Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 399–410

ELEMENTS GENERATING BALAYAGES

Sirkka-Liisa Eriksson-Bique

University of Joensuu, Department of Mathematics P.O. Box 111, FIN-80101 Joensuu, Finland

Abstract. We consider balayages in H -cones. Formerly balayages were characterized in terms of sets but a new approach looks at the elements instead of sets. Earlier, for example, we have proved an explicit formula for a balayage in an H -cone possessing a certain type of unit in terms of mixed envelopes formed relative to two partial orderings. Our problem is to describe those elements that generate a balayage. We state a necessary and sufficient condition for an element to generate a balayage in an H -cone possessing a special type of unit. We also give a relation between balayages and extreme points of a convex set of elements dominated by a fixed element.

Introduction

The theory of balayages is an integral part of potential theory. We consider balayages in an H -cone which is an axiomatic model of a convex cone of positive superharmonic functions on a harmonic space. A balayage is a mapping from an H -cone into itself which is additive, left order continuous, contractive and idempotent (see Section 2). Originally a balayage R̂_{u}^{A} of a superharmonic function s on a subset A of a harmonic space X is given by

R̂_{u}^{A}(x) = lim inf_{y→x} inf{ v(y) | v ≥ u on A, v is superharmonic }.

If A is open then the mapping u ↦ R̂_{u}^{A} is a balayage. Moreover, if a harmonic space X satisfies the axiom of polarity [6, Theorem 9.1.1] then the mapping u ↦ R̂_{u}^{A} is a balayage for any A ⊂ X.

Previously balayages were characterized in terms of sets. In [9, Theorem 2.9] we present an explicit formula for a balayage in terms of mixed envelopes defined relative to two partial orderings. Some related characterizations of balayages are also given by Popa [10]. In this paper we study which elements generate a balayage? Our main theorem gives a necessary and sufficient condition for an element to generate a balayage in an H -cone possessing a special type of unit (Theorem 2.9). We also present a relation between balayages and extreme points of a convex set of elements dominated by a fixed element (Theorem 2.13). We prove an interesting result that the set of balayages is an abelian semigroup with respect to some special addition. This result gives a new formula for the least upper bound of two balayages.

1991 Mathematics Subject Classification: Primary 31D05, 06A10; Secondary 46E99.

The author thanks Department of Mathematics of McGill University for the hospitality she enjoyed there while this work was done, and the Academy of Finland for the financial support that made it possible.

1. Preliminaries

We use the following definition of an H -cone which is equivalent with the original one ([8, Theorem 1.3]).

Definition 1.1. Let E be an ordered vector space and S be a convex subcone of E such that S ⊂ E^{+} and E = S − S. The cone S is called an H -cone if it possesses the following properties: (A_{1}) any upward directed and dominated subset F of S has a least upper bound

in E denoted by ∨F and ∨F ∈ S, (A_{2}) any subset F of S has a greatest lower bound in E denoted by ∧F and

∧F ∈ S, (A_{3}) for any elements s and t of S, the greatest lower bound of the set { u ∈ S |

s − t ≤ u }, denoted by R(s − t), satisfies the conditions R(s − t) ∈ S and s − R(s − t) ∈ S. A partial order called specific order, denoted by ≼, is defined in an H -cone

by

s ≼ t if and only if t = s + s′ for some s′ ∈ S.

Any pair of elements in an H -cone has mixed envelopes introduced by Arsove and Leutwiler in algebraic potential theory ([2]).

Theorem 1.2. Let S be an H -cone. Then for any elements s and t in S there exist a mixed lower envelope

s⌣\t = max{ x ∈ S|x ≼ s, x ≤ t } = s − R(s − t)

and a mixed upper envelope

s⌢/t = min{ x ∈ s|x ≽ s, x ≥ t } = s + R(t − s)

satisfying the equality

s⌣\t + t⌢/s = s + t.

Proof. See [2, Theorem 2.5].

We recall the definitions of special units which are important in the theory of H -cones.

Definition 1.3. Let S be an H -cone. An element e ∈ S is called a weak unit if s = ∨_{n∈N}(ne) ∧ s for all s ∈ S. An element p ∈ S is called a generator if s = ⋁_{n∈N}(np)⌣\s for all s ∈ S. An element s ∈ S is called a u-quasi-unit for u ∈ S if s = ⋁_{n∈N}(ns)⌣\u.

We apply the following characterization of quasi-units given by Arsove and Leutwiler in [1, p. 2499]:

Theorem 1.4. Let S be an H -cone and u, s be elements of S. Then the following conditions are mutually equivalent:

(i) An element s is a u-quasi-unit. (ii) s = (αs)⌣\u for all α > 1. (iii) s = (αs)⌣\u for some α > 1. (iv) R(s − αu) = (1 − α)u for all α < 1.

(v) R(s − αu) = (1 − α)u for some α < 1. (vi) An element s is an extreme point of the convex set C = { t ∈ S | t ≤ u }.

Let S be an H -cone and u ∈ S. We recall that an element s of an H -cone S is called u-continuous if for any ε > 0 and any upward directed family F ⊂ S with s = ∨F there exists an element f_{ε} of F such that s ≤ f_{ε} +εu. An element s in S is universally continuous, if it is u-continuous with respect to all weak units u in S.

Definition 1.5. An H -cone S is called a standard H -cone ([4, p. 104]) if it has a weak unit and a countable dense set of universally continuous elements.

For a reference to the theory of H -cones we mention [4].

2. Elements generating a balayage

We consider balayages in H -cones. Recall that a mapping B from an H -cone S into S is called

(a) left order continuous if for any s ∈ S the property B(s) = ⋁_{t∈F} B(t) holds for all upward directed subsets F of S,

(b) idempotent if B^{2} = B,

(c) contractive if B(s) ≤ s for all s ∈ S. A balayage is a mapping B: S → S which is additive, left order continuous,

idempotent and contractive. A potential-theoretic model for a balayage is the mapping s ↦ R_{s}^{U} where s is a positive superharmonic function on a harmonic space, U an open set and R_{s}^{U} the so-called reduced function. For further reference see [6, Section 4.2].

In the set of mappings from an H -cone S into itself we use the partial ordering given by ψ ≤ φ if ψ(s) ≤ φ(s) for all s ∈ S.

Balayages have the following important property as proved in [9, Lemma 2.3].

Lemma 2.1. Let S be an H -cone. If B: S → S is a balayage then

(2.1)  B(u)⌣\v = B(u)⌣\B(v)

for all u and v in S.

The value of a balayage at a point is obtained from its value at a generator [9, Theorem 2.9].

Theorem 2.2. Let S be an H -cone possessing a generator p, and B a mapping from S into S. Then B is a balayage if and only if B is left order continuous and satisfies the equality

(2.2)  B(x) = ⋁_{n∈N}(nB(p))⌣\x

for all x ∈ S.

Quasi-units and balayages have a close connection stated next.

Proposition 2.3. Let S be an H -cone and u ∈ S. If B: S → S is a balayage then the element B(u) is a u-quasi-unit and therefore an extreme point of the convex set { s ∈ S | s ≤ u }.

Proof. Let B: S → S be a balayage and u ∈ S. Applying Lemma 2.1 we obtain

(2B(u))⌣\u = B(2u)⌣\u = B(2u)⌣\B(u) = B(u).

Hence B(u) is a u-quasi-unit by Theorem 1.4.

A natural question is what values of B(p) in the formula (2.2) produce a balayage? For handling this we define the following concept.

Definition 2.4. Let S be an H -cone. An element u ∈ S generates a balayage if the mapping B: S → S defined by

B(x) = ⋁_{n∈N}(nu)⌣\x

is a balayage.

Applying [9, Theorem 2.10] we obtain directly the next result.

Proposition 2.5. An element u of an H -cone S generates a balayage if and only if the condition

(2.3)  u = ⋁_{{n∈N \atop f∈F}}(nu)⌣\f

holds for any upward directed family F with ⋁ F = u.

Note that the condition (2.3) does not hold generally for all u in an H -cone S. Indeed, it is possible that u = ⋁ F for some upward directed family F and u⌣\f = 0 for all f ∈ F. For example this property holds if u is a harmonic function and F the set of potentials with u = ⋁ F.

Lemma 2.6. Let S be an H -cone and u an element of S. If the function ψ: S → S defined by ψ(x) = u⌣\x for x ∈ S is left order continuous then the element u generates a balayage.

Proof. By virtue of the preceding proposition we only have to verify the condition (2.3). Assume that F is directed upwards with ∨F = u. Since ψ is left order continuous we have

u = ψ(u) = ⋁_{f∈F} u⌣\f ≤ ⋁_{{f∈F \atop n∈N}}(nu)⌣\f ≤ u,

completing the proof.

An example of elements generating a balayage are v -continuous elements for any element v ∈ S.

Lemma 2.7. Let S be an H -cone and v ∈ S. If u is v -continuous then u generates a balayage. Moreover, any element u ∈ S enjoying the property

(2.4)  ⋀{ R(u − t) | t ≤ u, t is v-continuous } = 0,

generates a balayage.

Proof. Assume that an element u of S is v -continuous for some v ∈ S. It is enough to prove that the mapping ψ defined by ψ(x) = u⌣\x for x ∈ S is left order continuous. Let a family F be directed upwards with ⋁ F = u. Since u is v -continuous for any ε > 0 there exists an element f_{ε} in F such that u ≤ f_{ε} +εv. This implies that u ≤ (u + εv)⌣\(f_{ε} + εv) and further by [2, p. 16]

u ≤ u⌣\f_{ε} + εv ≤ ⋁_{f∈F} u⌣\f + εv.

Since ε is arbitrary, the condition (2.3) holds.

Lastly suppose that the condition (2.4) is valid for u ∈ S. Denote by 𝒱 the set of v -continuous elements. Let F be directed upward with ⋁ F = u. By Theorem 1.2 and (2.4) we obtain

(2.5)  u = ⋁_{t∈𝒱} u⌣\t.

For any t ∈ 𝒱 with t ≤ u, there exists for each ε > 0 an element f_{ε} ∈ F such that t ≤ f_{ε} + εv. Hence we have

u⌣\t ≤ u⌣\f_{ε} + εv ≤ ⋁_{f∈F} u⌣\f + εv.

Combining this with (2.5) we obtain the condition (2.3).

In standard H -cones even the following stronger result holds.

Theorem 2.8. Let S be a standard H -cone. Any balayage B on S is generated by a u-continuous element for some weak-unit u in S. Conversely, a u-continuous element generates a balayage for any weak unit u ∈ S.

Proof. We only have to prove the first statement. Assume that B is a balayage on a standard H -cone S. Then the set B(S) = { B(s) | s ∈ S } is also a standard H -cone by [4, Corollary 5.2.6] and there exists a countable dense set (s_{i})_{i∈N} of universally continuous elements in B(S). Since S is a standard H -cone it has a generator p by [4, Lemma 4.3.7]. Moreover B(p) is a weak unit (even a generator) in B(S). Indeed, on account of Theorem 2.2 we have

B(x) = ⋁_{n∈N} B(np)⌣\x = ⋁_{n∈N}(B(np)⌣\B(x)).

By [4, Proposition 4.1.2] for every s_{n} there exists α_{n} ∈ R such that s ≤ α_{n}B(p) ≤ α_{n}p. Set

u = ∑_{n∈N} \frac{s_{n}}{2^{n}α_{n}B(p)}.

Elements s_{n} are p-continuous in S. Indeed, let F be directed upwards with ∨F = s_{n} = B(s_{n}) and ε > 0. Since B is left order continuous we have ⋁_{f∈F} B(f) = s_{n}. As s_{n} is B(p)-continuous we obtain

s_{n} ≤ B(f_{ε}) + εB(p) ≤ f_{ε} + εp

for some f_{ε} ∈ F. Hence s_{n} is p-continuous. Applying [4, Proposition 4.1.2] we easily see that u is p-continuous. Moreover, u is clearly a generator in B(S). Using Lemma 2.1 we find that

B(x) = ⋁_{n∈N}(nu)⌣\B(x) = ⋁_{n∈N}(nu)⌣\x.

Thus the balayage B is generated by a p-continuous element.

Let S be an H -cone possessing a generator p. If an element u in S generates a balayage B, then Proposition 2.3 implies that B(p) is a p-quasi-unit. Combining this observation with Proposition 2.5 we obtain the result.

Theorem 2.9. Let S be an H -cone possessing a generator p. Then every balayage is generated by some p-quasi-unit. Conversely, a p-quasi-unit u generates a balayage if and only if

u = ⋁_{{n∈N \atop f∈F}}(nu)⌣\f

for any upward directed family F with u = ∨F.

In some important cases there is a one to one correspondence between pquasi-units and balayages. In order to find sufficient conditions we first state two preliminary results.

Proposition 2.10. Let S be an H -cone in an ordered vector space E and f ∈ E. Then the mapping B_{f}: S → S defined by

B_{f}(x) = ⋁_{n∈N} R(x ∧ (nf))

is a balayage and B_{f} = B_{f^{+}} . Moreover, B_{f}(R(f)) = R(f).

Proof. Let x ∈ S and f ∈ E. Since by [4, Proposition 2.1.1] the set E is a vector lattice we infer

R(x ∧ (nf)) = R ((x ∧ (nf)) ∨ 0) = R(x ∧ (nf^{+})).

Applying [4, Theorem 2.2.9] the mapping B_{f} = B_{f^{+}} is a balayage. The second statement follows from B_{f}(R(f)) ∈ S and R(f) ≥ B_{f}(R(f)) ≥ f.

The following result is a stronger form of the result stated by Boboc [5, Lemma 3, p. 74].

Proposition 2.11. Let S be an H -cone and p ∈ S. Then the following assertions are equivalent:

(i) s is a p-quasi-unit; (ii) There exists a decreasing sequence of balayages (B_{n})_{n∈N} such that

s = B_{n}(s) for all n ∈ N and s = ⋀_{n∈N}(B_{n}(p)).

(iii) There exists a sequence of balayages (B_{n})_{n∈N} such that

s = B_{n}(s) for all n ∈ N and s = ⋀_{n∈N}(B_{n}(p)).

Proof. Assume that s is a p-quasi-unit. Let α < 1 and set f_{α} = s − αp. Define a mapping B_{α}: S → S by

B_{α}(x) = ⋁_{m∈N} R(x ∧ (mf_{α})).

Then by Proposition 2.10 the mapping B_{α} is a balayage and

B_{α}(R(s − αp)) = R(s − αp).

Applying Theorem 1.4 we obtain R(s − αp) = (1 − α)s and so

B_{α}(s) = \frac{1}{1 − α}B_{α}(R(s − αp)) = s.

We show that

s + (1 − α)p ≥ B_{α}(p).

According to [4, Lemma 2.2.8] it is enough to prove that

(p − (s + (1 − α)p)) ∧ (s − αp) ≤ 0.

But this is evident, since E is a vector space and therefore

(2αp) ∧ (2s) = s ∧ αp + s ∧ αp ≤ s + αp.

Setting B_{n} = B_{1−1/n} for n ∈ N we obtain (ii). The condition (ii) implies trivially (iii). Assume that the condition (iii) holds. Then by Lemma 2.1 we have

s ≤ (2s)⌣\p = B_{n}(2s)⌣\p = B_{n}(2s)⌣\B_{n}(p) ≤ B_{n}(p)

for all n ∈ N. Using the condition (iii) we infer that (2s)⌣\p = s and so by Theorem 1.4 the element s is a p-quasi-unit.

Lemma 2.12. Let S be an H -cone and v be u-continuous for some element u in S. If φ: S → S is additive, increasing and contractive then the mapping φ̃ defined by

φ̃(s) = ⋁_{n∈N} φ((nv)⌣\s)

is additive and left order continuous.

Proof. Similarly as in the proof of [9, Proposition 2.7] we deduce that φ̃ is additive. Assume that F is directed upwards and ∨F = v. Since v is ucontinuous for some element u there exists f_{ε} for any ε > 0 such that v ≤ f_{ε}+εu. Hence we have v = (nv)⌣\v ≤ (nv)⌣\f_{ε} + εu and therefore φ̃(v) = ∨_{f∈F} φ̃(f). Let s ∈ S and F ⊂ S be directed upwards with s = ∨F. Assume first that s ≼ v. As the set v − s + F is directed upwards towards v, we have

φ̃(s) + φ̃(v − s) = φ̃(v) = ⋁_{f∈F}(φ̃(f)) + φ̃(v − s).

Hence φ̃ is left order continuous for all s ∈ S such that s ≼ nv for some n ∈ N. Assume next that s is an arbitrary element in S. Since φ̃ is left order continuous at (nv)⌣\s for all n ∈ N we infer

φ((nv)⌣\s) = φ̃((nv)⌣\s) = ⋁_{f∈F} φ̃(((nv)⌣\s) ∧ f) ≤ ⋁_{f∈F} φ̃(f) ≤ φ̃(s)

for all n ∈ N. Hence φ̃ is left order continuous.

Theorem 2.13. Let S be an H -cone and v be u-continuous for some u ∈ S. Assume that the greatest lower bound in the set of left order continuous additive mappings for any decreasing sequence (B_{n})_{n∈N} of balayages in S is a balayage. Then for any v -quasi-unit s there exists a balayage B: S → S such that B(v) = s. Conversely, for any balayage B: S → S the element B(v) is a v -quasi-unit.

Moreover, if an H -cone S possesses a generator v which is u-continuous for some weak unit u in S there exists a one to one correspondence between balayages and v -quasi-units.

Proof. Assume that v is u-continuous for some u ∈ S. Let B: S → S be a balayage. Then B(v) is a v -quasi-unit by Proposition 2.3. Let s be a v -quasi-unit. Because of Proposition 2.11 there exists a decreasing sequence of balayages B_{n} such that s = ⋀_{n∈N}(B_{n}(v)). Since the sequence (B_{n}) is decreasing, the mapping φ: S → S defined by φ(t) = ⋀_{n∈N}(B_{n}(t)) for t ∈ S is additive, increasing and contractive. Using Lemma 2.12 we find out that φ̃ is left order continuous, additive and φ̃(v) = φ(v) = s. By the assumption the greatest lower bound ⋀ B_{n} in the set of left order continuous additive mappings is a balayage denoted by B. Hence B(v) = φ̃(v) = s. Taking into account Theorem 2.2, this correspondence is one to one if v is u-continuous (for some weak unit u) and a generator.

The condition of the preceding theorem is equivalent with the axiom of polarity in standard H -cones by [7, p. 188].

Theorem 2.14. Let S be an H -cone possessing a generator p. Let B: S → S be a balayage. Then there exists a lower directed family of functions f_{n} ∈ S −S such that B = ∧_{n∈N}B_{f_{n}}.

Proof. Let p be a generator of an H -cone S. Since by Proposition 2.3 the element B(p) is a p-quasi-unit there exists a decreasing sequence of balayages (B_{f_{n}})_{n∈N} such that B(p) = ⋀_{n∈N}(B_{f_{n}}(p)). Applying [9, Corollary 2.9] to the inequality B(p) ≤ B_{f_{n}}(p) we infer that B ≤ B_{f_{n}} for all n ∈ N. If s + s′ = np for some n ∈ N and s, s′ ∈ S then

B(s) + B(s′) = B(np) = ⋀_{n∈N} B_{f_{n}}(s + s′) = ⋀_{n∈N} B_{f_{n}}(s) + ⋀_{n∈N} B_{f_{n}}(s′).

Since B(t) ≤ ⋀_{n∈N} B_{f_{n}}(t) for all t ∈ S we have B(s) = ⋀_{n∈N} B_{f_{n}}(s) for all s ≼ np for some n ∈ N. Assume now that ψ: S → S is left order continuous, additive and ψ ≤ B_{f_{n}} for all n ∈ N. Then we obtain

ψ((mp)⌣\x) ≤ ⋀_{n∈N} B_{f_{n}}((mp)⌣\x) = B((mp)⌣\x)

for all m ∈ N. Hence we conclude

ψ(x) = ⋁_{m∈N} ψ((mp)⌣\x) ≤ ⋁_{m∈N} B((mp)⌣\x) = B(x),

establishing the assertion.

Lemma 2.15. Let S be an H -cone possessing a generator p and elements u_{1} and u_{2} in S each generate a balayage. Then the element u_{1} + u_{2} generates a balayage, and this balayage is also generated by (u_{1} + u_{2})⌣\p.

Proof. Assume that u_{1} ∈ S and u_{2} ∈ S generate a balayage. Set z = u_{1} + u_{2} and let F be an arbitrary upward directed set with ⋁ F = z. Proposition 2.5 states that it is enough to show that

(2.6)  z = ⋁{ (nz)⌣\f | f ∈ F, n ∈ N }.

Applying [4, Proposition 2.2.3] there exists upward directed families (g_{1f})_{f∈F} and (g_{2f})_{f∈F} such that f ≥ g_{1f} + g_{2f} for all f ∈ F and ∨_{f∈F} g_{if} = u_{i} for i = 1, 2. Hence we have

(nu_{1})⌣\g_{1f} + (nu_{2})⌣\g_{2f} ≤ (n(u_{1} + u_{2}))⌣\f

for all f ∈ F. Since u_{1} and u_{2} generate a balayage and ∨_{f∈F} g_{if} = u_{i} for i = 1, 2, we obtain

z = u_{1} + u_{2} ≤ ⋁_{n∈N}(n(u_{1} + u_{2}))⌣\f ≤ ⋁ F = z.

Thus the element z = u_{1} + u_{2} generates a balayage denoted by B.

We still have to prove that z⌣\p generates also B. Let x ∈ S be arbitrary. From (mp)⌣\x ≼ mp ≼ np for all m, n ∈ N with m ≤ n it follows by [2, Theorem 3.2] that

(nz)⌣\(np) ≽ (nz)⌣\((mp)⌣\x).

Hence we have

\begin{aligned}B(x) & ≥ ⋁_{{f∈F \atop n∈N}}((nz)⌣\(np))⌣\x ≥ ⋁_{{f∈F \atop m,n∈N}}(nz)⌣\((mp)⌣\x) \\ & = ⋁_{m∈N} B((mp)⌣\x) = B(x). \end{aligned}

Consequently the elements u_{1}+u_{2} and (u_{1}+u_{2})⌣\p generate the same balayage B.

Theorem 2.16. Let S be an H -cone with a generator p. Denote by ℬ the set of balayages from S into S. Then ℬ is an abelian semigroup with respect to the truncated addition defined by

(B_{1} ⊕ B_{2})(x) = ⋁_{n∈N}(n(B_{1}(p) + B_{2}(p)))⌣\x

for all x ∈ S. Moreover the equality (B_{1} ⊕ B_{2})(x) = (B_{1} ∨ B_{2})(x) holds for all x ∈ S.

Elements generating balayages 409

Proof. Let p be a generator of an H -cone S. Assume that B_{1}: S → S and B_{2}: S → S are balayages. By virtue of Lemma 2.15 the truncation addition is well-defined and B_{1} ⊕ B_{2} is a balayage generated by (B_{1}(p) + B_{2}(p))⌣\p.

Applying [2, Theorem 11.7] we note that (B_{1}(p)+B_{2}(p))⌣\p is a p-quasi-unit and so (B_{1} ⊕ B_{2})(p) = (B_{1}(p) + B_{2}(p))⌣\p. We show that

(B_{1} ⊕ B_{2})(p) = (B_{1} ∨ B_{2})(p).

Assume that w ≼ B_{1}(p) + B_{2}(p) and w ≤ p. Applying [4, Theorem 2.1.5] we find elements t_{1}, t_{2} such that w = t_{1} + t_{2} , t_{1} ≼ B_{1}(p) and t_{2} ≼ B_{2}(p). Since B(S) is specifically solid by [9, Lemma 2.3], we have B_{1}(t_{1}) = t_{1} and B_{2}(t_{2}) = t_{2} . Hence we obtain B_{1}∨B_{2}(t_{1}) = t_{1} and B_{1}∨B_{2}(t_{2}) = t_{2} . Reviewing to [7, Proposition 2.1] we infer

\begin{aligned}w & = B_{1}(t_{1}) + B_{2}(t_{2}) = B_{1} ∨ B_{2}(t_{1}) + B_{1} ∨ B_{2}(t_{2}) \\ & = B_{1} ∨ B_{2}(t_{1} + t_{2}) = B_{1} ∨ B_{2}(w). \end{aligned}

Setting w = (B_{1}(p) + B_{2}(p))⌣\p we see that (B_{1}(p) + B_{2}(p))⌣\p ≤ B_{1} ∨ B_{2}(p). On the other hand by [4, Corollary 2.1.3] we have

B_{1} ∨ B_{2}(p) = B_{1}(p) ∨ B_{2}(p) ≼ B_{1}(p) + B_{2}(p).

Combining this with the inequality B_{1} ∨ B_{2}(p) ≤ p we note that

(B_{1}(p) + B_{2}(p))⌣\p ≥ B_{1} ∨ B_{2}(p).

Now it is clear that ⊕ is associative.

References

[1] Arsove, M., and H. Leutwiler: Infinitesimal generators and quasi-units in potential theory. - Proc. Nat. Acad. Sci. U.S.A. 72, 1975, 2498–2500.

[2] Arsove, M., and H. Leutwiler: Algebraic potential theory. - Mem. Amer. Math. Soc. 23:226, 1980.

[3] Arsove, M., and H. Leutwiler: Quasi-units in mixed lattice structures. - Conference in Potential Theory, Copenhagen 1979. Lecture Notes in Mathematics 787, Springer-Verlag, Berlin–Heidelberg–New York, 1980, 35–54.

[4] Boboc, N., Gh. Bucur, and A. Cornea: Order and convexity in potential theory: Hcones. - Lecture Notes in Mathematics 853, Springer-Verlag, Berlin–Heidelberg–New York, 1981.

[5] Boboc, N., Gh. Bucur, and A. Cornea: Autodual H -cones. - Conference in Potential Theory, Copenhagen 1979. Lecture Notes in Mathematics 787, Springer-Verlag, Berlin–Heidelberg–New York, 1980, 64–77.

[6] Constantinescu, C., and A. Cornea: Potential theory on harmonic spaces. - Springer-Verlag, Berlin–Heidelberg–New York, 1972.

[7] Cornea, A., and H. Höllain: Morphisms and contractions on standard H -cones. - Hokkaido Mathematical Journal Special Issue 10, 1981, 157–212.

410 Sirkka-Liisa Eriksson-Bique

[8] Eriksson-Bique, S.-L.: Real-valued duals of H -cones. - Math. Scand. 71, 1992, 243–251.

[9] Eriksson-Bique, S.-L.: Characterizations of balayages. - Ann. Acad. Sci. Fenn. Ser. A I Math. 19, 1994, 59–66.

[10] Popa, E.: Balayages and morphisms in H -cones. - An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 36, 1990, 203–213.

Received 14 May 1993