tion F A (see [14]), while pA is the normal completion N. /pAp and. D v are .... (V) x-supZ= sup{x.z : z E Z } and SupZ.x = sup{z.x : z E Z } for all subsets Z of S ...
Semigroup Forum Vol. 34 (1987) 253-285 9 1987 Springer-Verlag New York Inc.
RESEARCH
ARTICLE
COMPLETIONS FOR PARTIALLY ORDERED SEMIGROUPS M.Ern~ and J.Z.Reichman Communicated by K.H. Ho~nann
ABSTRACT A standard completion Y assigns a closure system to each partially ordered set in such a way that the point closures are precisely the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations are Y-continuous (i.e., Y 6 YS implies { x 6 S : y.x 6 Y } E YS and { x 6 S : x.y 6 Y } E YS for all y E S ), then S is called a Y-semigroup. If S is a Y-semigroup, then the completion YS is a complete residuated semigroup, and the canonical principal ideal embedding of S in ~S is a semigroup homomorphism. We investigate the universal properties of Y-semigroup completions and find that under rather weak conditions on Y, the category of complete residuated semigroups is a reflective subcategory of the category of Y-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions associated with so-called "subset systems". Related facts are derived for conditional completions.
O. INTRODUCTION Various completions of partially ordered semigroups occur sporadically in the literature. Perhaps the best known is the Dedekind-MacNeille completion by cuts (see, for example, [ 3 ] , [ 5 ] , [ 1 5 ] ) , but also various types of "join-completions" are of interest. Some of them are studied in [2],[4],[13],[18],[19],[20],[21]. Both the constructions and the derived results often show a significant similarity; hence i t appears now to be high time to furnish the common background and to develop a unifying theory of completions for partially ordered semigroups, including the categorical aspects of such completions. Some of the primary ideas are indicated in the book of Fuchs [15], and the purely
ERNE AND REICHMAN order-theoretic foundations of a general completion theory have been provided to some extent in [9] and [11], so a brief survey on the main facts will suffice for our present purposes (see Section I). Let V be any standard comPletion, that is, a function assigning to every partially ordered set P (resp. po-semigroup S )a closure system of lower sets that contains all principal ideals. Basic for our investigations will be the category SV of V-semigroups and V-continuous semigroup homomorphisms, where a map f between posets P and P' is V-continuous i f f-1[y,] E VP for all Y' E VP', and a V-semi~roup is a po-semigroup whose translations are V-continuous. Apparently, these notions have both topological and order-theoretical applications, and a collection of typical examples is presented at the beginning of Section 2. Many standard completions V occurring in practise share the following useful property: every weakly V-continuous map, that is, every map f :
P ~ P' such that inverse images of principal ideals belong toVP,
is already V-continuous. This is true, for example, of the MacNeille completion by cuts, the completion by Frink ideals [14], the Aleksandrov completion by lower sets~and the completion by Scott closed sets. Standard completions V with this property are called compositive, because the class of weakly V-continuous maps is closed under composition. For any compositive standard completion V, we can prove that i t gives rise to a reflector from the category SV to the subcategory s
of complete residuated semi~roups (c_~l-semi~roups in the sense of
Birkhoff [3]; see 2.14). I f
V fails to be compositive (examples are
given in [9] and [11]) then the situation becomes a bit more complicated, and we must replace s with the category ST{ of so-called V-semi~roup cgmpletions. Its objects(!) are join-dense weakly V-continuous embeddings e: S ~ C of V-semigroups S in complete residuated semigroups C (these embeddings need not be V-continuous I).A morphism between two such $~s
e: S ~ C and e ' : S' ~ C' is
then, as one expects, a join-preserving semigroup homomorphism c : C ~C'
such that the "restriction"
co = ~-I o coe : S ~ S'
is V-continuous. Our main result in Section 2 is Theorem 2.8, stating that the domain functor from S7s to $V has a l e f t adjoint right inverse 7. On the object level, the "completion" functor ~ assigns to any V-semigroup S the natural principal ideal embedding
254
ERNE AND REICHMAN qS : S ~ VS ,
x ~ +x = { y E S: y ~ x } .
In case of a compositive standard completion map associated with the reflector
V, qS is the reflection
V: $V ~ C$ . Hence, nS or VS,
respectively, may be regarded as the free V-completion of S. Almost every standard completion V occurring in concrete situations is isomorphism-closed, i . e . , every order isomorphism is V-continuous (of course, this property is much weaker than compositivity). For such completions V, the objects of the category
$~C
may be replaced
with concrete V-completions, i.e. pairs (S,C) where C is a complete residuated semigroup and S is a V-subsemigroup, that is, a V-semigroup with respect to induced order and multiplication such that the inclusion S~-*C is weakly V-continuous. The resulting adjunction theorem is formulated in 2.9. For ordered algebraic structures like groups or cancellative semigroups, etc., one knows that they never can be complete (unless being t r i v i a l ) , so that only conditional completions are of interest. Therefore we study this type of modified completions in Section 3, and i t turns out that one has to be very careful when passing from standard completions V to their conditional modifications not hard to see that
V~
I t is
V~ is compositive whenever V is (3.3); but the
corresponding adjunction theorem (see 3.7) only works when V is conditionable, i.e. the inclusion maps V~ ~ VP are (weakly) V-continuous - a condition which is violated, for example, i f
V is Frink's
ideal completion, while the Dedekind-MacNeille completion by cuts turns out to be conditionable (3.4). Moreover, we can prove that all join-ideal completions arising from so-called subset systems are conditionable (3.8), and these special cases already give a lot of applications, for example to the Aleksandrov completion by lower sets, to the join-completion in the sense of Bishop [4], and to the Scott completion studied by Rosenthal [20] Some of our investigations are closely related to Aubert's fundamental theory of x-ideals [0]. In fact, i f
V is any standard comple-
tion and S is a V-semigroup then VS (resp. the corresponding closure operator) is an x-system in the sense of Aubert, provided the members of
VS are invariant under translations. Moreover, the par-
tial order is uniquely determined by VS. Thanks are due to Professor Bosbach for having called the author's attention to Aubert's paper.
255
ERNE AND REICHMAN 1. STANDARD COMPLETIONS FOR PARTIALLY ORDERED SETS The theory of standard completions originated with B.Banaschewski [ I ] and J.Schmidt [21]. A systematic analysis of such completions in a suitable categorical framework has been developed in [ 2 ] , [ 9 ] , and I l l ] . Let us recall some of the basic order-theoretic concepts. For any subset X of a poset P with order relation ~, define +X, the down set of X, by +X : = { y E P: there exists an x E X with y < x } . A subset Y with ~x = +{x} ,
Y = +Y is a lower set (or lower end). The sets x EP ,
are called principal ideals. Thus, arbitrary unions of principal ideals are lower sets. The collection of a l l principal ideals of
P is
denoted by MP, and that of a l l lower sets by AP. A standard extension V is an assignment to each poset P of a collection VP of subsets of P with MPc VP c AP. I f
V is a standard extension and i f
each VP is a closure system ( i . e . , closed under arbitrary intersection) then V is referred to as a standard completion. The greatest standard completion is the Aleksandrov completion A by lower sets, and the smallest standard completion is the Dedekind-MacNeille completion or normal completion N, where NP is the collection of all cuts, i . e . , intersections of principal ideals. For each subset X of P, AX = A { + Y : Y E P, X ~ +y} is the cut generated by X, and NP = { 6X : X c p } . A map f : P ~ P' for a l l all
x,y E P,
for a l l
between posets P and P' is said to be isotone i f
x,y E P, x ~ y implies f ( x ) ~ f ( y ) , an embedding i f for x ~y
said to be residuated for a l l
f(x)
is equivalent to
f(x) ~ f ( y ) , and join dense i f
y' E P ' , y' = s u p f [ f - 1 [ + y ' ] ] . Furthermore, f :
P ~ P'
is
i f there exists a map g: P' ~ P such that
x E P and x' E P ' , =< x'
iff
x =< g(x') .
In this case, f and g are isotone maps which uniquely determine each other;
f is called the lower (or l e f t ) adjoint of g, and g i s
called the upper (or right) adjoint of f . Notice that f
is residu-
ated i f f inverse images of principal ideals under f are again prin256
ERNE AND REICHMAN cipal ideals. The following weaker property is of fundamental importance f o r the study of standard extensions. Given an a r b i t r a r y standard extension V, a function tinuous
i f for all
f-lily,]
f:
P ~ P'
is said to be weakly V-con-
y' E P ' ,
E VP ,
and V-continuous
i f for all
Y' E VP',
f - i [ y , ] E VP . Note that "(weakly) M-continuous" means "residuated" while "(weakly) A-continuous" means "isotone". A residuated map is always weakly Vcontinuous, but i t need not be V-continuous! Clearly, all identity maps are V-continuous and the composition of V-continuous maps is again V-continuous. We may thus speak of the category ~V of posets and V-continuous maps. A standard extension
V is called compositive
i f the class of all
weakly V-continuous maps is closed under composition. I t has been demonstrated in [9,2.5] that this is tantamount to saying that each weakly V-continuous map is already V-continuous. The largest standard completion A and the smallest standard completion N are compositive. Many further examples of compositive (and also of non-compositive) standard extensions are to be found in [9]. A slightly weaker condition than compositivity is union completeness. A standard extension V is union complete provided X E VVP implies
U X E VP for each poset
P . As observed in [9,2.2], this condition is f u l f i l l e d i f f the embeddings n p : P ~ VP ,
x ~ +x
are V-continuous (qp is weakly V-continuous for arbitrary standard extensions
V). Notice that qp preserves arbitrary meets whereas
joins are not, in general, preserved. If
V is a standard completion then every subset X of a poset P is
contained in a smallest member of VP, denoted by X- and called the (V-)closure of X. I f
X has a join in P
then this is also the join
of X- (and conversely), because VP contains all principal ideals. Furthermore, V-continuity of a map f :
P ~ P'
may be characterized,
as in the context of topological spaces, by the property f[X-] c f [ X ] -
( X c P ). 257
ERNE AND REICHMAN
Many standard completions arise as so-called ideal completions. Let Z be an arbitrary assignment to each poset P of a collection
ZP of
subsets of P (not necessarily lower sets). Then we may define two standard extensions
ZA and Zv by setting
ZAP = MP U { ~Z : Z E ZP } , Z P = { Z E zAP: Z has a join in P}. v The collection zVP = {Y E AP: i f
Z c Y and Z E Z P then supZE Y} --
V
defines a standard completion Zv , the join-idea] completion
associ-
ated with Z. A related standard completion is the A-ideal completion ZA (in [9] denoted by Z~) where ZAP = {Y E AP: i f Obviously,
Z c Y and Z E ZP then AZ c Y } .
ZAP is always contained in
zVP, and i f
P is Z-complete,
i . e . , each Z E ZP has a join, then ZAP coincides with
zVP.
Many authors call an assignment Z of the above type a subset system provided for all isotone maps f :
P ~ P'
and all
Z E ZP the image
f[Z] belongs to ZP' ( c f . [ 2 ] , [ 1 9 ] , [ 2 2 ] ) . In order to exclude inconvenient but t r i v i a l exceptions, one usually assumes (and we shall do so) that at least one ZP contains a nonempty member. This weak assumption already has the consequence that for each poset P, all singletons {x} with x E P belong to ZP, whence zAP = { @Z:Z E ZP }. I t follows from [9,4.1] that for any subset system Z of this kind, both Zv and ZA are compositive standard completions, while ZA is a compositive standard extension i f f i t is union complete (see also [11]). Three of the most important subset systems are P, F, and P, where PP is the collection of all subsets of P (power set), FP is the collection of all f i n i t e subsets of P, and PP is the collection of all directed subsets of P. (By a directed set we mean a subset D containing an upper bound of each f i n i t e subset of
D.) By definition,
pap is the collection AP of all lower sets, FAp consists of all f i n i t e l y generated lower sets, DAP consists of all directed lower sets.
258
ERNEAND REICHMAN pa,F ^ ,
and O^ are compositive, hence union complete, but only
P~ = A is a standard completion. The join-completion 3 = pv has been studied by various authors,
and its categorical aspects have
been mentioned in several contexts (see, e.g., [ 4 ] , [ 1 3 ] , [ 1 8 ] , [ 2 1 ] ) . The join-ideal completion Fv frequently occurs in the setting of (join-semi-)lattices, where i t coincides with Frink's ideal completion FA (see [14]), while pA is the normal completion N. /pAp and Dv are topological closure systems (in general different, but equal for posets in which every directed subset has a j o i n ) . The corresponding topologies have been investigated in [7] and [12] (see also [16]). pv is often referred to as the Scott completion (see, for example, [10],[13] and also [20]). For a map f : C ~ C' lattices
between two complete
C and C ' , the following four conditions are equivalent:
(a)
f is residuated
(b)
f is N-continuous
(c)
f is ]-continuous
(d)
f preserves joins, i . e . , f(sup X)= supf[X] for all
X~ C.
More generally, for an arbitrary subset system Z, a map f : P ~ P' between posets P and P' is (weakly) zV-continuou.s i f f i t preserves Z-joins, i . e . , Z E ZP and x = supZ imply
f(x) = supf[Z]
(see [9,3.3] and [21]). Now suppose V is any standard completion. An important universal property of the maps n p : P ~ VP ,
x ~ Cx
was pointed out f i r s t by J.Schmidt [21]; this property plays a crucial role for many adjunction theorems in the theory of partially ordered sets. 1.1 PROPOSITION. (1) Amap g: P ~ C from a poset P into a complete lattice
C i s weakly V-continuous i f f there exists a unique j o i n -
preservin.~ map gv : VP ~ C with g = gV onp , viz 9 gV(y) = sup g[Y] (2) A map f : P ~ P'
between two posers P and P' is V-continu-
ous i f f there exists a uniq.ue join-preserving maR. h: VP ~ VP' with honp = q ~ o f , v.i.z, h = Vf with
259
Vf(Y) = h[Y]
for all
ERNE AND REICHMAN YE VP. f P np !
P ~
C
~p'
npJp . . . . . . . . .
gV
,, Jp, rip,
h = Vf
Denoting the category of complete lattices and join-preserving maps by { , we may regard V as a functor from the category ~V of posets and V-continuous maps to the category { . I f the completion V happens to be compositive then { is a subcategory of FV , and Proposition 1.1 shows that the inclusion
g ~ PV is a right adjoint to V, i . e . ,
{
is a reflective subcategory of FV , with reflection maps qp (cf.[9, 4.15] and [11,2.11]). However, for general standard completions V, the category { may even fail to be a subcategory of FV . An adequate substitute for C is then the category ~ {
of V-completions which is
defined as follows: The objects of ~ C are weakly V-continuous joindense embeddings of posets in complete lattices. A morphism between two V-completions e: P ~ C and e ' : P' ~ C' is a pair (Co,C) , where c is a join-preserving map c: C ~ C' and co is a V-continuous map from P to P' such that
c oe = e' o co . Notice that in this
equation co is uniquely determined by c and vice versa (see [11, Section 3]). With the help of 1.1 it is not hard to show that the forgetful functor with
Pe = domain of e for l~-~{-objects e
and
Pc = co for ]P~vC-morphisms c is right adjoint to the "completion" functor ~: PV ~ ~ g ~P = np for FV -objects
P
defined by
and
Vf = Vf for FV-morphisms f . This adjunction characterizes the canonical embeddings np as free V-completions in the following sense. For each V-completion e: P ~ C there is a unique join-preserving surjection e
:
e v onp
ev: VP ~ C such that
9
For further background on standard extensions, see [9] and [11].
260
ERNEAND REICHMAN 2. COMPLETIONS FOR PARTIALLY ORDERED SEMIGROUPS We are now going to extend the completion theory alluded to in Section 1 and performed in [11] from posets to p a r t i a l l y ordered semigroups. For basic facts and terminology the reader is referred to the monographs [6] and [15]. For lattice-theoretical notions, see [3] and [17]. A po-semigrouP ( p a r t i a l l y ordered semigroup) is a semigroup S equipped with a partial order such that the l e f t translations lx : S ~S,
yHx.y
(x E S)
and the right translations r
: S ~S , x ~ x.y ( y E S) Y are isotone. More generally, given an arbitrary standard extension V, we call a semigroup together with a partial order a V-semi~roup i f l e f t and right translations are V-continuous (compare the notion of ideal systems in[15,V.17].) Of course, i f
S is a po-semigroup then
we put VS = VP where P is the underlying poset. I t w i l l not be necessary to distinguish between P and S. 2.1 EXAMPLES. ( I ) An A-semigroup is nothing but a po-semigroup. (2) M-semigroups are usually referred to as residuated semi~roups, in view of the fact that their translations are residuated. Thus a semigroup S equipped with a partial order ~ is residuated i f f there exist binary operations y < z "x m
iff
" and ". such that
x'y ~ z
iff
x < z ".y
( c f . [ 6 ] ) . Notice that the residuation operations
." and "
are u-
niquely determined by the multiplication 9 and the partial order relation ~. A residuated semigroup is Abelian i f f the operations
."
and ". agree. (3) I f
V is the join-completion J = pV (see Section 1) then a V-
semigroup S is characterized by the identities (V)
x-supZ= sup{x.z : z E Z } and SupZ.x = sup{z.x : z E Z }
for a l l subsets Z of S possessing a join. This condition is equivalent to: 261
ERNE AND REICHMAN (W)
x : supX and y = supY imply x.y = sup{u.v : u E X , v E Y } .
Similarly, i f P P denotes, for any poset P, the collection of all o nonempty subsets of P then ]o = PVo is a join-ideal completion, and a ]o-semigroup is a po-semigroup such that (V) holds for all nonempty subsets Z possessing a join. Partially ordered semigroups with this property are also called lower semicontinuous (cf.[15,XI.7]). Recall that ]C = MC for every complete l a t t i c e
C. Hence a complete
semigroup is residuated i f f (V) holds for arbitrary subsets Z. (4) The examples in (3) can be generalized in the obvious way. Given an arbitrary subset system Z, i t is easy to see that the ZV-semigroups are those in which the identities (V) are valid for all subsets Z E ZS which have a join (use the fact that "Z-join preserving" means "ZV-continuous"). (5) In a join-semilattice
S, considered as a po-semigroup, the iden-
t i t i e s (V) hold for all nonempty subsets possessing a join. Thus every join-semilattice may be regarded as a ]o-semigroup. However, (V) fails for
Z = ~ unless x is the least element of S.
(6) Now suppose S is a meet-semilattice, considered as a po-semigroup. We discuss four special choices for (a)
V.
S is an M-semigroup i f f i t is a Brouwerian or relatively pseudo-
compl.emented semilattice ( c f . [ 3 ] , [ 6 ] ) . (b)
S is a PA-semigroup i f f i t is a distributive semilattice, i . e . ,
for all
x,y,z E S with XAy ~Z there exist xi ~ x and Yl ~ Y
such that x1^Yl = z ( c f . [ 8 ] , [ 1 7 ] ) . (Since this equivalence is not completely evident and seems to be widely unknown, we sketch a proof. Suppose S is a pA-semigroup, i . e . , each of the sets
M(x,z) =
{ y E S: xmy ~ z } is a directed (lower) set. Then x ^ y ~ z implies y,z E M(x,z) , so we find a Yl ~ y,z with
xAyl ~ z ; hence,
x,z E M(yl,z ), and there exists an element x I ~ x,z with
x i A y l ~ z;
but then x 1 ^ y l = z because x I ~ z and Yl ~ z . Conversely, assume S is distributive and choose y,y' E M(x,z). Then we find
x I ~ x and
Yl ~ Y with x I ^Yl = z . I t follows that z ~ Yl and thus x^ y' ~ Yl, so there exist x~ ~ x and y~ ~ y'
such that x ll a y l ! = Yl "Hence Yll
is a member of M(x,z) with y ~y~ and y' ~ y~. As z belongs to M(x,z) and M(x,z) is obviously a lower set, we obtain M(x,z) E pAS, and S is a I)A-semigroup.) 262
ERNE AND REICHMAN (c)
S is a DV-semigroup i f f i t is an upper continuous or meet-con-
tinuous semilattice, i . e . , (^)
x^supZ= sup { x ^ z : z E Z }
holds for a l l directed (lower) sets Z possessing a join ( c f . [ 3 ] , [ 1 6 ] ) . (d)
S is a pV-semigroup i f f i t is a sup-distributive semilattice,
i . e . , (A) holds for all subsets Z possessing a join ( c f . [ 8 ] ) . Complete sup-distributive lattices are also referred to as complete Heytin~ algebras, frames or locales. Notice that the following three conditions are equivalent for a complete l a t t i c e C (because MC = 17AC n DVc = pVc ) : (a)
C is Brouwerian.
(b)
C is distributive and meet-continuous.
(c)
C is sup-distributive.
(7) I f
S is a group equipped with a partial order then the following
conditions are a l l equivalent, because the translations are orderisomorphisms whenever they are isotone: (a) (b)
S is a residuated (semi)group. S is a Y-semigroup for one, resp. each compositive standard extension V.
(c)
S is a po-(semi)group.
Henceforth l e t V be anj standard completion. We can assign to every V-semigroup S the system VS, where S is considered as a posetonly. For X,Y E VS we set X . Y = { z E S: z = x.y for some x E X , y E Y} X|
= (X.Y)- = n { z
E VS : X . Y ~ Z } .
V-conti nui ty of the translations yields (cf. Aubert [0]) : X.Y- ~- (X.Y)-
and X-.Y c (X.Y)-
for a l l
X,Y c S,
and i t follows that (X.Y)- = ( X . Y - ) - -- ( X - - Y ) - = ( X - . Y - ) "
(x,YES);
in particular, X O ( Y e Z ) : ( X . ( Y . Z ) - ) " : ( X . ( Y . Z ) ) - : ((X.Y).Z)- : ((X.Y)-.Z)= (X|
oz 263
ERNE AND REICHMAN
for all
X,Y,Z E VS, and consequently VS is a semigroup with re spec.t
to the operation|
. Furthermore, for
X,Y,Z E VS we have the follow-
ing equivalences: XGY E Z i f f iff
X.Y ~ Z
iff
Zx[Y] ~ Z for all
x EX
Y ~ n { z ~ 1 [ Z ] : x E X } = Z." X E VS.
This, together with the opposite calculation for the right translations, leads to 2.2 PROPOSITION. I f
S is a V-semi~roup then VS is a complete re-
siduated semigroup, and the natural embedding n S : S ~ VS ,
x ~ ~x
i s not on1~ a V-completion but a l s o a semigroup homomorphism. I f
S is a residuated semi~roup then nS preserves the residuation operations, too. 2.3 REMARKS. (1) I f
S is a p a r t i a l l y ordered monoid then so is
VS.
(2) Completions of Abelian V-semigroups are again Abelian. (3) The completion VS of a meet-semilattice S, considered as a Vsemigroup, is again a meet-semilattice. In fact,
VS is a frame with
respect to set inclusion (see 2.1(6)), and the meet-semilattice operation induced from S is set intersection: X AY = { X^ y : X E X , y E Y } = X n Y for (4) The completion
VS of a join-semilattice
X,Y E VS. S which is a V-semi-
group need not be a semilattice with respect to the induced operation, although i t is a complete l a t t i c e with respect to inclusion and set intersection. For example, i f
V is the Aleksandrov completion A then
may f a i l to be idempotent, as the following simple example shows: i S = o/~ x
A = {x,y} E AS, A(~A = S ~ A y
However, i f
V is a standard completion contained in the join-ideal
completion
Fv (see Section 1) then VS will carry a semilattice op-
eration (~ inherited from S. But, observe that the corresponding partial order relation ~ defined by
264
ERNE AND REICHMAN X< Y
iff
X~ Y = Y
will in general be distinct from set inclusion, since ~- is not the least but the greatest element with respect to ~! Indeed, X9
= (X v B)- = B-
For example, i f
for
X E VS.
V is the normal completion N and S is the preceding
three element semilattice then the extended semilattice (VS, v) has the diagram
2, while VS, considered as a poset with respect to set inclusion, has the diagram
These examples show that the interpretation of join-semilattices as po-semigroups is a b i t delicate, because the order-inherited natural join-semilattice structure of the completion VS is not always the semigroup structure induced from S. Now l e t us consider homomorphisms between J-semigroups. 2.4 LEMMA. Let f : C ~ C'
be a j o.in-preserving map between two J-
semigroups C and C' (see 2.1(3)). I f there exists a join-dense subset S of C with
f(x.y) = f ( x ) . f ( y )
for all
x,y E S, then
f is a semigrou.p homomorphism. PROOF. For x,y E C put
X = +XNS , Y = ~yNS . Then we have x =
supX and y = supY,whence f(x-y) = f(sup(X-Y)) sup(f[X]-f[Y]) ,and
supf[X.Y] =
f ( x ) . f ( y ) = f(supX).f(supY)= supfLX]-supf[Y] =
sup(f[X].f[Y]) , i . e . , f(x-y) = f ( x ) . f ( y ) " o Now i t is easy to extend Proposition 1.1 to the level of V-semigroups. 2.5 PROPOSITION. For each V-s.emig.rou.p S, t h.e embeddin.~ ns: S ~ VS, x ~ +x has the following universal Property: Amap g: S ~ C from S into a complete residuated semigroup C is a weakl~ Ycontinuous semigroup homomorphism i f f there exists a unique j o i n pres.ervi.n.g semi.group homomorphism gV: VS ~ C with g = gVo ns .
265
ERNE AND REICHMAN PROOF. Suppose g: S ~ C is a weakly V-continuous semigroup homomorphism from S into a complete residuated semigroup C. By 1.1, gV : VS ~ C ,
Y ~ supY
is the unique join-preserving map from VS into
C with g = gVon S.
Furthermore, the restriction of gV to MS satisfies the equation gV(+x).gV(+y) = g(x).g(y) = g(x.y) : gV(+(x-y)) = gV(+x.+y),and as MS is join-dense in the complete residuated semigroup VS (see 2.2), we conclude from Lemma 2.4 that gV is a semigroup homomorphism. Conversely, i f phism with
h: VS ~ C is any join-preserving semigroup homomor-
g = hon S then from 1.1 and 2.2 we infer that g must be
a weakly V-continuous semigroup homomorphism (but observe that g need not be V-continuous i f
V f a i l s to be compositive!) []
2.6 COROLLARY. A map f : S ~ S'
between V-semigroups S and S' i s
a V-continuous semigroup homomorphism i f f there exists a unique join-preservin 9 s.emigroup homomorphism Vf: VS ~ VS'
with
vf Oqs = nS'~ f " PROOF. Apply 2.5 to the map g = nS,o f which is weakly V-continuous iff
f is V-continuous ( c f . [ 9 , 2 . 4 ] ) . m
Now l e t us introduce the adequate categories. The V-semigroups together with V-continuous semigroup homomorphisms form a category $V" For example, SA is the category of po-semigroups and is 9
semi-
group homomorphisms. By CS we denote the category of complete residuated semigroups together with join-preserving ( i . e . , residuated) semigroup homomorphisms. Notice that CS is a subcategory of $A but not necessarily a subcategory of the smaller category SV" However, if
V is compositive then C$ turns out to be a reflective subcate-
gory of $V; we shall come back to that situation later. In the general case of an arbitrary standard completion V we have to replace the category s
by another category of so-called "V-semigroup com-
pletions" in order to obtain the desired adjunction theorems. We start with a somewhat larger category, the category
S~E of V-semi-
group embeddings. The objects of this category are all weakly V-continuous embeddings of V-semigroups into complete residuated semigroups. If
e : S ~ C and e ' : S' ~ C'
are two of these V-semigroup embed-
,dings, then a morphism between e and e'
266
in the category $~E is a
ERNEAND REICHMAN join-preserving semigroup homomorphism c: C ~ C' exists a V-continuous map co: S ~ S'
with
such that there
c o e = e ' o c o . The " i n -
duced" map co is then an $V-morphism and is uniquely determined by c. The category $~E may be regarded as a generalized "arrow category" (see [18,4.16] and [11]). By a V-semi~roup completion we mean a V-semigroup embedding which is join-dense. A V-semigroup completion i f for a l l (*)
x E S and a l l
e(x) ~ supe[Y]
e : S ~C
is free or ~ r i m i t i v e
Y E VS,
implies
x EY.
By 2.2, each of the natural embeddings qS is a V-semigroup completion, and the subsequent lemma w i l l demonstrate that they are, up to isomorphism, the only free V-semigroup completions. Moreover, we shall characterize (free) V-completions "V-liftings"
e: S ~ C by means of the unique
eV: VS ~ C (defined in 2.5).
2.7 LEMMA. Let e: S ~ C be a V-semigroup embedding. Then the f o l lowing three conditions are equivalent: (a) e i s join--dense, ~ . e . , a V-.semigrouP completion. (b) ev is surjective. (c) There exists a (unique) surjective C$-morphism c: VS ~ C with
e = COqs "
Furthermore, the f.ollowi.n~ three conditions are equivalent: (a') e is a free V-semigro.up completion. (b') ev is b i j e c t i v e , i . e . , a C$-isomorphism. (c') There exists a (unique) s
c : VS ~ C with
e = coq S . PROOF. (a) ~ (b): For y E C, we have Y = e-Z[+y] E VS and y = supe[e-1[+y]] = eV(y). Hence e is surjective and therefore a s epimorphism. (b) - (c): See 2 . 5 . (c) - (a): For y E C there exists an Y E VS with
y = c(Y) =
c(supns[Y] ) = supc[~s[Y]] = supe[Y]. Hence e is join-dense. (a') ~ ( b ' ) : By what we have shown before, ev must be surjective.
267
ERNE AND REICHMAN Suppose eV(y) = eV(Z) for some Y,Z E VS. Then , by condition (*), we obtain x E Y i f f e(x) ~ eV(y) i f f e(x) ~ eV(Z) i f f x E Z, whence Y = Z. (b') -, (c'): See 2.5 . (c') ~ (a'): Since (c) implies (a), we already know that e must be a V-semigroup completion. Furthermore, for x E S and Y E VS with e(x) ~ supe[Y], we conclude c(~x) = (COns)(X) ~ supc[qs[Y]] = c[suPns[Y]] = c(Y), and as c is an isomorphism, i t follows that +x ~ Y,
i.e.
x E Y.o
REMARK. I t is easy to see that the epimorphisms (resp. monomorphisms) in the category C (see Section 1) are precisely the surjective (resp. injective) C-morphisms. But i t is not evident whether the same holds for the category CS instead of C. However, the isomorphisms in s are certainly the bijective {$-morphisms. The V-semigroup completions are the objects of a full subcategory$~C of the category S~E and the free V-semigroup completions are the objects of a full subcategory S~F. Suppose there is given an Sv-morphism f : S ~ S' and a V-semigroup embedding e: S' ~ C' . Then e o f is weakly V-continuous, and by 2.5, the map ( e o f ~
is the unique
Sv-~E-morphism c: nS ~ e with Co= f . This proves the main part of the following result.
2.8 THEOREM. Let V be an a r b i t r a r y standard completion, (I) The domain functor ~ : $v-'E~ SV with Pe : domain of e for S~E-objects e and ~c = c o for S~E-morphisms c has a l e f t adjoint r i g h t inverse V : $V ~ Sv-~E with VS = ns for $v-objects S and ~f = Vf for Sv-morphisms f . (2) P restricts to a forgetful (i.e. faithful) functor from $~{ t o $V which i s right adjoint
left
inverse to the corestric-
tion of the functor ~ t o S~{. (3) ~ r e s t r i c t s to an equivalence between the categories S~F and $V ; the inverse equivalence is obtained by corestrictin~ the functor ~ t o S~F.
268
ERNEAND REICHMAN (4) In each of these adjunctions, the counit ce : VS ~ C ,
c is~.iven by
Y ~ sup e[Y]
for all V-semigroup embeddings e: S ~ C . In almost all practical applications, the standard completions V are isomorphism-closed, i . e . , every order-isomorphism is V-continuous. In this case i t is convenient to replace the categories
$~E , $~C and
SV~f with certain subcategories of the product category $VxC$. This can be done as follows: Objects of the category SVE are pairs (S,C) where C is a complete residuated semigroup and S is a V-subsemigroup of C, i . e . , a subsemigroup of C which is a V-semigroup with respect to the induced partial order relation and a V-s.ubposet, i . e . , +y n S E VS for all
y E C.
In other words, (S,C) is an object of the category $VE i f f the inclusion map iS,C: S ~ C is a V-semigroup embedding. A morphism between two $vI-objects (S,C) and (S',C') is a join-preserving semigroup homomorphism c: C ~ C' inducing a V-continuous map Co:S ~ S' (which is then a semigroup homomorphism between S and S' ). There is an obvious equivalence functor
I : $VE ~ $V~I with
I(S,C) = is, c for $vI-objects (S,C) and Ic = c for SvI-morphisms c. This functor induces equivalences between the full subcategory $VC of concrete V-semi~roup completions (defined in the obvious way) and the category $~s as well as between the full subcategory $VF of free concrete V-semigroup completions and the category $~F. An $vI-object (S,C) belongs to $V{ i f f
y = sup(+ynS)
for all yEC.
The objects of $VF are characterized by the additional property that each element of S is V-prime in (S,C), i . e . , x E S , Y E VS and x ~ supY imply x E Y . Now the ~oncrete" version of Theorem 2.8 reads as follows: 2.9 THEOREM. Let V be anu i.somo.rphism-closed standard comple.t!on. Then the " l e f t projection" functor
PE: SVI ~ SV with
PE(S,C) = S for $vI-objects and PEc = co for $v[-morphisms c i s right adjoint to the "V-embedding" functor VE: $V ~ $VE with
269
ERNEAND REICHMAN VES = (MS,VS) for $v-objects
S and
VEf = Vf for $V-morphisms f . The unit of the adjunction is an isomorphism, and the components of the counit are the join maps ~(S,C) : VS ~ C ,
Y ~ supY.
2.10 COROLLARY. The adjoint pair VE,PE restricts to an adjoint pair of functors
: sy
syc,
Pc : svc sv"
The projection functor Pc is faithful because the counit maps ~(S,C) are surjective for SvC-objects (S,C). Furthermore, Pc restricts to an equivalence between $V and SVF; the inverse equivalence is obtained by corestriction of the functor
Vc to
the category SVF, As an immediate consequence of 2.4 we notice: 2.11 LEMMA. A join-preserving map c: C ~ C'
is an $vs
be-
tween (S,C) and (S',C') i f f i t induces an SV-morphism between S and S ' . I t follows from 2.8 that $~F is a coreflective subcategory of $~C and of $~E; similarly, from 2.9 we infer that SVF is a coreflective subcategory of SVr and $VE . Moreover, we can prove: 2.12 THEOREM. $~{ is a coreflective subcategory o~f $~E, and $Vs i s a coreflective subcategory o__f_f SVE . PROOF. We only consider the second statement; the f i r s t one is treated similarly. Given an arbitrary $vE-object (S,C), we define CS = { supX: X E S }
(the joins formed in C).
CS is closed under arbitrary joins in C and is a subsemigroup of C (because C is a complete residuated semigroup). Hence CS is a complete residuated semigroup also, and the inclusion map iCs,C :CS ~ C is a join-preserving semigroup embedding. On the other hand, the inclusion map is, c
: S ~ CS turns out to be a V-semigroup completion s whence (S,Cs) is an object of the category SVc . In order to show that SVc is a coreflective subcategory of SvE with coreflection maps 270
ERNEAND REICHMAN C(s,c ) : iCs,C : (S,Cs)~ (s,c) , i t remains to verify the following couniversal property: For each $v{-object (S,C) and each $vI-morphism c: C ~ C'
between (S,C) and
an SvE-object (S',C'), there exists a unique $vC-morphism ~ between (S,C) and (S',C's,) such that c = E(S,,C,)O ~ . I t is easy to see that the corestriction of c to C' S, (which is well-defined because c preserves joins and S is join-dense in C) is the only $vs with the required property. Summarizing some of the adjunction theorems derived before, we obtain the following diagram of adjoint functor pairs for an arbitrary isomorphism-closed standard completion V.
~ inclusion functor Vc
E
~ coreflector !
$V
~ equivalence F
Now let us make the stronger assumption that V is a compositive standard completion. Under this hypothesis the preceding definitions can be simplified a bit, and the adjunctions derived before are supplemented by a theorem characterizing
V as a reflector from the cat-
egory $V to the subcategory CS. 2.13 PROPOSITION. S.up.pose V i s a
compositive standard completion
and S is a V-semigroup. Then: (1) The completion VS is a residuated V-semigroup, and the embedding qS: S ~ VS is an $V-morphism. (2) I f
S' i s a su.bs.emi~r.o.up of S with
~y~S' E VS' for all
y E S then S' is a V-subsemigroup of S. (3) Let (S,C) and (S',C') be SvC-objects. Then for a join-preserving map c: C ~ C' , the following conditions are equivalent: (a) c is an $vt-morphism between (S,C) and (S',C'). (b) c i s a semigroup homomorphism between C and C' with
c[S] ~ S'. 271
ERNE AND REICHMAN (c) c induces a semig.roup homomorphism from S into S' . PROOF. (1) See 2.2. (2) Let x,z E S' and +z = { y E S: y
