Eugene M. Norris. Communicated by Robert McFadden. I. Introduction. In [I] Clifford and Miller deduced the structure of those endomorphlsms on the ...
Semigroup Forum Vol. 15 (1977) 169-179 RESEARCH A R T I C L E
UNION PRESERVING HOMOMORPHISMS OF SEMIGROUPS OF CLOSED RELATIONS
I.
Eugene M. Norris Communicated by Robert McFadden Introduction. In [I] Clifford and Miller deduced the structure
of those endomorphlsms on the s e m l g r o u p ~ ( X )
of all relations on a
set X which preserve unions and map symmetric relations onto symmetric relations.
The work of Clifford and Miller was followed by that
of Magill and Yamamuro, who generalized from endomorphisms o n ~ ( X ) to homomorphlsms between ~ ( X )
andS(Y)
where in general ~ ( Z )
the semlgroup of all closed relations in a ~ -
is
space Z, i.e. a T l-
space on which the composition of two closed relations is again closed [2]. In the present paper we study homomorphisms which preserve unions but we replace the condition that they preserve symmetry by the more relaxed condition that they map the empty relation and the universal relation onto relations which are quaslorders on subsets. Our main result, Theorem 2.5, completely describes these homomorphlsms.
We have therefore described a class of homomorphisms which
properly contair~those of Clifford-Miller and Magill-Yamamuro. If ~ and 8 are any two relations we define their composition as in [I], i.e. ~08 = {(x,y):
for some z, (x,z) g ~ and (z,y) e 8}.
If G S XxY, A & X and B & Y then we put A~ = {y:
(x,y) e e for
some x in X} and dually deflne.eB to he B~ -I, where - I verse of ~, i.e.
(x,y) e e -I Just in case (x,y) E @.
domain of ~ and X~ is the range. simpllclty's sake.
We write xe in place of {x}e for
(If the expression for @ is complicated we may
write domain ~ for ~Y and so forth.) subset E of Y is b E = ((x,x): If 8:
~
is the conThen ~Y is the
(X) ÷ ~ ( Y )
The identity relation on a
x e E}.
is a homomorphlsm, 169
© 1977bySpringer-VedagNew Yorklnc.
i.e. if (~o8)@ = ~@o8@
Norris
for all s,8 e ~ ( X ) , s e ~(X)
and S =
then we call O union preserving if, whenever
Uyer8Y, where
each By e ~ ( X ) ,
then s0 =
Oyer(Bv)O.
We say that O is bounded above by p if p = (XXX)@ and s0 ~ P for all s e
~(X)
and dually that O is bounded below by ~ if O = ~ and
o c s0 for all s e ~ ( X ) .
Of course the relations P and o mentioned
above are idempotent because O is a homomorphlsm. Among the Idempotent relations in ~ ( X ) tial equivalence relations,
are the so-called par-
i.e. relations which are symmetric and
transitive and hence are equivalence relations on a subset of X. union- and symmetry-preservlng
The
homomorphlsms studied by Clifford and
Miller and by Magill and Yamamuro are bounded by partial equivalences. More generally,
quasiorders on subsets of X are idempotent; we recall
that a quaslorder on a set E is a reflexive, tained in EXE.
transitive relation con-
We will refrain from calling these "partial quasi-
orders." One can establish immediately that if p is any idempotent then -i p n p is a partial equivalence. In general one has that (P O P-I) x is subset of pX n Xp.
We shall call p a nea___.~tidempotent
if (p n p-I)x = pX n Xp. Any partial equivalence relation O on a set X is a neat idempotent, for p-i = P and consequently QX = Xp; therefore pX = pX o Xp. P n
P -1
(p ~ p-I)x =
Similarly all quaslorders p on a set X are neat, for
is an equivalence relation on X, so that (0 n
0X n Xp; neatness follows.
O-1)X
= X
Neat idempotents other than these
"stock" finds will be discussed in Example 2.6. There are simple (and easily generalized) idempotents:
½
examples of non-neat
Let X = [0,i] be the closed unit interval,
1
[ ,i] x [~,I], let r = {(x,y) e XxX:
let o =
0 ~ x < y < 7} i and put p = O U T .
One sees easily that ~ and T are idempotents, ooT = Too = @ and consequently
(~ u T)o(O u T) = OoO U OoT u ToO U ToT = SUT, so p is -I However p N p =~ but pX = Xp = [0,I] and consequently
indempotent°
I 1 ], so p isn't neat. (p n p-I)x = OX = [~,
Finite examples of non-
neat idempotents abound, but n o t on sets of two e l e m e n t ~ a s
straight-
forward case-by-case computation shows. B. M. Schein has given a characterization of idempotent relations in [5] which says roughly that every Idempotent on a set X is a quaslorder relation on X less a certain piece of AX"
170
Given an
Norris
Idempotent 0, Scheln's quasiorder turns out to be 0 u ~X and his characterization
is therefore "external" in that it is in terms of
the least containing quasiorder.
The present author has an "internal"
characterization of idempotents,
in terms of the maximal contained
quasiorder and three other relations, which is too long to be given here [4].
There is a related paper by S. Schwarz [6] characterizing
Idempotents on finite sets in terms of
Boolean matrices.
have a characterization ofneat idempotents
We do
in [4].
We shall adhere to the following convention:
in t h i s
paper, x
and Y shall alway@ denote T~-spaces containin~ more than one point. An important property of~..- spaces frequently needed in what is to follow is this: If X is ~ ~ -
spac ~ then the firs_~t projection o f
X×X onto X is a closed mapping. To see this, let ~ be a closed subset of X×X and observe that the first projection of ~ is Just ~X, the domain of ~.
Then ~ and X×X
are in "~(X) and therefore so is ~X×X = ~o(X×X); hence ~X×X is closed, implying ~X is closed in X. We recall the definition and lemma below from [I]. {~x,y:
A family
(x,y) ~ X×X} is called a matricial family if f
~nx, w if y = Z q
o~
x,y Lemma (i.I)
z,w
i~ '
if y ~ z.
(Clifford and Miller).
Let ~nx,y:
matrlcial family of relations on a set X.
family i__ssempty, then every member is empty. and Rx the range .of nx, . x..
if 8: ((x,y)8:
~
~(x)
Let D x be the domain
Then . .the . domain . .of ~x,y is D x and its
range is R ; and, for all x and y, D y
(x,y) ~ X×X} b__£e~
If one member of the
n R x
÷ ~(Y)
= ~ if and only if x ~ y° y
~
~
is a homomorphlsm then it is clear that
(x,y) g X×X} is a mtricial family, and that if E
then the collection {E :
=
x
D
x
n
R
x
(I.i.i)
xgX} is a family of pairwise disjoint sets,
x
each nonvoid if 8 is not constantly ~.
These remarks are due also
to Clifford and Miller [I]. The bnmomorphlsms
studied in [1,2], one recalls, are all des-
cribable by an equation 17!
Norris
aO
:
(p
n
~o~oU- 1 )
u
(1.1.2)
for appropriate partial equivalence relations p and ~ and a function ~.
It is not clear that one can construct nonconstant homomorphisms
of this general form, involving two arbitrarily chosen idempotents -I ; the first lemma below is our most
and a '~agill signature" po~o~
general result in this direction, and is the starting place for our investigations. 2.
quasiorder-Bounded
Lemma (2.1). suppose ~:
Homomorphisms.
Let p b__£ea_.nnidempotent i__nn~ ( Y ) ,
put E = pY n Yp and
E + X is a continuous function for which (p 0 p-l)o~ = ExX,
and define ~ m a p p i n g
8:
~(x)
+ ~(Y)
~O = p n ~o~°p
for each ~ e ~
(X).
(2.1.1) by -i
(2.1.2)
Then O is a union-preserving
homomorphism
bounded below by ~.
,. Proof.
-i
op ~ AXll Then ~ e o s o = (p n ~ o ~ o ~ - l ) o ( p n ~oBo~ - 1 ) ~ Oop o ~o~o~ o~oBo~ - 1 ~ 0 n Uo~oAxoBO~-I Since p is a function, p
= (eo6)O. (eo8)8.
To s e e t h e c o n v e r s e c o n t a i n m e n t , Then f o r some t ,
(u,t)
potent, when it follows t e E.
e p and ( t , v )
suppose that
(u,v) e
c p s i n c e p i s idem-
Also, for some t', (u,t') e po~
and (t',v) £ SoU -I, so that (u~,t') E ~ and (t',v~) E 8.
Condition
(2.1.1) applied to (t,t') e ExX implies that for some z, (t,z) e -i -i p A P and (z,t') E ~, i.e. t' = z~; then (u,z) e pA~o~o~ and (z,v) E P o UoSo~ -I and hence (u,v) e ~8o8@.
@ is therefore a
homomorphism into ~ ( Y ) .
We shall be done if we show that ~@ is a -1 closed subset of Y×Y for each a E ~ ( X ) . But ~o@op is easily seen to be the inverse image of the closed set ~ under the continuous -i function ~x~: EXE ÷ Xxx. Hence P A ~0~o~ is closed in EXE which is closed in yxy. We remark that if p is any idempotent and ~ is any function satisfying
(2.1.1) then p is necessarily neat and ~ is necessarily
surjectlve.
To see this, suppose x E X and a E E. By (2.1.1), -i there is some b for which (a,b) E p A P and (b,x) E D. The first
of these implies that E ~ (p o O-I)Y and consequently p is neat; the
172
Norris
second of these gives that ~ is surJective. We also remark that the homomorphism of lemma (2.1) guarantees only that (XxX)8 ~ p ;
we give in Example
for which the containment is proper.
(2.6) below an idempotent
However we can easily see that
(XxX)e itself is always a quasiorder on E.
In fact, since (XxX)8 -I is idempotent it is transitive, and if x e E then (x,x) ~ 0 n P so A E & p n po(XXX)o~ -I = (X×X)%. Thus we are led to study homomorphisms bounded above by quasiorders. Theorem (2.~). on E and ~:
Let E be a closed subset of Y, p a closed quasiorder
E + X a continuous function for which (P n 0-I)
and define a mapping
e : ~(X)
= m x X
(2.2.1)
+ ~ ( Y ) by -I
s0 = P n ~oE~o~ f o r each ~ s ~ ( X ) .
(2.2.2)
Then 0 i s a u n i o n - p r e s e r v i n g hpmomorphism
bounded above b ~ p
and below b y 0 .
Conversely, every union-preserv-
ing homomorphism from ~ ( X ) i n t o ~ ( Y ) which i s bounded ahove b y ~ quasiorder and below by 0 is obtained in this way. Proof.
Since every quasiorder on E is clearly a neat idempotent,
all the assertions concerning e follow from lenuua (2.1), except (XxX)@ = p.
But P ~ ExE so p = P 0 ExE = pn~o(XxX)o~ -I = (XXX)O.
We proceed to the converse assertion.
To simplify notation in what
follows, the image of a one-point relation {(x,y)} will be denoted by (x,y)8 rather than {(x,y)}O. To see the converse, suppose that O: unlon-preservlng homomorphism bounded below (XxX)@ is a quasiorder on a subset E of Y. family {Ex:
~(X)
÷
~(Y)
is a
by O, for which p = In order to see that the
x £ X} of nonvoid pairwlse disjoint sets defined in
(I.I.i) is a partition of E, we need only show that E = {E x :
x e X} .
To this end let t e E x for some x; then there exists
some r for which (t,r) ~ (x,x)O ~ p, so t e E. then, 0 being reflexive, we have (t,t) g p.
Conversely if t e E
Since O is unlon-pre-
serving and p = (XxX)@ there exist x,y e X for which (t,t) E (x,y)@, so that t e D
N R . Thl8 implies x = y, i.e. t e E . From this it y x will follow that E = D = R for each x e X, for if t e D then for x
X
X
X
X
173
Norris
some r, (t,r) c (x,x)O hence t ~ E.
Then for some z, we have t e E
and consequently t g Dx n Rz, so by lemma
z
(I.I) z = x, i.e. t e Ex.
The converse containment is immediate from (I.i.I) so Ex = Dx; in a similar way one sees E = R . These equations were crucial to x x the proof of Theorem I in [i], and we are able to carry that proof over to our more general homomorphisms without difficulty. If w e set ~ = {(y,x):
y E Ex, x c X} then it is immediate that
is a function from E onto X.
(x,y)e for if (u,v) g P n E and hence u E E
x
× E
y
Next we see that, just as in [I], = p n E x × Ey,
(2.2.3)
then (u,v) g (z,w)e for some (z,w) g X × X
and v E E n Ew, whence x = z and y = w, so z y (u,v) ~ (x,y)8. Hence p n E × E ~ (x,y)e. The converse containx y ment is immediate from the definition of D and R and the hypothesis x y that e is bounded above by p. x
n E
The verification of (2.2.2) now proceeds exactly as in [i], but since the argument
is brief we include it here in the interest of
self-containment.
In view of (2.2.3), we can verify easily that
(u,v) ~ ~e if and only if there is some (x,y) g ~ so that each of the following equivalent assertions in turn holds:
(u,v) ~ (x,y)O; -I
(u,v) ~ p, u c Ex and v c Ey; Hence
(u,v) g p,
(u,x) ~ ~ and
(u,v) E ~0 if and only if (u,v) g p n (poaop-l),
(y,v) g p i.e. if and
only if (2.2.2) holds. To see that (2.2.1) holds we observe that since p n p and ~ ~ E x X,
(p n p-l)op ~ E x X.
suppose u e E and y s X.
-I
~ E x E
For the converse containment
Putting x = up w e have u e E x so that
(u,u) s p n E x E . Since this last set is Just (x,x)0 (in light x x of (2.2.3)) and since 8 is a homomorphlsm, we have that (u,u) (x,y)So(y,x)O and hence there is some v for w h i c h it is the case that -I (u,v) g (x,y)0 and (v,u) s (y,x)8. Consequently, (u,v) c p n p and v~ = y so that (u,y) E (p n p-l)o~. and w e have established
(2.2.1).
Therefore
(p 0 p-l)
= E x
It remains only to show that E is
a closed set and p is continuous. As we mentioned earlier,
in a n y ~ -
of every closed relation are closed sets.
space the domain and range In particular,
E is closed
since p is closed. To see that ~ is continuous,
let F be any closed subset of X,
174
Norris
so that F x X e ~ ( X ) .
Then (F × X)0 = p n ~o(F x X)oB -I = O n
(~F x E) ~ F x E, so the domain of (F × X)8 is a subset of ~F.
Con-
versely if t e ~F, then, since E is the domain of ~ and p is reflexive, (t,t) g O n ~F x E; we conclude ~F = domain (F × X)@ and ~F is therefore closed.
Hence ~ is continuous.
The next lemm~ and its corollary are needed for the proof of Theorem 2.5, which gives the structure of all union-preserving quasiorder-bounded eemma
(2.3).
homomorphlsms.
Let ~ (X) b~e any subs emi~roup o f ~ ( X )
all the one-element relations and let 8: ~ preserving homomorphism bounded below b y a
(X) ÷ ~ ( Y )
that contains be a union-
quasiorder o on a sub-
set F of Y, and define, for each x and y i_~nX, ~x,y = (x,y)8\~. and v ~ F, and (ll) {~,y: (i) (u,v) e ~x,y implie s u ~ F ..............
Then:
(x,y) e X x X} is a matricial family. Proof.
We remark that since O = ~8 and 8 is a homomorphlsm,
~o~8 = G = ~SoG for all ~ e ~ ( X ) . (u,u) e ~ and therefore
If (u,v) E ~x,y and u e F, then
(u,v) e oo(x,y)0 = ~, which is absurd.
Similarly we can see that v ~ F either, proving To see (ii)
,
(i).
observe first that if y ~ y' then Nx,yOny,,z -C
(x,y)Oo(y',z) = ~, so there remains only to show nx,yODy,z = nx,z' for all x,y,z g X.
Let (u,v) e Nx,z = (x,z)@\O = (x,y)Oo(y,z)O\O.
Then for some t, (u,t) ~ (x,y)8 and (t,v) e (y,z)@.
Conclusion
(i)
implies that (u,t) ~ F x F and (t,v) { F x F; since o ~ F x F, and (t,v) c n . Hence (u,v) ~ n o~ , proving that (u,t) e nx,y y,z x,y y,z nx,z~Dx,yOny,z. To see the converse containment, suppose that for some t, (u,t) e ~x,y and (t,v) e ny,z. appeal to (i) gives (u,v) ~ G, i.e. Corollary
(2.4).
Then (u,v) e (x,y)O; another
(u,v) g nx,z' proving (il).
With the hypotheses of lemma (2.3), let 8':
(x) ÷ ~(Y) he ~iven by ~O' = u { n x , y : Then:
(x,y) ~ ~}.
( i ) 0' i s a u n i o n - p r e s e r v i n g homomorphism bounded
(ll) aO = aO' U 0 for all ~ e J ( X ) , ( i l l ) e d(X),
and (iv) i_~f, additionally,
below by ~,
~O' n F × F = ~ for all
(X x X)@ is ~ quaslorder on a
set E 0 then the relation ~ = (X x X)@' is ~ quaslorder on the set E = Eo\F.
175
Norris
Proof.
Conclusion
(i) is immediate from (ii) of lemma (2.3).
To
see (ii), it is clear for any (x,y) e X x X that (x,y)@ & q and hence that e@ ~ @@' u @. order-preserving,
~ u (x,y)@' ~ (x,y)8 and therefore
It follows from (i) of Lemma g~
(X).
u x,y since ~ = @@ and @ is
Conversely,
(ii) follows.
(2.3) that ~8' A F x F = @ for all
To see (iv), observe first that p is idempotent because
8' is a homomorphism and p is therefore transitive.
If u e E then
(u,u) s (X × X)@\~ = (X × X)@' because of (iii) above.
Finally if
(u,v) e p, then (u,v) g (X x X)8 by part (ii) if the corollary. additionally,
If,
u e F then (u,u) e ~ = ~8, so (u,v) e ~o(X × X)8 = ~,
so that (u,v) e p A F x F, contrary to (iii).
Similarly, v # F
either, so (u,v) e E × E, i.e. p is a quaslroder on E x E. Theorem (2.5).
Let E and F be disjoint subsets of Y, le___ttp and o be
quasiorders on E and F respectively suc~ that O and o u p are closed i_.nnY x y, and let ~:
E + X b_.ee~ continuous function satisfying (p n Q-I)ou = E x X.
Then the mapping @ defined for ~ e ~ ( X )
by
~O = (p n p o e o p - 1 ) i_~s~uni0n-preservin$
homomorphism fr°m ~(X)
above by the quasiorder O u ~ and below b y o . union-preservin~ '~(Y) Proof.
(2.4.1)
U O into ~ ( Y )
(2.4.2) bounded
Conversely , every
quasiorder-bounde d homomorphism from ~ ( X )
into
is obtained in this wa__~y.. For the moment we let X d and Yd denote X and Y with the dis-
crete topology.
If we define ~@d = p n ~o~o~ -I for ~ e ~ ( X d) and
if @' is the restriction of @d t o ~ ( X ) ,
then it is immediate from
Theorem (2.2) that @' is a union-preserving ~_(X) into ~ ( Y )
homomorphism mapping
= ~ ( Y d ) which is bounded above by p and below by ~.
Then we define ~8 = ~8' u ~ for all ~ e ~ ( X ) .
Since E 0 F = ~,
poo = ~ = Oop, and from here it is immediately seen that @ is a homomorphism,
@ preserves unions and is bounded by p u ~ and ~.
There remains only to verify that @ maps ~ ( X ) any relation,
into ~(Y). If a is -I is the inverse
then, as we mentioned earlier, Bo~o~
image of e under the function p x B, so if ~ is closed in X x X and -I is continuous, then ~o~o~ is closed In E x E and hence, by lemma (3.3.7) of [2],
~'
(p n ~o~o~ -I) u ~ = ~0 is closed in Y x y, i.e. ~8 e
(y). 176
Norris
We proceed to establish the converse. ~(Y)
Suppose that 8: ~
(X) +
is a union-preserving homomorphism bounded above by a quasior-
der P0 with domain E 0 and bounded below by a quasiorder O defined on a subset F.
Let P = ~\~.
Define 8' as in Corollary
is a union-preserving homomorphism f r o m ~ ( X )
(2.4).
into ~ ( Y )
Then 8'
which is
bounded below by ~ and satisfies ~8 = ~8' u G for all ~ g
(X).
Referring to the definition of 8' and to part (ill) of Corollary (2.4) we calculate that (X × X)8' = ~ {~x,y:
(x,y) e X × X} =
Ux,y((X,y)8\o ) = (Ux,y(X,y)8)\~ = (X × X)8\~ = 9\0 = P.
Then Theorem
(2.2) applied to 8', X d and Yd produces a function ~ satisfying (2.4.1) and (2.4.2).
The proof of the theorem will he complete after
we have shown that ~ is continuous and both ~ and O u ~ are closed. To see that ~ is continuous, we will show that if G is a closed subset of X then G~ -I = E n domain
(G × G)8.
(2.4) it follows that eO\o = sO'.
From (ili) of Corollary
Then t g G~ -I implies t e E, the
domain of ~, and (t,t) g p since p is a reflexive relation.
Hence
(t,t) e 0 n G~ -I x G~ -I = O A ~o(G x G)o~ -I = (G × G)O, so that t g E A domain (G x G)O.
Conversely if t e E N domain (G × G)O
then t E F~so for some s, (t,s) g (G x G)8' = p n ~o(G × G)o~ -I, i.e. t E G~ -I.
Since Y is a ~ -
space and (G × G)8 is closed in
y x y, domain (G x G)@ is closed in Y and therefore G~ -I is closed in E, proving that ~ is continuous.
Finally, since G = @8 and
p u~=(X x X)@, it follows that o and p u ~ are closed in Y.
This
concludes the proof of the theorem. Example (2.6).
Take Y to be a n y ~ -
space whatsoever and choose
three closed subsets A, B and C of Y subject to the conditions: (I) A n B # @
(2) A n C = @
and (3) either B n C = ~ or C & B.
Then define P = (A × B) u (C x C). P is idempotent.
Because of the three conditions
Moreover, O n p-I = [(A n B) × (A n B)] u (C × C)
so that O is a neat idempotent.
Now let X he a n y ~ -
a continuous image of both the subspaces A n B and C.
space which is Then there is
a continuous function ~ from (A n B) u C onto X whlch maps each of the subspaces A n B an~ C onto X.
The condition
(0 n p-l)o~ = E x X -I
will hold for any such function ~ where E is the domain of p n p which, in this case, coincides with PY n YO and is equal to (A ~ B) u C.
According to Lemma (2.1) the mapping @ defined by e@ = O n
(~°~°~ -I) is a union-preserving
homomorphlsm f r o m ~ ( X ) 177
into~(Y)
Norris
for which (X x X)8 = O n E × E.
One verifies that 0 n E × E is
(A 0 B) × (A N B) when B N C = ~ and (A x E) o (C × C) when C ~ B. In case B n C = ~, the domain of O N E × E is A N B, else the domain of 0 n E x E is A u C; in either case the domain of p is A o C, so
that p N E × E is a proper subset of O, provided B N C = ~, or provided C ~ B and E is a proper subset of A u C.
On the other hand if
E = A u C, then it may be seen that 0 0 E × E = 0, so that O ~ E ×' E and A E m O, making 0 a quasiorder on E.
That this may occur can be
seen by taking A o C = ~ and B = A o C. Exampl@ (2.7).
Let I be the closed unit interval, and for each n ~ 1
let. An = [~n'l 2n-ll].
Let E = o {An I n = 1,2 .... }, o = {(0,0)} and
define
P =%'2{An x Am
I n~m}.
Then 0 is a quasiorder with domain E, 0 is not closed in I
x I,
is a closed quasiorder with domaln {0} and o u 0 is closed in I x I. Each A
maps homeomorphlcally on
I via a function h
n
and we may n
then define a continuous function ~ from E onto I by x~ = xh and -I n verify that 0 n 0 = u {An × An: n = 1,2,...} and consequently (0 n 0-I)o~ = E x I.
Theorem (2.5) then applies so that the mapping
sO = [0 n ( ~o~o~-l)] u o is a union-preservlng quaslorder-bounded endomorphlsm o n ~ ( 1 )
which does not preserve symmetric relations.
(This example is a modification of an example in [2].) Since ~ ( X )
is an r-semigroup (as defined in [3]) it is a con-
sequence of Proposition (2) of [3] that every nonconstant homomorphlsm of the sort described by Theorem (2.5) is injeetive.
The en-
domorphism of Example (2.7) is not an automorphism, however, since (I x l)O is a proper subset of I x I and s0 ~
(I x I)8 for all e e
~(I)1
This shows the exist-
hence I x I is not in the range of O.
ence of endomorphisms of ~ ( X ) when X is a nondlscrete
which fail to be automorphisms, even
(in fact compact, Hausdorff) ~ -
space and
is in this sense an improvement of an example in [3]. Finally, we remark that Theorem (3.3) of [2] is a corollary of our Theorem (2.5); one simply requires
in addition to the hypotheses
of (2.5), that the relations 0 and o be symmetric.
The additional
conclusion of Theorem (3.3) of [2], that O preserves symmetry, is verified quickly and depends only on the relatlon-theoretlc 178
Norris
observation that the union and intersection of symmetric relations are again symmetric. The author thanks the referee for his helpful comments, including Example (2.6), which generalizes the author's original example. REFERENCES I.
2.
3. 4. 5. 6.
Clifford, A. H. and D. D. Miller, Union and symmetry preserving endomorphisms of the semigroup of all binary relations on a set, Czech. Math. J. 20 (1970), 303-314. Magill, K. D., Jr., and S. Yamamuro, Union and symmetry preserving homomorphisms of semigroups of closed relations, Proc. London Math. Soc. 24 (1972), 59-78. McAlister, D. B., Homomorphisms of Semigroups of Binary Relations, Semigroup Forum 3 (1971), 185-188. Norris, E. M., An internal construction of idempotent binary relations, (in preparation). Schein, B. M., A construction for idempotent binary relations, Proc. Japan. Acad. 46 (1970), 246-247. Schwarz, S., On idempotent binary relations on a finite set, Czech. Math. J. 20 (1970), 696-702.
Department of Mathematics and Computer Science University of South Carolina Columbia, SC 29208
Received 12 November,1976; in revised and final form,19 August,1977.
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