R O C ZN IK I P O L SK IE G O T O W A R Z Y S T W A M A T E M A ÏY C Z N E G O Séria I : P R A C E M A T E M A T Y OZNE X V I I I (1975)
Przemyslaw Kranz (Poznan)
Sandwich and extension theorems on semigroups and lattices
Abstract. Several results concerning existence of certain types of functionals on semigroups appeared in the literature since the pioneer work of R. Kaufman in this field. Most of them give sufficient conditions of existence of an additive functional on a semigroup satisfying some requirements. Few results, however, are known con
cerning similar problems on yet more general algebraic structures. Adopting the ori
ginal method of R. Kaufman we were able to give a handful of such theorems. In particular, the existence of some general mappings on lattices as well as their exten
sion is established. This is applied to the problems concerning measures and sub
measures, a result resembling the classical theorem of Hahn and Banach is then derived for measures. Some other applications are indicated.
1. Section 1 is devoted to studying the existence of additive mappings on semigroups and their Hahn-Banach extension properties. We shall use the technique already succesfully exploited in such situations ([3],
[4], [6], [8], [10]) but here the proof differs from the original one given by E. Kaufman in [6]. The method being more direct and constructive applies to real functionals as well as to mappings taking their values in ordered spaces (in fact in complete vector lattices — this follows from a result of Ting-On To [12], and will be published by the author separately), where
as the Kaufman’s proof fails to work in such a case. This method has wider applications as it is shown in Section 2.
We begin by recalling some definitions and notations. Les 8 be a semigroup with an additive composition (i.e. a set closed with respect to a single associative binary addition). Assume that T is an ordered vector space. It is called a vector lattice (YL) if to every pair й?, у e T there corresponds a (unique) element x v y e T called their supremum, and hence a unique х л у е Т (infimum). It is an order complete vector space (OCYS) if every bounded from above subset in T has the least upper bound (and therefore every bounded from below subset of T has the greatest lower bound). T is an order complete vector lattice (OCYL) if it is YL and OCYS.
A mapping w : 8->T is said to be subadditive if w(st + $2) < w (si) + w(s2) f°r
A mapping u: 8-^T is said to be superadditive if —u is subadditive A mapping which is both subadditive and superadditive is called additive.
Let w be a subadditive mapping on 8 with range in T and и a superad
ditive mapping from 8 into T. Define
(1.1) r = sup — {u(na0 -fs) — w?(s)}, VI
seS
and
(1.2) ' r' = inf — {w (s + ma0) — и (s)}, 1 Vfl
seS
a0e 8,
a0e 8.
If w(s) ^ u(s) for every s in 8, then it is easily verified that both the supremum and the infimum exist and w(a0) > r, r' > u(a0). Equally easy is to check that r' > r. In fact, by our assumption on w and и the following relation holds for every a0, se 8 and m, n > 1.
w(m na0 -{-ms + ns) > u (m n a0 -f-ms -{-ns).
Now making use of subadditivity of w and superadditivity of и we get nw(ma0-\-s)-{-mw(s) > mu(na0 + s)-\-nu(s)
from which the required relation follows.
1.1. Le m m a. I f w(a0) > u(a0), then either w(a0) > r or r' > w(a0).
Indeed w(a0) = r and u(a0) = r' imply w(a0) = u(a0). Hence as w(a0) > r and r' > и (a0) at least one of the inequalities must be strict.
Lemma 1.1 is a crucial step in proving the following theorem.
1.2. Th e o r e m. Let w and и he mappings from 8 into T such that (i) w is subadditive on 8,
(ii) и is superadditive on 8, (iii) u(s) < w(s), s e 8.
There exists an additive mapping 8->T such that u(s) < |(s) < w(s) for all se 8 if and only if T is an^OCVS.
P roof, a. Sufficiency. Observe first that if w(s) = u(s) for all se 8, then these two mappings are already additive. Let us exclude this and assume that there is a point a0 in 8 such that w(a0) > u{aQ). Then by Lemma 1.1 either w(aü) > r or r' > и (a0). Similarly as in the proof of Theo
rem 1 in [8] we construct mappings w' and u' satisfying (i) and (ii) and such that at least one of w > и' > и or w > w' > и holds. Let us adopt the convention 0 • s + 1 t. The mapping u' is defined as
(1 .3 ) u'(t) = s u p { m r ' + w ( s ) } ,
where supremum is taken over all representations t = s-f-ma0, m > 0, setS'. If there is no s in this representation we replace u(s) in (1.3) by 0.
Similarly
{1.4) w'(t) = inf {nr + w(s)},
where infimnm is again taken over all representations t — s + a^n, n > 0, s e 8, with the same convention as before.
Application of the Zorn-Kuratowski Lemma shows that there exists the maximal element uQ in the class {V } of all snperadditive mappings satisfying w > и' > и and the minimal element w0 in the class {w'} of all subadditive mappings satisfying w > w' > u.
The details of the proof being completely analogous to that of the proof of Theorem 1 in [8] are omitted here. The reader is referred to this paper.
From Theorem 1 follows a number of corollaries some of which we shall list here for the sake of complèteness.
1.3. Co r o l l a r y (Mazur-Orlicz-Kaufman). Let 8 be a semigroup and T an 0GV8, w a subadditive map 8->T. Let further и be another mapping S-+T such that if st , ..., sn is a finite sequence in 8, then
П w{s1+ ... + S n) > J£ u(s{).
i= 1
Then there exists an additive mapping £: 8->T such that £ > r. 1.4. Co r o l l a r y (Fuchssteiner [3]). Let 8 be a preordered semigroup, u: 8-^-T a super additive function and w: 8->T a monotone subadditive function. Assume that u(s) < w(s) for all s e8. Then there exists a mono
tone additive function £ on 8 satisfying u(s) < £(я?) < w(x) for all xe 8. This follows immediately from the easy observation that if w is a mono
tone function, then also w' given by (1.4) is such and hence the infimum w0 is not only additive but also monotone.
1.5. Co r o l l a r y. Let on 8 be given a subadditive mapping w: 8-+T and on a subsemigroup 8 0 of 8 an additive mapping £0 such that £0(s) ^
^ w ( s ) for all s in 80.
I f T is an 0CV8, then each of the three following conditions assures existence of an additive mapping £ on 8 such that £ < w and the restriction of £ to 8 0 £/£0 = £0
(i) Sje 8q, $o=>S2e tf0.
(ii) e Sq, s24 8q =>w($i £2) ^ £0(^1) 2)?
(iii) s1e 8 0, sz4 80, sx + s2e80 implies that £0(вг + s2) < £0(si) + w (sa).
Several other theorems of similar form are presented in [13], p. 45-50, where the reader is referred too.
2. The method developped in the preceding section proves nsefnl in more general situations. Using similar procedure as in the proof of 1.1 we are able to prove existence and extension theorems for a more general class of mappings. Prom these we derive result on extension of measures from a boolean ring to a larger boolean ring.
Following G-. Birkhoff ([2], p. 74) we shall call a map defined on a lat
tice L with values in a vector space T a valuation if it satisfies the following relation :
V(x) + v ( y ) = v(ævy) Jr v(XAy), X, y e L .
Probability and measure functions on sets are valuations so is dimen
sion in projective geometry; any function on a chain with values in a semigroup is valuation. These functions were investigated in connec
tion to the theory of integration by Alfsen [1] and Kappos-Mallios [5], [9].
Analogously to subadditive functions and superadditive functions we define tipper semivaluation w: L-+T
w(x) + w(y) ^ w ( x v y ) + w ( x A y ) and lower semivaluation u: L ^ T
u{x) -\-и{у) < u(x v y ) -\-ixAy) .
Upper semivaluations and lower semivaluations will be denoted hence
forth usv and lsv.
Let L be a lattice and let w : L-+T be an usv and и : L^-T a lsv. We are interested in mappings satisfying certain conditions as follows
(2.1) (2.2)
(2.1')
(2.2')
where n, m, n', m' are non-negative real numbers and x x, x 2, x[, x2 are members of L. Such numbers and points always exist as e.g. we can take in (2.1) n — 0, x x — x0vx.
These conditions are obviously motivated by the properties of semi
valuations. Let us f urther restrict to the case (2.3) w(a?i) > w(x), w{xz) ^ w { x ) , (2.4) u{x) > u{x[), u(x) ^ u(x'ÿ).
w(x0v x ) < nw(x0) -\-w(x1), w (x0 л ж )< mw (x0) + w (x2) , u(x0v x ) > n'u(x0) +и( х [) ,
u(x0a x) > m'u{x0) -{-u(x2),
It is again not unreasonable to take such functions as for every usv and lsv there exist appropriate members of L satisfying these requirements.
We assume that both и and w are non-decreasing (T is an OCVL), and that u(x) ^ w(x) for x e L :
Consider now the following quantities:
rx = s u p i(w (# 0v x) — w(aq)), O^n П
XxeL
Г2 —SUp — ( u ( X 0 A X) — w(x2))l 0<m m
x%eL
r[ = inf —T (w{x0v x ) — u{x'x)\, 0<n' w
XjcL
r2 = inf ~^r(w(x0a x) — u(x’2)\, 0<m’ m
x2eL
where n, m, n , m , x lf x2, x[, x2, are such that they satisfy the corres
ponding (2.i) condition.
It is clear on account of (2.1) through (2.2) that all these suprema and infima exist and they satisfy the relations:
rt ^ w { xo ) , i = 1 , 2 ,
(*) r'i > U(x0), i = 1 , 2.
Note, moreover, that r[ > 0 and r2 < 0 .
2.1. Le m m a, (i) Assume w(x0) = 0 . Then rx < rx and in consequence if w(Xq) > u(x0), then either rx< w(x0) or r’x > u(x0).
(ii) Assume u(x0) = 0. Then r2 < r2 and in consequence if w{xQ) > u(x0), then either r2< w ( x 0) or r2> u ( x0).
P roof, (i) If w{x0) = 0, then rx < 0 what implies that rx ^ r [ . As
sume now w(x0) = rx, r'x = u(x0). This gives w(x0) = u(x0).
Hence if w(x0) > u(x0), then either w(x0) > rx or u(x0) < r[ . (ii) Proof the same as above.
If T is an ordered vector space we can add one point defined as a class of all unbounded from below decreasing sequences in T. This point having all properties of — oo in the real line completes in order sense T. This completion will be denoted Td.
2.2. Th e o r e m. Let L be a distributive lattice, T an OCVL, w: L->T6 an increasing usv, and u: L-+Ts an increasing lsv such that u(x) ^ w ( x ) for all x e L . Then there exists an increasing valuation v: L-+T such that u(x)
< v(x) < w(x).
P roof. If u(x) = w{x) for every x in L, then it is already an increasing valuation. Let us therefore assume that there exists a point x0 in L such
that u( x0) < w(x0).
Without loss of generality we may take w(xQ) = 0. Indeed if w(x0)
— к Ф 0, then taking w' — w — k, u' = u — k we get again the same situation with w(x0) = 0.
If w(x0) > rx let us define
w'(y) inf [nr1 -\-w(xx)) for y = x v x0,
w (x) otherwise.
Infimum in the above definition is taken with respect to all n > 0 and w(xx) satisfying (2.1).
By the definition of rx and on account of Lemma 2.1 it is clear that w'{x0) < rx< w{x0),
w ' ( y X m ( y ) ' , y e L . Besides
(2.7) w'(x0v x) > u(XqVx).
Moreover, w' is an increasing usv. To prove it take y x = x0v x x, y2
— x0 v x 2. Then
y xv y2 = {xxv x2) v x 0,
Ух л У2 — iæxл œz) v æo by the distributivity of L.
Therefore in accordance with (2.1), (2.2)
nrx -\-w(x! v x 2) + mrx -(- w (xx л x 2) < nrx + w (xx) + mrx + w (x2) . Taking now infima on both sides we get
w ' { y i v y i) + w ' ( y1A yi) < w'(yx) + w ' { y 2).
If only one of y x and y2 is representable in the form х{у х0, then this relation is clearly satisfied.
It remains to prove that w'(y) ^ u(y), ye L.
If у = x0vx, then from (2.7) we get it immediately. If у is not repre
sentable in this form, then w'(y) = w ( y ) ^ u(y). In short if w(x0) > u(x0), then w is not a minimal element in the class {w'} of all usv such that w' ^ u.
It should be noted that from the definition of rx follows that any r > rx can replace rx provided w(x0) > r. Therefore even in the case u(x0) e T0\T we can take re T.
If w(x0) = rx, then by Lemma 2.1 rx > u( x0) and
SU J>{nr1+u{x'l))j y — X QV X j
и (x ) otherwise,
u'{y) =
where the infimum is taken with respect to all n > 0 and и (x[) satisfying (2.1) is again an increasing lsv, u'{y) < w(y), y e L, and u( x0) < u'(x0).
Therefore u' is not the minimal element in the class of all increasing 1st
majorized by w. Application of Zorn-Kuratowski Lemma yields now the existence of the minimal element in {w'} and the maximal element in {u’}. In view of the above considerations these elements must be equal and therefore it is the required valuation.
'2.3. Remark. It is worth to note that it suffices to assume that only one of the semivaluations is monotone to get that the valuation is mono
tone. It is so because we can always construct the sandwich valuation based on the increasing one due to the both forms of Lemma 2.1.
The proven theorem as well as the above remark allows us to establish the Hahn-Banach type theorem on extension of valuations.
2.4. Th e o r e m. Let L be a distributive lattice, L0 its sublattice, T — an OCVL. Let further w: L->T6 be an increasing usv and v0: L0->Td an increas
ing valuation dominated on L0 by w.
I f L0 has the property
(i) x e L 0, x v y , x л у e L Q=>y€ L 0,
then v0 has an extension to an increasing valuation v: L->Td such that V ^ w.
P roof. Assume that the range of is not included in T0\T. (In that case defining v in such a fashion that its range remains in T0\T gives an increasing valuation with required properties.)
Define
v' (x) V0(x) for X e L 0,
— oo for xe L \ L0
Note first that if (i) holds, then also xe L 0, x v y e L0 imply that x a у e L q.
It is so because x — ( x A y ) v x .
Clearly v is an lsv although it is not monotone. However, as v < w it is possible to find then an increasing valuation v such that v < v < w.
Obviously v ^ v0 on L 0.
In view of condition (i) there exists another increasing valuation v' on L, v > v' > v unless already v = v0 on L 0. This can easily be verified.
Indeed if there is a point x 0e L0 such that ^0(^0) < ^ (^o)> then taking r = sup — (v(xQ v x) — v(xx))
n we see that r ^ v0(x0).
This follows from the fact that the sup is attained on L0 and there
^ = vQ is a valuation. Easy calculation shows that for r = < v(x0)
\
by the proof of Theorem 2.2 there exists an increasing valuation v on L such that v(x0) = v0(Xq). This accomplishes the proof.
Every Boolean ring satisfies condition (i) of the theorem. Therefore it is a generalization of the following theorem proved by Kelley [7] (see also a related result of Pettis [11]).
2.5. Co r o l l a r y (Kelley [7], Theorem 14). Let A be a Boolean algebra, let p be a non-negative monotonie real-valued function on A such that p(a) +
-\-p(b) ^ p (avb) "t p (a/\b) for alb members a and b of 4 and let m be a measure on a subalgebra В of A such that m(b) < p (b) for b in B. Then there is a mea
sure n on A which is an extension of m such that n{a) < p(a) for all ae A.
This follows easily from 2.4 if we assume that vo(0) = 0, because it is a well-known fact that then the notions of a measure and a valua
tion on a Boolean ring are identical.
References
[1] E. M. A lfs e n , Order theoretic foundations of integration, Math. Ann. 149 (1962), p. 419-461.
[2] G. B ir k h o ff, Lattice theory., 2nd ed., Amer. Math. Soc. Coll. Publ., Yol. 26, New York 1948.
[3] B. F u c h sste in e r , Monotone additive Funlctionale auf Commutative Halbgrup- pen, Technische Hohschule Darmstadt, Fachbereich Mathematik, Preprint
No. 14, 1972.
[4] — Sandwich theorems and lattice semigroups, J. Funct. Anal. 16 (1974), p. 1-14.
[6] D. A. K a p p o s und A. M a llio s, Der Integralbegriff in o-topologischen Verbanden, Bull. Soc. Math de Grèce, N .S.l Fasc. 1 (1960), p. 98-105.
[6] R. K a u fm a n , Interpolation of additive functionals, Studia Math. 28 (1966), p. 269-272.
[7] J. K e lle y , Measures on Boolean algebras Pacific J. Math. 9 (1959), p. 1165-1177.
[8] P. K ra n z, Additive functionals on abelian semigroups, Comm. Math. 16 (1972), p. 239-246.
[9] A. M a llio s, Isotone modular mappings in lattices (in Greek), Bull. Soc. Math, de Grece, N .S .l, Fasc. 2 (1960), p. 1-45.
[10] S. M azur et W . O rlicz, Sur les espaces métriques linéaires (II), Studia Math.
13 (1953), p. 137-179.
[11] B. J. P e ttis , Linear functionals and completely additive set functions, Duke Math. J. 4 (1938), p. 552-565.
[12] T. O. To, The equivalence of the least upper bound and the Hahn-Banach property in ordered linear spaces, Proc. Amer. Math. Soc. 30 (1971), p. 287-295.
[13] E. W a e lb r o e c k ed., Summer School on Topological Vector Spaces, Springer Verlag, Lecture Notes, Yol. 331, 1973.
IN S T IT U T E OF M A TH EM A T IC S,
P O L IS H A C A D E M Y OF SCIENCES, P O ZN A N
