative diagrams ^ inac which

3,fAC)} ^{BAC -*(B0rA,C1}

|1' eqUAC L ( B,(AC)) -iS A C _ » ( B 8 A .C ) (c o e q u a l)

and next, a family pBac which close th e co m m u ta tiv e trian gle

(B.IAC}}

'B A C - Ï

(eSoA.o Finally, the following three facts m u st be ch eck ed •

(a) Pb^c and Pbac actu ally are m orphism s of left i?-m od ules over V ; (b) pbac and pbac are m utually in v e rse ;

(c) the family

p

\>BAC

As for (a), we show for in stan ce th a t pBAC actu ally , is a morphism of left .R-modules over V. The co m m u ta tiv ity of th e d iag ram

ro iea ^a $ — kixrh— Api

iepb a c

R0{B,

BAC

¡ÀC\\--- y fB,fACfl ,»(b, (ACh

reduces by left composition with equ

to

M]

,C {

} • &{

)< S>P

= P

• Y(a®

, c) ° r> applying

, to verify ip

, 1) • ¿{/c}, {

c} • «.{

} — (1, P

) • «-{

®

,

}» equality which ° ^ ^ g " . by left composition with (l,(l,equ4c)) by a new „enriched diagram c

For (b) we choose p • p— 1 = 1; indeed we show that

(1 , e q u x c ) • e q U f i^ x c } • Pb a c • Pb a c — (1» e q U A c ) * ^

We have __ _i __

(1, equ^c) • cquB, {¿cj • Pbac • Pbac = ( 1» cqu^c) • Pbac ____

= P

• (coequ^, 1) • equ*c •

= P

(coeqUBA. 1) PB

= P

• P

• (l,equ^c) • equB, {AC} = .(1* eqttAc).* e<1Ua''{

ON AN ENRICHED THEORYOF MODULES (II)

17 F o r (c) we choose the naturality of uBAC in (A „ v \ • ..

m utativity of the following diagram ’ A’ ^A' ’ ls’ com*

{00RA,Cj-

UBAC.

u r n

(1 , e q u ^ 'c ) • e q u f l.^ c } • Pb^'C • { 1 ® * / > 1 } = (l.e q u ^ -c) • Pba'c ■ { l ( g ) j j / , l } =

= Pba'c • (c o e q u a l, 1) • e q u Be)j^ , c - , { 1 ® * / , ! } = N

= Pb a'c * ( c o e q u a l , 1) • ( 1 ® * / . 1 ) • e q u x c = ■ ' - •’

— Pba-c • ( 1 ® / , 1 ) • (coequB ii, 1) • e q u * c = ( 1 ,( / ,1 ) ) • Pbac • (co e q u fly,, I) • e q u * c =

= ( 1 , ( / , 1 ) ) • (1 , eq u^ c) • e q u fi< {AC} • ^ bac = " "

= (1 , equ^-c) • ( 1 ,{ / » ! } ) • e q u B, {ac> • Pb^c =

= (1, equ^c) • eqUB.^-c} • { L { / , 1}} ■ Pb^c

In this way all the symmetric monoidal closed structure for „MV is establis­

hed. One can complete the proof of our theorem verifying axioms [MCC2],

[MCC3], [ATCC3'] and [AfCC4]. . ,

{Received July 6, 1978)

r e f e r e n c e s

1. B é n a b o u , J ., Algèbre élémentaire dans les catégories avec multiplication, C. R. Acad. Sci.

Paris, 258 (1964), 7 7 1 -7 7 4 .

2. B u n g e , M., Relative functor categories and categories of algebras, / . of Algebra, 11 (1969), 6 4 - 1 0 1 .

3. C à l u g â r e a n u , Gr., On an enriched theory of modules, ( I ) , Studia Univ. Babe^-Bolyai, Math.,

XXV, 2 (1979), 2 5 - 3 8 .

4. E l l e n b e r g , S.. G. M. K e l l y , Closed categories, Proc. Conf. on Categorical Algebra (La

Jolla, 1965), Springer Verlag. ; - - ‘ ■

5. M a c L a n e , S., Categorical Algebra, Bull. Amer. Math. Soc„ 71 (1965), 40—106.

ASUPRA UN EI TEO RII ÎMBOGĂŢITE A MODULELOR (II)

( Rezumat ) ‘ r . ,

Utilizînd noţiunile preliminare studiate în partea întîia a aceluiaşi articol, autorul stabileşte rezultatele principale privitoare la partea închisă şi monoidal închisă a teoriei modulelor peste un monoid fixat, rezultate care conduc în final la teorema: dacă V este o categorie simetric monoidal închisă, categoria subiacentă V 0 are egalizatori şi coegalizatori, R este un monoid comutativ peste L functorul de subiacenţă V păstrează egalizatori şi R ® - păstrează coegalizatori, atunci cate­

goria modulelor este simetric monoidal închisă. * .,

2 — Matheraatica ^3/1980

s t u d i au n i v. b a b h ş-b o l y a i, m a t h e m a t i c a. XXV, 3, 198Ó

PÂLCULUI, PULSAŢIILOR

CAL DE GAZ ÎN ROTAŢIE

N. LUNGU

r T în această lucrare prezentăm metoda de calcul utilizaţi . f' S tíS a re ale unei sfere în rotaţie uniformă. Acestea apar în cazul in pulsaţiile nelini

reprezintă aproximativ

din raza totali.

t caau

simetriei sferice, au fost integrato púa.»

. t o S S S l Ş E a Ü uşor adaptată la calculator. îu fmal se M l rezultatele numerice.

f„m incu “ S ' ^ ”t L f co“ t " în ecuaţiile mişcării fotón mu Şi coordonată Lagrime*- Masa corespunaătoare raaer r este [3]

m(r) = ^47rr2p (r)dr

unde p(r) este densitatea corespunzătoare razei r. Volumul specific este V = — = 47rr2 —

p(r)

dm

( i )

Ecuaţia lui Newton, considerînd şi rotaţia, are forma [7],

L « Ï + 2 UV _

■ 1 A. Saaj

dr' dmT

unde P este presiunea corespunzătoare masei vn,\ radiaţi' S1 Fluxul caloric se consideră difuzat in întregim

fluxului devine:

(21

ecuaţ**

(3

medie-paC1

unde ax este constanta Ştefan-Boltzmann, iar ^(^’^Hlează, ecuaţia ^ neglijăm generarea energiei nucleare în învelişul care o

caloric este:

(4

dt ' E fiind energia internă pe gram.

ÉË + + J L ^ o

•Ij '

dt

dmr