Union preserving homomorphisms of semigroups of ... - Springer Link

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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