Embed Size (px)
Citation preview
Semigroup Forum Vol. 36 (1987) 293-313 �9 1987 Springer-Verlag New York Inc.
R E S E A R C H A R T I C L E
ON EQUALIZER-FLAT AND PULLBACK-FLAT ACTS
Peeter Normak
Communicated by N.R. Reilly
INTRODUCTION
In [4], [5] and [13] M. Kilp and B. Stenstr~m proposed
various concepts of flatness of S-acts: principally weak
flatness, weak flatness, flatness, strong flatness. The
classes of these acts are linearly ordered and pairwise
different. In [6] it is proved that all flat S-acts are
strongly flat if and only if S has only one element.
Hence the property of "flatness" is essentially weaker
than "strong flatness". Consequently in [63 there is ask-
ed for an intermediate concept between flatness and
strong flatness. In this paper we introduce pullback-flat
and equalizer-flat S-acts and find some basic properties
with respect to homological classification. As we see
from results below (especially from Lemma i.ii, Theorem
2.4, Theorem 3.4 and Proposition 3.10), a more convenient
property between flatness and strong flatness under the
aspect of homological classification seems to be the fol-
lowing property (P): for all elements s,t in a monoid S
and a,b in a left S-act A, if sa= tb, then there are ele-
ments u,v in S and a I in A such that su = tv, ua I = a and
va I = b.
293
NORMAK
i. PRELIMINARIES
In the following, S will always stand for a monoid. A
left S-act is a set A on which S acts unitarily from the
left in the usual way, that is, to say
(st) a = s(ta), la = a, for s,t 6 S, a 6A,
where 1 denotes the identity of S.
By S-Act we denote the category of all left S-acts,
and by Act-S the category of all right S-acts.
Let A and B be right and left S-acts, respectively. The
tensor product o_~f A and B, denoted by A| is the quo-
tient (A• B) ~ , where ~ is an equivalence on A • B, gener-
ated by the set { ((as,b), (a,sb)) I a 6 A, s 6 S, b 6 B}. For
A= S, we have an isomorphism S| ~ B.
For a fixed left S-act B, tensoring by B is a functor
from Act-S into the category of sets.
The following two lemmas are easy to show and hence
they can be considered as the definitions of equalizer
and pullback, respectively, in S-Act.
i.I. LEMMA. Let
< ~ ~ Y (1) K 1 > X 8
be a commutative diagram i__nn S-Act. Then this diagram i__ss
a_~n equalizer if and only i_~f whenever e(x) = B(x) there
exists a unique k i__nn K 1 such that <(k) = x.
1.2. COROLLARY. I_~f (i) is an equalizer, then K 1 is iso-
morphic to the S-act K = {x 6 X I ~(x) = 8(x)} and instead of
(i) we can consider the equalizer
K i > x ~ Y (2) 8
where i (k) =k for every k 6 K.
294
NORMAK
1.3. COROLLARY. A__nn equalizer of two arbitrary homomor-
phisms is a monomorphism.
1.4. LEMMA ([ii], Lemma 3.8). Let
f
Y
B Y *Z
(3)
be a commutative diagram of S-acts. Then this diagram i_~s
pullback if and only if whenever ~(x) = B(Y), then there
exists a unique k i__nn K 1 such that x = f(k), y = T(k).
1.5. COROLLARY. If (3) is a pullback, then K 1 is isomor-
phic to the S-act K = { (x,y) 6 X • YI e(x)= ~(y)} and in-
stead of (3) we can consider the pullback
K
Y
7 1 >X
(4)
where ~l(x,y) = x and z2(x,y) = y for every (x,y) 6 K.
A left S-act A is called flat if the functor | pre-
serves monomorphisms, and strongly flat if | preserves
equalizers and pullbacks (cf. [4], [131).
1.6. DEFINITION. We call a left S-act A equalizer-flat if
the functor | preserves equalizers; and we call A pull-
back-flat if the functor | preserves pullbacks.
1.7. REMARK. Every strongly flat S-act is equalizer-flat
and pullback-flat.
Note that the coproduct I I in the category of acts is
the disjoint union.
An epimorphism 9: B § is called pure if for every
295
NORMAK
= tiasi family al,...,a m 6 A and relations sia i
i= l,...,n, there exist bl,...,b m 6 B such that ~(bi) = a i
and s bl ~i= tib~i for all i.
The definition of free acts, projective acts and pro-
jective generators can be found, for example, in [61 and
[9].
In the sequel we often use the following properties (E)
and (P) of a left S-act A:
(E) If sa = ta with s,t 6 S, a 6 A, then there exist u 6 S,
a' 6 A such that su = tu and a = ua'.
(P) If sa = tb with s,t 6 S, a,b 6 A, then there exist
u,v 6 S, a' 6 A such that su = tv, a = ua' and b' = va'.
We recall the following facts:
1.8. LEMMA ([13], Theorem 5.3). The followin~ properties
of a left S-act A are equivalent:
(a) A is strongly flat.
(b) A has properties (E) and (P).
(c) Every epimorphism B § A i_ss pure.
(d) There exists a pure epimorphism F + A where F i__ss
free.
(e) Every homomorphism B + A, where B i__{s finitely pre-
sented may be factorized through a finitely gener-
ated free act.
(f) A is a direct limit of finitely generated free
acts.
1.9. REMARK. In the proof of Lemma 1.8 it is shown that
property (E) follows if A is equalizer-flat, and property
(P) follows if A is pullback-flat.
A left act with one generating element is called cyclic.
If p is a congruence relation on the left S-act A, then
the equivalence class containing the element a 6 A is de-
noted by a.
296
NORMAK
i.i0. COROLLARY. Every finitely generated S-act having
property (P) is a coproduct of cyclic S-acts.
i.ii. LEMMA ([8], Lemma 1.3). Every cyclic S-act having
property (E) is strongly flat.
In the following we shall frequently use a condition
used by many authors and formulated in [i], Lemma 1.2.
1.12. LEMMA. Let A 6 Act-S, a,a' 6 A, B 6 S-Act and b,b' 6 B.
Then a| = a' | i__nn A| if and only if there exist
al,...,a n 6 A, b2,...,b n 6 B, Sl,...,s n 6 S and tl,...,t n 6 S
such that
a = als 1
alt I = a2s 2 Slb = tlb 2
a2t 2 = a3t 3 s2b 2 = t2b 3
a t = a' s b = t b' n n n n n
Using the proceding lemmata and corollaries, we can con-
struct examples showing that conditions (E) and (P) do
not imply equalizer-flatness or pullback-flatness of a
left S-act A, respectively.
1.13. EXAMPLE. Let S = (~,') be the set of natural numbers
under multiplication. Let A= (~i I I ~2)/p , where
~i II ~2 is the coproduct of two copies of ~ and p is
the congruence relation generated by the pair (21,22).
Let sla= s2a. Without loss of generality we can assume
that a = n 1 I. Define now the homomorphism
9': ~i I I ~2 + ~, setting ~' (n i) = n, i = 1,2. Then
9'(21 ) = 2 = 9'(22 ) and hence p~ker 9' Then there exists
a homomorphism 9: A § ~ such that the diagram is commuta-
tive,
297
NORMAK
where ~ is the canonical projection. Then sln= sln ~' (i I) =
= sln~(l I) = sln ~(I I) = ~(Slnll) = ~(sla) = ~(s2a) = ~(s2nll) =
= s2n and therefore A fulfills condition (E). But A is
not equalizer-flat. For, consider the equalizer-diagram
T IN r
where IN/21 N is the right Rees factor of ~q by 21N, ~ the
canonical projection and T (n)= ~ for all n 6 I~. Tensoring
by A gives 2|174 in IN| and (~| (2| I) =
= (T| (2| in l~/2~q| Assume that 2|174 in
21~| Then by Lemma 1.12 there exist elements
nl,...,np 6 21N, a2,...,ap 6A, Sl,...,Sp 6 IN and
tl,...,t p 6 ~ such that
2 = nls 1
nlt I = n2s 2 Sl~ = tla 2
n2t 2 = n3s 3 s2a 2 = t2a 3
n t =2 s a = t 12 P P P P P
Because A is generated by the elements 11 and 12 , there
must exist an index k such that a k 6 ~'i I and ak+ 1 6 ~.i 2.
Then Sk~(ak) 6 2~. We have 2 = nls I = nl~(Sl.ll) = nl~(tla2 )=
= nltl~(a 2) = n2s2~(a 2) = n2~(s2a 2) = n2~(t2a 3) = n2t2~(a 3) =
= n3s3~(a 3) = ... = nkSk~(a k) 6 (2~) �9 (2~) c 4~, a contra-
298
NORMAK
- - m
diction. Hence 2 | 11 # 2 | 12 and A is not an equalizer-
flat S-act by Lemma i.i.
1.14. REMARK. As we see from Example 1.13 the functor |
constructed there for A E S-Act does not preserve monomor-
phisms and hence A is not flat. Hence condition (E) does
not yield flatness.
In the following we shall denote by 0 the singleton
S-act.
1.15. EXAMPLE. Consider the monoid S = {0,s,lJ s 2 = i} and
the left S-act A= {z,a I sa = a, 0a = sz = 0z = z}. Then A ~S~,
where Sl0S 2 if and only if s I = s 2 or {Sl,S 2} = {l,s}. Let
us I = vs 2 for u,v 6 S, Sl,S 2 E A. If us I =vs2, then Sl = sit
and s= s2~. If us I # vs2, we can assume without loss of
generality that us I = 1 and vs 2 = s. Then USlS = is = vs 2
and SlST= SlS= Sll = Sl, s21 = s 2. Hence condition (P) fol-
lows for A. But A is not pullback-flat. For, tensoring the
pullback
71 SxS ~S
72[ ]~i
~2 S 7%
with A we get the diagram
(S x S) | 71|
~2|
~2 | S|
�9 S|
' r l |
,O|
299
NORMAK
Then I@| = i and (~i| | i| =
= (~i|174 (~2|174 = i| = i| =
= (7 2 | i) ((l,s) | Assume that (i,i) | = (l,s) | in
(S • S) | Then by Lemma 1.12 there exist elements
(Ul,V I) .... ,(Un,V n) 6 S • S, a 2 ..... a n 6 S, Sl,...,s n 6 S,
tl,...,t n 6 S such that
(i,i) = (Ul,Vl)S 1
(Ul,Vl)t I = (u2,v2)s 2
(u2,v2)t 2 = (u3,v3)s 3
sit = tla 2
s2a 2 = t2a 3
(Un,Vn) tn = (l,s) sa =tY n n n
From the second column we get for every i 6 {l,...,n} that ki
there exists k i 6 {0,i} such that sia i = tiai+is . Now we
kl have (i,i) = (Ul,Vl)S 1 = (Ul,Vl)tla2 skl= (u2,v2)s2a2s =
kl+k 2 kl+...+kn- 1 = (u2,v2)t2a3s = ... = (Un,Vn)SnanS =
kl+...+k n skl+...+kn = (Un,Vn)tnS = (l,s) , a contradiction.
Hence (i,I) | # (l,s) | and A is not a pullback-flat
S-act by Lemma 1.4.
1.16. DEFINITION. We say that an epimorphism ~: B +A is
1-pure if for every element a 6A and relations sia= tia,
i = l,...,n, there exists an element b 6B such that
~(b) = a and sib = tib for all i.
A left act A is called finitely presented if it is iso-
morphic to F~ , where F is a finitely generated free S-act
and p a finitely generated congruence on F (see [103).
The following two propositions are directly inspired
by [13, Proposition 4.3] and by Lemma 1.8.
1.17. PROPOSITION. An epimorphism ~: B+A in S-Act is l-
pure if and onl~ if for every cyclic finitely presented
300
NORMAK
C and every homomorphism e: C +A there exists a homomor-
phism 8: C§ such that e = ~ �9 ~.
PROOF. Necessity. Let ~: B§ be a 1-pure epimorphism and
~: C+A a homomorphism where C is cyclic and finitely
presented. Let C ~s~ , where p is generated by a set
{ (si,ti) I i = 1 ..... n}. Then in A we have si~(Y) = ~(si ) =
= ~(~i ) = ti~(Y), i = l,...,n. By hypothesis there is an
element b 6 B such that s b= t.b for all i. Then B: C+B, 1 l
8(S) = sb is a correctly defined homomorphism such that
= ~ �9 8.
Sufficiency. Let ~: B +A be an epimorphism and let
{sia = tia I si,t i 6 S, a 6 A, i= 1 .... ,n } be a finite set of
equations in A. Let p be a left congruence on S, generated
by the set { (si,ti) I i = 1 ..... n}. Then ~: S~ ~A, ~(~) = sa
is a correctly defined homomorphism. By hypothesis there
exists a homomorphism ~: S~ + B such that ~ =~ �9 8. Then
we have siB(Y) = ~(si ) = 8(~ i) = tiE(Y) for all i. Hence
is 1-pure.
2. EQUALIZER-FLAT ACTS
By induction we easily get the following
2.1. LEMMA. Let a left S-act A have property (E). Then if
s.al = t.al with si,t i 6 S, i = l,...,n, a 6 A, there exist
elements u 6 S , a 1 6 A s u c h t h a t u a 1 = a a n d s i u = t i u f o r
all i.
For the sake of completeness we prove now the following
2.2. PROPOSITION. The following properties of a left S-act
A are equivalent:
i) A has property (E).
2) Every epimorphism B § A i_~s 1-pure.
301
NORMAK
3) There exists a 1-pure epimorphism B § A where B i__ss
equalizer-flat.
4) Every homomorphism B § A where B is a finitely pre-
sented cyclic S-act may b_ee factorized through a free
S-act.
5) Every homomorphism B + A where B is a finitely pre-
sented cyclic S-act may be factorized through a_nn
equalizer-flat S-act.
PROOF. i) ~ 2). Let ~: B +A be an epimorphism and let
{sia = tia I si,t i 6 S, a 6 A, i= l,...,n} be a finite system
of equations in A. By Lemma 2.1 there exist elements u 6 S,
a' 6 A such that siu = tiu, i= l,...,n, ua' = a. Let b 6 B !
be such that ~(b') = a'. Then si~(ub') = siu~(b') = siua =
= t.a = t.ua' = ti~(ub') , i= l,...,n, ~(ub') = a. Hence = sia 1 l is a 1-pure epimorphism.
2) ~ 3) is clear.
3) ~ 4). Let ~: B§ be a homomorphism with finitely
presented cyclic S-act B. By hypothesis there exists a l-
pure epimorphism ~: C+A with equalizer-flat S-act C. By
Proposition 1.17 there exists a homomorphism ~': B§ such
that ~ = ~ �9 ~'. Let B~S~ , where p is generated by the
set { (si,ti) I i = 1 ..... n}. Then si~' (Y) = ~' (si) = ~' (~i) =
= tim'(i) for all i. By Remark 1.9 and Lemma 2.1 there
exist elements u 6 S and c 6 C such that s.u= t.u, l l
i= l,...,n and uc = ~' (T). Then the mappings e': B§ S,
e: S § defined by ~' (s) = su and e(s) = ~(sc), are homo-
morphisms such that ee' = ~.
4) ~ 5) is clear.
5) ~ i). Let sa= ta for some s,t 6 S, a 6 A. Consider
the left congruence Q on S generated by the pair (s,t).
Then ~: S~ ~A, ~(p) =pa is a correctly defined homomor-
phism. By hypothesis there exists an equalizer-flat S-act
E and homomorphisms ~': S~ + E, ~": E § such that
= ~"~'. We have s~' (Y) = ~' (~) = ~' (~) = t~' (Y). By Remark
1.9 there exist elements u 6 S, e 6 E such that su = tu and
ue = ~' (Y). Then u~"(e) = ~"(ue) = ~".~' (Y) = ~(Y) = a.
302
NORMAK
By Remark 1.9 we get the following
2.3. COROLLARY. For an equalizer-flat S-act A the proper-
ties i) - 5) in Proposition 2.2 hold.
A left S-act A is called indecomposable if A# @ and for
every a',a" E A there exist al,...,a n E A such that a' 6 Sa I,
a" E San, and Sa i N Sai+ 1 # ~ for i = l,...,n-l. It is well
known that every left S-act A is the unique disjoint union
of indecomposable S-acts called the indecomposable com-
ponents of A.
2.4. THEOREM. The following properties of a monoid S are
equivalent:
i) All left S-acts are equalizer-flat.
2) All left S-acts have property (E).
3) All cyclic left S-acts have property (E).
4) All cyclic left S-acts are strongly flat.
5) S = {i} or S = {0,i}.
PROOF. The implication i) =~ 2) follows by Remark 1.9. Im-
plication 2) ~ 3) is clear. 3) =~ 4) is proved in [8, Lem-
ma 1.3] and 4) =~ 5) is proved in [4, Theorem 4]. We will
now prove 5) => i). If S = {i}, then all S-acts are free
and hence equalizer-flat. Let S= {0,1}, A is a left S-act
and let
K i > X---~> Y B
be an equalizer diagram. By Corollary 1.2 we can take
K={x6 Xl e(x) =8(x)}. Consider now a diagram
~| K| i| > X| >>Y| . (5)
B|
Obviously (e | i) (i | i) = (B | i) (t | i) . Let now
(~| I) (x| a) = (B| i) (x| for some x| 6X| Then
~(x) | = 8(x) | in Y| and hence by Lemma 1.12 there
303
NORMAK
exist elements yl,...,y n 6Y, a2,...,an6 A, Sl,...,Sn,
tl,...,tn6 S such that
e(x) = YlSl
Yltl = Y2S2
Y2t2 = Y3t3
sla = tla 2
s2a 2 = t2a 3 (6)
Yntn = t a = B(x) Snan n
If s i= t i= 1 for all i, then e(x) =yl =Y2 = "'" = B(x) and
hence x 6K and x | 6 K| Assume that s. = 0 or t = 0 l l
for some i. Suppose that (6) is the shortest sequence of
equalities which imply the equality ~(x) | = B(x) | If
s I = t n= 0, then ~(x) = Yl-0 = Y2.0 = Y3.0 = ... =Yn-0 = B(x)
and hence x 6 K and x| 6 K| If now s I = 1 or t n= i,
then t I = 0 or s n = 0, respectively, while otherwise the
sequence (6) can be shortened. Then a= 0-a. By equalities
(6) we have ~(x.0) =Yl.0 = Y2.0 = ... = B(x.0). Hence
x| =x| 0.a =x.0 | a 6 K| Since S is commutative and
regular it follows by [4, Theorem 2] that A is flat and
hence the homomorphism i | 1 is a monomorphism. By Lemma
i.i diagram (5) is an equalizer-diagram.
2.5. COROLLARY. There exists a_nn equalizer-flat S-act that
is not pullback-flat.
PROOF. Let S ={0,I} and A={x,y,z I 0x = 0y = 0z = z, ix = x,
ly =y, iz = z} be a left S-act. Then A is equalizer-flat
by Theorem 2.4, but A is not pullback-flat by Corollary
1.10.
By induction we easily get the following
2.6. LEMMA. Let a left S-act A have property (P). Then if
sia i= tiai+ 1 with si,t i6 S, a i6 A, i= l,...,n, there ex-
ist elements ui,v i6 S, i= l,...,n, a6 A such that
siu i= tivi, uia = ai, via= ai+ 1 for all i.
304
NORMAK
2.7. PROPOSITION. A left S-act A having property (E) i_ss
pullback-flat (strongly flat) if and only if there exist
elements u,v 6 S, a 6A, such that a' = ua, a" = va, whenever
a' and a" belong to the same indecomposable component o_~f
A.
PROOF. Necessity. Let the left S-act A having property
(E) be pullback-flat and let a',a" belong to the same
indecomposable component of A. Then there exist elements
al,...,an_ 1 6A, Sl,...,Sn,tl,...,t n 6 S such that
sla' = tla 1
s2a I = t2a 2
s3a 2 = t3a 3
Snan_ 1 = tna"
Then by Lemma 2.6 there exist elements Ul,...,Un,
Vl,...,v n 6S, a 6A such that
SlU 1 = tlu I , a' = ula , a I = vla
s2u 2 = t2v 2 , a I = u2a , a 2 = v2a
SnUn = tnVn ' an_ 1 = Una , a" = Vna
Sufficiency. Let A have property (E) and let
sla I = s2a 2 for some Sl,S 2 6S, al,a26 A. Then by hypoth-
esis there exist elements tl,t 2 6S, a'6 A such that
a I = tla' and a 2 = t2a' Then Sltla' = sla I = s2a 2 = s2t2a'.
By hypothesis there exist elements u 6S, a 6A such that
SltlU = s2t2u, a' = ua. We have now a I = tla' = tlua and
a 2 = t2a' = t2ua. Then A is strongly flat by Lemma 1.8 and
hence pullback-flat.
305
NORMAK
2.8. COROLLARY. A finitely generated equalizer-flat S-act
i__ss pullback-flat (strongly flat) if and only if it is a
coproduct o_~f cyclic S-acts.
2.9. PROPOSITION. Every equalizer-flat S-act is flat.
PROOF. Let A be an equalizer-flat S-act and let ~# XcY be
an inclusion of right S-acts. Define on Y the congruence p
by ylpy 2 if and only if yl =y2 or yl,y 2 6X. Let ~: Y~Y/p
be the canonical epimorphism and let 8: Y § Y/p be a homo-
morphism such that 8 (y) =x for all y 6 Y. Then by Corollary
1.2
e
is an equalizer diagram. Then by condition
~| X| l >Y| > Y/p |
e|
is also an equalizer diagram. Moreover i | 1 is an monomor-
phism by Corollary 1.3.
2.10. REMARK. The converse to Proposition 2.9 is not true.
For, if G is a nontrivial group, then all G-acts are flat
by [4, Theorem 23, but not all G-acts are equalizer-flat
by Theorem 2.4. Furthermore, for the 1-element G-act con-
dition (E) does not hold. Hence flatness does not yield
condition (E).
With respect to the homological classification of
monoids the following proposition is interesting.
2.11. PROPOSITON. The followin 9 properties of a monoid S
are equivalent:
i) All S-acts having property (E) are free.
2) All equalizer-flat S-acts are free.
3) S is a group.
306
NORMAK
PROOF. The implication I) ~ 2) follows by Remark 1.9, the
implication 2) ~ 3) by [9, Theorem 2.6].
3) ~ i). Let S be a group. Then every S-act is a co-
product of cyclic acts by [12, Theorem 4]. Then every S-
act having property (E) is strongly flat by Lemma i. Ii and
hence free by [9, Theorem 2.6].
3. PULLBACK-FLAT ACTS
From Lemma 1.12 we get by straightforward calculation
the following
3.1. LEMMA. Let S be a group, A 6 Act-S and B 6 S-Act. Then
a | = a' | i__nn A| if and only if there exists an ele-
ment g 6 S such that a = a'g and gb = b'.
3.2. LEMMA. Let S be a group. Then the singleton left S-
act 8 i__ss pullback-flat if and only if S = {i}.
PROOF. Sufficiency is obvious.
Necessity. By Corollary 1.5 the diagram
l SxS "'S
S B ),,O
(8)
of right S-acts with I (s,t) = s, j(s,t) = t, ~(s) = B(s) = 8
and 8 a singleton right S-act, is a pullback. By hypoth-
esis the diagram
(S x S) | 8 I|
~|
j|
S|
S|
, 0 |
307
NORMAK
is a pullback. Because of S | e ~ 8 we have for every element
(s,t) | @ 6 (S • S) | 8 that (i | i) ((s,t) | = 8 and
(j | I) ((s,t) | = 8. By Lemma 1.4 we have (S x S) | 8 ~ 8.
Hence (l,g) | = (i,i) | 8 for an arbitrary g 6 S. Then by
Lemma 3.1 there exists an element gl 6 S such that
(l,g) = (l,l)g I. It follows that gl = 1 and g = I.
3.3. LEMMA. If S is a monoid which is not a group, then
there exists a finitely generated indecomposable S-act
which is not a coproduct of cyclic S-acts.
PROOF. By assumption there exists an element s 6 S such
that Ss # S. Next we derive a contradiction using a con-
struction in analogy to [9]. Let x,y,z be three elements
not in S and let A= {(x,t) I t 6 SkSs} 0 {(y,t) I t 6 S~Ss} 0
U {(z,t) I t 6 Ss}. Define an action of S on A by
(p,ut) if ut 6 S \ Ss u(p,t) = for p 6 {x,y}
(z,ut) if ut 6 Ss
u(z,t) = (z,ut) for all u 6 S .
Then A is an indecomposable left S-act with two genera-
tors (x,l) and (y,l).
3.4. THEOREM. All left S-acts are pullback-flat if and
only i_~f S = {i}.
PROOF. Sufficiency is obvious.
Necessity. By Corollary 1.10 and Lemma 3.3 it follows
that S is a group. Hence S = {I} by Lemma 3.2.
By the following Proposition 3.5 we see that condition
(P) in contrary to condition (E) (Remark 1.14) implies
flatness.
3.5. PROPOSITION. If a left S-act A has property (P), then
A is flat.
308
NORMAK
PROOF. Let XcY be an inclusion of right S-acts and let
x| a= x' | in Y| Then by Lemma 1.12 there exist ele-
ments yl,...,y n 6 Y, a2,...,a n 6 A, Sl,...,Sn,tl,...,t n 6 S
such that
x = YlSl
Yltl = Y2S2 sla = tla 2
Y2t2 ? Y3S3 s2a 2 ? t2a 3 (9)
Yntn = t a' = x' Snan n
We prove by induction on n that x| =x' | in X |
Let n = i. Then we have the following equalities: x = YlSl ,
= x' sla= tla'. By hypothesis there exist elements Yltl
u,v 6 S, a16 A such that SlU= tlv, a=ual, a' =va I and
hence
x = x.l
x-u= (YlSl)U=Yl(SlU) =Yl(tlv) =x'.v
x'-i = x'
l.a = u.a 1
v-a I = l.a'
Let now the assumption hold for every k < n. By condition
(P) there exist elements u,v 6 S, a I 6 A such that SlU = tlv,
a = ual, a 2 =va I. Replacing the second equalities in (9)
by xu=YlSl u =Yltl v=y2s2 v and s2va I = s2a 2 = t2a3, re-
spectively, we get by induction, that xu| I = x' | in
X| Hence x| =x| I =xu| I =x' | in X|
3.6. REMARK. The converse of Proposition 3.5 is not true.
For, the S-act A= {x,y,z I 0x = 0y = 0z = z, ix = x, ly = y,
Iz = z}, S = {0,i} is flat by Theorem 2.4 but has not prop-
erty (P), because A is not a coproduct of cyclic S-acts.
3.7. COROLLARY. All pullback-flat S-acts are flat.
The following Proposition 3.8 follows immediately from
Lemma 1.8 and Remark 1.9.
309
NORMAK
3.8. PROPOSITION. A pullback-flat S-act A i__ss equalizer-
flat (strongly flat) if and only if there exist elements
s 6 S, a' 6 A such that SlS= s2s , a = sa' whenever sla = s2a.
In respect to the homological classification of monoids
the following propositions are interesting:
3.9. PROPOSITION. All pullback-flat S-acts are free if and
only i_~f S is a group.
PROOF. Necessity. If all pullback-flat S-acts are free,
then also all strongly flat S-acts are free. Hence S is a
group by [9, Theorem 2.6].
Sufficiency. Let S be a group. Then by [12, Theorem 43
all S-acts are coproducts of cyclic S-acts. Hence it suf-
fices to show that all cyclic pullback-flat S-acts are
free. Suppose that A ~ S~ , where D is a left congruence
on S. Suppose that SlPS 2. Then, tensoring the diagram (8)
by A, we have (I | ((Sl,l) | = s I | 1 | = 1 | =
=s2| (l | ((s2,1) | and (j| ((Sl,l) | =I|
= (j | i)((s2,1) | Hence by Lemma 1.4 we have
(Sl,l) | = (s2,1) | in (S • S) | A. Then by Lemma 3.1
there exists an element g 6 S such that (Sl,l) = (s2,1)g.
But then g = 1 and s I = s 2.
3.10. PROPOSITION. All left S-acts have property (P) if
and only i_~f S is a group.
PROOF. Necessity follows from Lemma 3.3 and Corollary
1.10.
Sufficiency. Let S be a group and let sa= tb for ele-
ments s,t 6 S, a,b 6 A. Then we have s.s-lt = t-l,
(s-lt)b = s-l(tb) = s-l(sa) = a, l-b = b. Hence A has proper-
ty (P).
3.11. PROPOSITION. All left S-acts having property (P) are
free if and only i_~f S = {i}.
310
NORMAK
PROOF. Sufficiency is obvious.
Necessity. From Remark 1.4 it follows that all strongly
flat S-acts are free. Hence S is a group by [9, Theorem
2.63. By Proposition 3.10 all left S-acts have property
(P). This means by assumption that all left S-acts are
free. Then by [4, Theorem 5] S = {i}.
4. CONCLUSION
Denote the properties investigated in this paper as
follows.
SF - strong flatness
PF - pullback-flatness
EF - equalizer-flatness
F - flatness
Then we have the following implications
F
All inclusions are strong, except (E) A (P) ~ SF and
maybe SF ~ PF. Note that in [11 and [23 it is proved (in-
dependently) that for every inverse monoid S all (left and
right) S-acts are flat. Hence in view of Lemma i. II, Theo-
311
NOIhMAK
rem 2.4, Theorem 3.4 and Proposition 3.10 it seems that a
convenient condition between flatness and strong flatness
is conditon (P).
The author would like to thank Professor U. Knauer for
his encouragement and much valuable help and advice in the
preparation of the paper. I would also like to thank the
Deutsche Akademischen Austauschdienst and the Fachbereich
Mathematik of the Universit~t Oldenburg for excellent
working conditions.
REFERENCES
i. Bulman-Fleming, S. and K. McDowell, Absolutely flat semigroups, Pacific J. Math. 107 (1983), 319-333.
2. Fleischer, V., Completely flat monoids, Tartu Riikl. UI. Toimetised 610 (1982), 38-52 (Russian).
3. Gr~tzer, G., Universal Algebra, Van Nostrand 1968.
4. Kilp, M., O__nn homological classification of monoids, Siber. Math. J. 13 (1972), 578-586.
5. Kilp, M., Characterization of monoids by properties of their left Rees factors, Tartu Riikl. UI. Toimetised 640 (1983), 29-37 (Russian).
6. Kilp, M. and U. Knauer, On free, projective, and strongly flat acts, Arch. Math. 47 (1986), 17-23.
7. Knauer, U., Projectivity of acts and Morita equiva- lence o_ff monoids, Semigroup Forum 3 (1972), 359-370.
8. Knauer, U., Characterization of monoids by properties o_~f finitely generated right acts and their right ide- als, Lecture Notes in Mathematics 998 (1983), 310-332.
9. Knauer, U. and M. Petrich, Characterization of monoids by torsion-free, flat, projective and free acts, Arch. Math. 36 (1981), 289-294.
10. Normak, P., O_~n Noetherian and finitely presented S- acts, Tartu Riikl. UI. Toimetised 431 (1977), 37-46 (Russian).
ii. Renschaw, J., Flatness and amalgamation in monoids, J. London Math. Soc. 33 (1986), 78-88.
12. Skornjakov, L., O_nn homological classification o_ff monoids, Siber. Math. J. i0 (1969), 1139-1143.
312
NORMAK
13. Stenstr6m, B., Flatness and localization over monoids, Math. Nachr. 48 (1971), 315-335.
Tallinna Pedagoogiline Instituut matemaatika kateeder Tallinn 200102 USSR
Received 23 April 1987 and, in final form, 24 September 1987.
313
