Reference number TRAN2915 July 8, 2001

A semigroup of operators in convexity theory

Christer O. Kiselman

Contents:

1. Introduction

2. Closure operators and Galois correspondences

3. An abstract semigroup of order eighteen

4. The semigroup generated by the three closure operators

5. The multiplicative structure of the semigroup

6. The order structure of the semigroup

7. Comparison between the supremum and the product of two elements References

Resumo: Duongrupo de operatoroj en la teorio pri konvekseco

Ni konsideras tri operatorojn kiuj aperas nature en la teorio pri konvekseco kaj plene determinas la strukturon de la duongrupo generita de ili.

Abstract: We consider three operators which appear naturally in convexity theory and determine completely the structure of the semigroup generated by them.

1. Introduction

The simplest convex functions on a vector space are the affine functions, and it is of interest to represent a convex function as a supremum of these simple functions. Such a representation is possible only if the function has three properties: it must be convex; it must be lower semicontinuous with respect to an appropriate topology; and it cannot take the value minus infinity unless it is identically minus infinity. Conversely, using the Fenchel transformation one can prove that these three properties are also sufficient for such a representation to hold; indeed, the function is then equal to its second Fenchel transform, or second conjugate function.

There are thus three operations naturally associated with the Fenchel transformation: that of taking the largest convex minorant; that of taking the largest lower semicontinuous minorant; and that of taking the constant minus infinity if the function attains that value and leaving it unchanged otherwise. These three operations generate a semigroup in the semigroup of all operators on the set of functions on the vector space in question. How many elements are there in this semigroup? What structure does it have with respect to composition and with respect to the natural order? Is it a lattice under this order? Is there a representation of the semigroup as a semigroup of matrices? The purpose of this note is to provide answers to these questions. A quick look at Figure 1 will reveal the order structure of the semigroup.

We now define the three operations more precisely. Let E be a real vector space. To any function f on E with values in the extended real line R ∪ {−∞, +∞} = [−∞, +∞] we associate its convex hull or largest convex minorant c(f), defined as the supremum of all its convex minorants. It can be shown that

(1.1)  c(f)(x) = inf ( ∑_{1}^{N} λ_{j}f(x_{j}); N ⩾ 1, ∑_{1}^{N} λ_{j}x_{j} = x),  x ∈ E,

where the infimum is taken over all representations of x as a barycenter of finitely many points x_{j} with f(x_{j}) < +∞, j = 1, ..., N, N ⩾ 1, and taking positive weights λ_{j} with sum equal to 1.

The second operator is that of taking the largest lower semicontinuous minorant of the function, thus the supremum of all its lower semicontinuous minorants for some topology τ on E, also given by the formula

(1.2)  l(f)(x) = lim inf_{y→x} f(y),  x ∈ E.

It is not difficult to see that this operation corresponds to taking the closure of the epigraph of the function, i.e., epi(l(f)) = \overline{epi f}, where the closure is taken with respect to the Cartesian product of τ and the usual topology on R.

In Fenchel duality we consider a vector space F of linear forms on E, thus a subspace of the algebraic dual E^{∗} of E. The topologies that are of interest are σ(E, F), the weakest topology on E such that all linear forms in F are continuous, and σ(F, E), the weakest topology on F such that all evaluation mappings F ∋ ξ ↦ ξ(x), x ∈ E, are continuous. We can for instance choose F = E^{∗}, or F = E′, the topological dual of E under a given topology. We do not require that E and F be in duality; it is even allowed to take F = {0}.

The third operator is the operator m defined as

(1.3)  m(f)(x) = \{ \begin{matrix}f(x), & if f is everywhere > −∞; \\ −∞, & if f assumes the value − ∞. \end{matrix}

The three operators c, l, m generate a semigroup G(E) with composition as multiplication.

The plan of the paper is as follows. First we study operations on functions defined on vector spaces (Section 2). Next we define an abstract semigroup 𝒢 of eighteen elements generated by the identity and three elements c, l, m subject to the relations c^{2} = c, l^{2} = l, m^{2} = m, and clc = lcl = lc, cmc = mcm = mc, lml = mlm = ml (Theorem 3.2). We find a matrix representation of this semigroup using 3 × 3 matrices (Theorem 3.3). For every vector space E, the semigroup G(E) is then a homomorphic image of 𝒢, and we determine this homomorphic image for spaces of dimension zero, one, two and higher (Theorem 4.1).

The order of G(E) can be 1, 6, 15, 16, 17, or 18, depending on the dimension of E and its topology. The order is 15 for E = R, 16 for E = R^{n}, n ⩾ 2, and 18 for every normed space E of infinite dimension.

I am grateful to Sten Kaijser for helpful comments concerning matrix representation of semigroups, in particular for bringing the paper by Hewitt and Zuckerman [1955] to my attention. Svante Janson carefully read an early draft of the paper and saved me from some embarrassing errors; I am very grateful to him.

2. Closure operators and Galois correspondences

An order relation in a set X is a relation (a subset of X × X) which satisfies three conditions: it is reflexive, antisymmetric and transitive. This means, if we denote the relation by ⩽, that for all x, y, z ∈ X,

(2.1)  x ⩽ x;
(2.2)  x ⩽ y and y ⩽ x implies x = y;
(2.3)  x ⩽ y and y ⩽ z implies x ⩽ z.

An ordered set is a set X together with an order relation. (Sometimes one says partially ordered set.)

A basic example is the set P (W) of all subsets of a set W, with the order relation given by inclusion, thus A ⩽ B being defined as A ⊂ B for A, B ∈ P (W).

A closure operator in an ordered set X is a mapping X ∋ x ↦ \overline{x} ∈ X which is expanding (or extensive), increasing (or isotone, order preserving), and idempotent; in other words, which satisfies the following three conditions for all x, y ∈ X:

(2.4)  x ⩽ \overline{x};
(2.5)  x ⩽ y implies \overline{x} ⩽ \overline{y};
(2.6)  \overline{\overline{x}} = \overline{x}.

The element \overline{x} is said to be the closure of x. Elements x such that \overline{x} = x are called closed (for this operator). An element is closed if and only if it is the closure of some element (and then it is the closure of itself).

A basic example of a closure operator is of course the topological closure operator which associates to a set in a topological space its topological closure, i.e., the smallest closed set containing the given set.

Another closure operator of great importance is the operator which associates to a set in R^{n} its convex hull, the smallest convex set containing the given set.

A Galois correspondence is a pair (f, g) of two decreasing mappings f: X → Y, g: Y → X of two given ordered sets X, Y such that g ∘ f and f ∘ g are expanding. In other words we have f(x) ⩾ f(x′) and g(y) ⩾ g(y′) if x ⩽ x′ and y ⩽ y′, and g(f(x)) ⩾ x and f(g(y)) ⩾ y for all x ∈ X, y ∈ Y.

The notion of Galois correspondence originates in Galois theory, which deals with the correspondence between all intermediate subfields of a field extension and the set of subgroups of the Galois group of the extension. The notion was introduced in the general setting of ordered sets by Oystein Ore [1944: 495] under the name of Galois correspondence or Galois connexion. (Sometimes he calls each of the mappings f and g a correspondence, the two of them together a connexion.) See also Everett [1944: 516], Birkhoff [1948: 56, 57], and Kuroš [1962: 6: 11].

Proposition 2.1 (Ore [1944: 496]). Let f: X → Y, g: Y → X be a Galois correspondence. Then g ∘ f: X → X and f ∘ g: Y → Y are closure operators. Moreover, f ∘ g ∘ f = f and g ∘ f ∘ g = g.

Proof. That g ∘ f and f ∘ g are expanding is part of the definition of a Galois correspondence; that they are increasing follows from the fact that they are compositions of two decreasing mappings. We know that f ∘g is expanding, so (f ∘g)(f(x)) ⩾ f(x); thus f ∘ g ∘ f ⩾ f. On the other hand, also g ∘ f is expanding, i.e., g ∘ f ⩾ id_{X}, so f ∘ g ∘ f ⩽ f ∘ id_{X} = f, hence f ∘ g ∘ f = f. By symmetry, g ∘ f ∘ g = g. From either one of these identities we easily obtain that g ∘ f and f ∘ g are idempotent.

It is now natural to ask whether the closure operators one obtains from Galois correspondences have some special property. The answer is no: every closure operator comes in a trivial way from some Galois correspondence.

Proposition 2.2. Let x ↦ \overline{x} be a closure operator defined in an ordered set X. Then there exist an ordered set Y and a Galois correspondence f: X → Y, g: Y → X such that \overline{x} = g(f(x)) for all x ∈ X.

Proof. We define Y as the set of all closed elements in X with the opposite order; thus y ⩽_{Y} y′ shall mean that y ⩾_{X} y′. Let f: X → Y and g: Y → X be defined by f(x) = \overline{x} and g(y) = y. Then both f and g are decreasing, and g ∘ f(x) = \overline{x} ⩾_{X} x, f ∘ g(y) = y ⩾_{Y} y. So g ∘ f and f ∘ g are expanding, and \overline{x} = g(f(x)) as desired.

Proposition 2.2 is, in a sense, completely uninteresting. This is because the Galois correspondence is obtained from X and the closure operator in a totally trivial way. However, there are many Galois correspondences in mathematics that are highly interesting and represent a given closure operator. This is so because they allow for important calculations to be made or for new insights into the theory.

We now ask whether the composition of two closure operators is a closure operator. It is for instance well-known that if we first take the topological closure of a set and then its convex hull, we get an operator which is not idempotent, thus not a closure operator.

Proposition 2.3. Let f, g: X → X be two closure operators. The following properties are equivalent:

(a) g ∘ f is a closure operator;

(b) g ∘ f is idempotent;

(c) g ∘ f ∘ g = g ∘ f;

(d) f ∘ g ∘ f = g ∘ f;

(e) g(x) is f-closed if x is f-closed. If one of these conditions is satisfied, then g ∘ f is the supremum of the two closure operators f and g in the ordered set of all closure operators; moreover f ∘ g ⩽ g ∘ f.

Proof. That h = g ∘ f is expanding and increasing is true for any composition of expanding and increasing mappings, so it is clear that (a) and (b) are equivalent. It is also easy to see that (d) and (e) are equivalent. If (c) holds, then h∘h = g ∘f ∘g ∘f = g ∘ f ∘ f = g ∘ f = h, so h is idempotent. Similarly, (d) implies (b). Conversely, if (b) holds, then

g ∘ f ⩽ g ∘ f ∘ g ⩽ h ∘ h = h = g ∘ f and

g ∘ f ⩽ f ∘ g ∘ f ⩽ h ∘ h = h = g ∘ f,

so we must have equality all the way in both chains of inequalities, which proves that (c) and (d) hold. The last statement is easy to verify.

Corollary 2.4. Two closure operators f and g commute if and only if both g ∘f and f ∘ g are closure operators.

Proof. If g ∘ f = f ∘ g, then (c) obviously holds, so g ∘ f is a closure operator. Conversely, if g ∘ f is a closure operator, then (c) applied to f and g says that g ∘ f ∘ g = g ∘ f; if also f ∘ g is a closure operator, then (d) applied to g and f says that g ∘ f ∘ g = f ∘ g. Thus f and g commute.

When two closure operators f and g are given, it may happen that f ⩽ g. Then the semigroup generated by f and g consists of at most three elements: id_{X}, f, g. If both g ∘ f and f ∘ g are closure operators, then the semigroup generated by f and g has at most four elements: id_{X}, f, g, and g ∘ f = f ∘ g. If precisely one of g ∘ f and f ∘ g is a closure operator, then the semigroup generated has exactly five elements, id_{X}, f, g, g ∘ f, and f ∘ g, of which four are closure operators. When none of g ∘ f and f ∘ g is a closure operator, the semigroup of all compositions f_{m} ∘ ⋯ ∘ f_{1}, with f_{j} = f or f_{j} = g, m ∈ N, may be finite or infinite.

Applying Proposition 2.3 to the case of the two operators f(A) = cvx A and g(A) = \overline{A}, we see that the operation of taking the topological closure of the convex hull, A ↦ \overline{cvx A}, is a closure operator. Indeed, it can be proved that the closure (with respect to a reasonable topology; see Proposition 2.5 below) of a convex set is always convex, so condition (e) is satisfied. (On the other hand the convex hull of a closed set is not always closed.) We call \overline{cvx A} the closed convex hull of A. This is a case where the semigroup generated by f and g consists of five elements.

Next we consider three closure operators f, g, h such that one of f ∘ g and g ∘ f is a closure operator, and similarly for the other pairs {g, h}, {h, f}. Up to renaming the operators there are then only two cases. Either g ∘ f, h ∘ g and h ∘ f are closure operators (alphabetical order), or g ∘ f, h ∘ g and f ∘ h are closure operators (cyclic order). It turns out that the operators defined by (1.1)–(1.3) belong to the first case. More precisely we have:

Proposition 2.5. Let c, l, m denote the three operators defined by (1.1)–(1.3) on a vector space E equipped with a topology τ. They are all closure operators, in particular they are idempotent:

(2.7)  c ∘ c = c,  l ∘ l = l,  m ∘ m = m.

Moreover, they satisfy

(2.8)  c ∘ m ∘ c = m ∘ c ∘ m = m ∘ c,  l ∘ m ∘ l = m ∘ l ∘ m = m ∘ l,

which means that the m-closure of a convex function is convex and that the m-closure of a lower semicontinuous function is lower semicontinuous. If the topology τ is such that all translations x ↦ x − a, a ∈ E, and all dilations x ↦ λx, λ ∈ R, are continuous, then we also have

(2.9)  c ∘ l ∘ c = l ∘ c ∘ l = l ∘ c,

which means that the largest lower semicontinuous minorant of a convex function is convex.

Proof. All statement except perhaps the last have routine proofs. Let f be a convex function on E and define g = l(f). To prove that g is convex, we fix x_{j} ∈ E such that g(x_{j}) < +∞, j = 0, 1, and a number t ∈ ]0, 1[, and shall then prove that g((1 − t)x_{0} + tx_{1}) ⩽ (1 − t)g(x_{0}) + tg(x_{1}). Given any numbers A_{j} > g(x_{j}) and any neighborhood U of x = (1 − t)x_{0} + tx_{1}, we can find a neighborhood U_{0} of x_{0} such that (1 − t)U_{0} + tx_{1} ⊂ U. By the definition of g = l(f), there exists a point y_{0} ∈ U_{0} such that f(y_{0}) < A_{0}. Next we can find a neighborhood U_{1} of x_{1} such that (1 − t)y_{0} + tU_{1} ⊂ U. By the definition of g = l(f) again, there exists a point y_{1} ∈ U_{1} such that f(y_{1}) < A_{1}. Since f is assumed to be convex, we know that f((1 − t)y_{0} + ty_{1}) ⩽ (1 − t)f(y_{0}) + tf(y_{1}) < (1 − t)A_{0} + tA_{1}. Hence inf_{y∈U} f(y) < (1 − t)A_{0} + tA_{1}. Since this is true for all choices of the A_{j}, we see that inf_{y∈U} f(y) ⩽ (1 − t)g(x_{0}) + tg(x_{1}). Finally, since this is true for any neighborhood U of x, we get

g(x) = l(f)(x) = lim inf_{y→x} f(y) = sup_{U} inf_{y∈U} f(y) ⩽ (1 − t)g(x_{0}) + tg(x_{1}).

This shows that l(f) is convex.

It follows from (2.8) and (2.9) that m ∘ c, m ∘ l, and l ∘ c are closure operators; cf. Proposition 2.3. We shall see later that in general c, l and m do not commute (Theorem 4.1).

We form all possible compositions of these operators, f_{k} ∘ ⋯ ∘ f_{2} ∘ f_{1}, where k ⩾ 0 and the f_{j} = c, l, or m. Thus we get a semigroup G(E) in the set of all operators on [−∞, +∞]^{E}. We shall now investigate its structure. We shall do so by defining first an abstract semigroup 𝒢, of which each G(E) is a homomorphic image.

3. An abstract semigroup of order eighteen

Definition 3.1. We let 𝒢 denote the semigroup generated by the identity id = 1 and three other elements c, l, m which are subject to the relations

(3.1)  c^{2} = c,  l^{2} = l,  m^{2} = m;

(3.2)  clc = lcl = lc,  cmc = mcm = mc,  lml = mlm = ml.

Theorem 3.2. The semigroup 𝒢 defined by Definition 3.1 has eighteen elements, which can be represented by products of the generators c, l and m as follows.

(0) One element using zero factors: id = 1;

(i) Three elements using one factor: c, l, m; (ii) Six element using two factors: cl, cm, lc, lm, mc, ml; (iii) Six elements using three factors: clm, cml, lcm, lmc, mcl, mlc = 0; (iv) Two elements using four factors: lcml, lmcl. Of these, eight are idempotent, viz. id = 1, c, l, m, lc, mc, ml, mlc = 0, while the other ten, viz. cl, cm, lm, clm, lcm, cml, lcml, lmc, mcl, lmcl, are not.

Proof. After the elements representable by zero, one or two factors we form all threeletter combinations. Since it does not pay to repeat a letter immediately in view of (3.1), there are only 3 · 2 · 2 = 12 combinations that could possibly represent a new element:

clc,  clm,  cmc,  cml,
lcl,  lcm,  lmc,  lml,
mcl,  mcm,  mlc,  mlm.

Of these twelve combinations, six are equal to elements representable by two factors:

clc = lcl = lc; cmc = mcm = mc; lml = mlm = ml.

The remaining six are listed under (iii).

So far we have found sixteen elements, not yet known to be distinct. Using (3.1) and (3.2), we can prove by induction on the length of a that cac = ac and mam = ma for all a ∈ 𝒢, thus that there is no use in repeating c or m. If we multiply the six elements listed under (iii) by l from the left, we get lclm = lcm, lcml, llcm = lcm, llmc = lmc, lmcl, lmlc = mlc. So possibly lcml and lmcl represent new elements. Five or more letters will not give anything new.

Thus we see that 𝒢 has at most the eighteen elements listed in (0)–(iv). It remains to be proved that these elements are in fact all different. This will follow from the next theorem, where we represent 𝒢 as a semigroup of matrices with exactly eighteen elements.

Any finite semigroup with a neutral element admits a faithful representation as a semigroup of n × n matrices, where n is the order of the semigroup; see Hewitt and Zuckerman [1955, Theorem 3.21]. Thus 𝒢 admits, by this general theorem, a representation in terms of 18 × 18 matrices. However, we shall see that we can find a representation in terms of 3 × 3 matrices.

Theorem 3.3. Define

I = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , C = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{matrix} ) , L = ( \begin{matrix}1 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , M = ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

Then C, L and M satisfy the relations corresponding to (3.1) and (3.2). They generate a semigroup ℋ of matrices with eighteen elements; in particular the products corresponding to (0)–(iv) are all different.

Proof. It is easy to prove the relations corresponding to (3.1) and (3.2); it follows that ℋ is a homomorphic image of 𝒢.

We may calculate all elements of ℋ. In addition to I, C, L, M they are:

CL = ( \begin{matrix}1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) , CM = ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{matrix} ) , LM = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) ,

CLM = ( \begin{matrix}0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) ,

LC = ( \begin{matrix}1 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , MC = ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{matrix} ) , ML = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) ,

LCM = ( \begin{matrix}0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , LCML = ( \begin{matrix}0 & 0 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , CML = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) ,

LMC = ( \begin{matrix}0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , LMCL = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) , MCL = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) ,

MLC = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

Since they are all different, it follows that ℋ has eighteen elements. Therefore ℋ is isomorphic to 𝒢.

One forms the abelization of a group by taking the quotient over the subgroup generated by all commutators g^{−1}h^{−1}gh. For a semigroup this procedure is not available. Instead we may define the abelization of a semigroup by adding all commutative laws to the defining relations of the semigroup. In this case we obtain a commutative semigroup 𝒢_{0} of order eight: it has three idempotent generators c, l, m satisfying the relations cl = lc, cm = mc and lm = ml, and consists of the elements 1, c, l, m, cl, cm, lm, clm = 0.

It is easy to represent 𝒢_{0} by matrices: we define

C_{0} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} ) , L_{0} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , M_{0} = M = ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

These matrices define orthogonal projections of R^{3} onto the two-dimensional coordinate planes x_{3} = 0, x_{2} = 0, and x_{1} = 0, respectively; C and L map R^{3} obliquely onto the same planes. Obviously C and L are perturbations of C_{0} and L_{0}, and the semigroup 𝒢 can be viewed as a perturbation of its abelization 𝒢_{0}.

It is not possible to use 2 × 2 matrices in the representation in Theorem 3.3. This is because for 2 × 2 real or complex matrices A and B such that A and BA are idempotent, either AB is idempotent or BA = 0. Therefore, with 2 × 2 matrices C, L, and M such that C, MC, and LC are idempotent whereas CM and CL are not, we must have MC = LC = 0: the representation cannot be faithful.

4. The semigroup generated by the three closure operators

As stated in Proposition 2.5, the three operators c, l, m defined on the set [−∞, +∞]^{E} of all functions on a topological vector space E are idempotent, i.e., they satisfy (2.7). They are closure operators, since the other requirements for being a closure operator are obviously satisfied. Moreover, they satisfy the relations (2.8) and (2.9). The semigroup G(E) they generate consists of all compositions f_{k} ∘ ⋯ ∘ f_{2} ∘ f_{1}, where k ⩾ 0 and f_{j} = c, l, m for j = 1, ..., k. This semigroup is a homomorphic image of the abstract semigroup 𝒢 constructed in Section 3. We shall now determine its structure for all finite-dimensional topological vector spaces as well as for all normed spaces.

If E is a topological vector space of finite dimension, then \overline{\{0\}}, the closure of the origin, is a vector subspace and the quotient E/\overline{\{0\}} is separated (Hausdorff). The space is isomorphic to \overline{\{0\}} × (E/\overline{\{0\}}) ≅ (R^{k})_{χ} × R^{n−k}, where the index χ indicates that we equip R^{k} with the chaotic topology, i.e., the topology such that the only neighborhood of the origin is the whole space. The dimensions of E and \overline{\{0\}} determine the semigroup G(E); equivalently, the dimension of the algebraic dual E^{∗} and that of the topological dual E′ determine the semigroup, since dim E′ is equal to the codimension of \overline{\{0\}} in the finite-dimensional case.

Theorem 4.1. On any real topological vector space E the three closure operators c, l, m defined by (1.1)–(1.3) generate a semigroup G(E) of at most eighteen elements. They can be represented as compositions of c, l, m as follows (for simplicity we write composition as juxtaposition):

(0) One element with zero factors: id = 1;

(i) Three elements with one factor: c, l, m; (ii) Six element with two factors: cl, cm, lc, lm, mc, ml; (iii) Six elements with three factors: clm, cml, lcm, lmc, mcl, mlc = 0; (iv) Two elements with four factors: lcml, lmcl. Of these, the elements represented by id = 1, c, l, m, lc, mc, ml, mlc = 0 (not necessarily distinct) are closure operators. (A_{1}) If E has dimension zero, then G(E) consists of only one element. (A_{15}) If E = R with the usual topology, then G(E) has fifteen elements; we have

lmc = lmcl = mcl = 0 and all other elements are different from 0 and each other.

(A_{16}) If E = R^{n}, n ⩾ 2, with the usual topology, then G(E) has sixteen elements; we

have lmc = lmcl = 0 and all other elements are different from 0 and each other. (B_{6}) If E is nonzero and equipped with the chaotic topology, then G(E) has six ele-

ments, 1, c, m, cm, mc, and l, the last being equal to all other products. (B_{16}) If E is of finite dimension and \overline{\{0\}} has codimension one and dimension at least

one, then G(E) has sixteen elements; we have mcl = lmcl = 0 and all other elements are different from 0 and each other.

(B_{17}) If E is finite-dimensional and \overline{\{0\}} has codimension at least two and dimension at least one, then G(E) has seventeen elements; we have lmcl = 0 and all other elements are different from 0 and each other.

(C) Finally, if E is a normed space of infinite dimension, then G(E) has eighteen elements and so is isomorphic to 𝒢 as defined in Definition 3.1.

To sum up, the order of G(E) in the finite-dimensional case is 1, 6, 15, 16 or 17 according to the following table, where n = dim E and k = dim \overline{\{0\}}.

k 0 1 2 3 4 5
n
0  1
1  15 6
2  16 16 6
3  16 17 16 6
4  16 17 17 16 6
5  16 17 17 17 16 6

Proof. The proof is organized as follows. We first present examples to show that the number of elements in G(E) is a least as large as that indicated in the statement of the theorem. Then we shall prove that in finite-dimensional spaces, the number is not larger than stated. Example 4.2. Define a function f_{1} on the real line by putting f_{1}(x) = +∞ for x < 0, f_{1}(0) = 1, and f_{1}(x) = −x^{2} for x > 0. Then G(R)(f_{1}) consists of the following six different functions:

f_{1} = m(f_{1}),  c(f_{1}) = cm(f_{1}),  l(f_{1}) = lm(f_{1}) = ml(f_{1}),
cl(f_{1}) = clm(f_{1}) = cml(f_{1}), lc(f_{1}) = lcm(f_{1}) = lcml(f_{1}),
mc(f_{1}) = lmc(f_{1}) = lmcl(f_{1}) = mcl(f_{1}) = mlc(f_{1}) = −∞.

Example 4.3. Let E = R with the usual topology and define f_{2}(x) = +∞ for x < 0, f_{2}(0) = 0, and f_{2}(x) = log x for x > 0. Then f_{2} does not assume the value −∞, so m(f_{2}) = f_{2}, cm(f_{2}) = c(f_{2}). We see that G(R)(f_{2}) consists of five different functions, viz.

f_{2} = m(f_{2}),  c(f_{2}) = cm(f_{2}),  l(f_{2}) = lm(f_{2}),
cl(f_{2}) = clm(f_{2}) = lc(f_{2}) = lcm(f_{2}),

ml(f_{2}) = mc(f_{2}) = cml(f_{2}) = lmc(f_{2}) = mcl(f_{2}) = lcml(f_{2}) = lmcl(f_{2}) = mlc(f_{2}),

where the last function is equal to minus infinity identically. Example 4.4. Define f_{3} on the real line by putting f_{3}(x) = +∞ for x < 0; f_{3}(0) = 1, and f_{3}(x) = 0, x > 0. Then G(R)(f_{3}) consists of two functions, viz.

f_{3} = c(f_{3}) = m(f_{3}) = cm(f_{3}) = mc(f_{3}),

and the function l(f_{3}), which is also the result of applying all other operators to f_{3}.

Example 4.5. Define f_{4} on R by putting f_{4}(x) = +∞ for negative x, f_{4}(0) = −∞, and f_{4}(x) = 0 for x positive. Then G(R) consists of three functions, viz.

f_{4} = l(f_{4}), c(f_{4}) = cl(f_{4}) = lc(f_{4}),

and the function −∞, which is the result of applying all other elements to f_{4}.

Combining Examples 4.2–4.5 we see that there are at least fifteen elements in G(R), the only elements which have not been isolated being the four elements lmc, mcl, lmcl and mlc = 0. Indeed, these four elements always give the same result when E = R as we shall prove in Proposition 4.11. Let us now look at examples in two variables. Example 4.6. Let E = R^{2} with the usual topology and define g_{1} by the requirement that g_{1}(x) = 0 when x_{2} > 0 and |x_{1}|x_{2} = 1, and with the value +∞ otherwise. Then g_{1} is lower semicontinuous, so l(g_{1}) = g_{1}, but c(g_{1})(x) is zero when x_{2} > 0 and +∞ otherwise, thus lc(g_{1}) ≠ c(g_{1}). We see that G(R^{2})(g_{1}) consists of three different functions, viz.

g_{1} = l(g_{1}) = m(g_{1}) = lm(g_{1}) = ml(g_{1}),
c(g_{1}) = cl(g_{1}) = cm(g_{1}) = clm(g_{1}) = mc(g_{1}) = cml(g_{1}) = mcl(g_{1}),
lc(g_{1}) = lcm(g_{1}) = lmc(g_{1}) = lcml(g_{1}) = lmcl(g_{1}) = mlc(g_{1}).

In particular, lmcl and mcl yield different results. Together with Examples 4.2–4.5 this shows that there are at least sixteen elements in G(R^{2}). Example 4.7. Let E = R^{2} be equipped with the topology generated by the seminorm x ↦ |x_{2}|. Define a function on E by putting g_{2}(x) = +∞ when x_{2} ⩽ 0, and g_{2}(x) = x_{1} when x_{2} > 0. Then g_{2} is not lower semicontinuous, and l(g_{2}) takes the value −∞ without being −∞ identically. We see that G(E)(g_{2}) consists of three different functions, viz.

g_{2} = c(g_{2}) = m(g_{2}) = cm(g_{2}) = mc(g_{2}),
l(g_{2}) = cl(g_{2}) = lm(g_{2}) = clm(g_{2}) = lc(g_{2}) = lcm(g_{2}) = lmc(g_{2}),
ml(g_{2}) = cml(g_{2}) = lcml(g_{2}) = mcl(g_{2}) = lmcl(g_{2}) = mlc(g_{2}) = −∞.

The function g_{2} shows that lmc is not equal to lmcl nor to mlc in this case.

Let us finally look at an infinite-dimensional space. Example 4.8. Let E be a normed space of infinite dimension and fix a closed hyperplane H in E. There exists a strictly increasing sequence (E_{j})_{0}^{∞} of subspaces of H. Pick points p_{j}, q_{j} ∈ E_{j} ∖ E_{j−1} of norms ∥p_{j}∥ = 1/j and ∥q_{j}∥ = j, j ⩾ 2. We define a function g_{3} as being equal to −k at the points p_{2k} ± q_{2k+1} and equal to +∞ everywhere else, k ⩾ 1. Since the set of all points p_{2k} ±q_{2k+1} is closed, this function is lower semicontinuous: l(g_{3}) = g_{3}. We note that the midpoint p_{2k} = \frac{1}{2}(p_{2k} + q_{2k+1}) + \frac{1}{2}(p_{2k} − q_{2k+1}) tends to the origin as k → +∞, and that c(g_{3})(p_{2k}) ⩽ −k → −∞; hence lc(g_{3})(0) = −∞. On the other hand, the origin does not belong to the convex hull of the points p_{2k} ± q_{2k+1}, so c(g_{3})(0) = +∞, which shows that c(g_{3}) cannot be lower semicontinuous: we must have lc(g_{3}) ≠ c(g_{3}). Moreover, lc(g_{3})(x) = +∞ when x ∉ H, so that lc(g_{3}) ≠ mlc(g_{3}).

To sum up, G(E)(g_{3}) consists of four different functions:

g_{3} = l(g_{3}) = m(g_{3}) = lm(g_{3}) = ml(g_{3}),
c(g_{3}) = cl(g_{3}) = cm(g_{3}) = clm(g_{3}) = mc(g_{3}) = cml(g_{3}) = mcl(g_{3}),
lc(g_{3}) = lcm(g_{3}) = lmc(g_{3}) = lcml(g_{3}) = lmcl(g_{3}), mlc(g_{3}),

where the last is equal to minus infinity identically.

Thus we have finally found a space where the zero element mlc can be separated from all the other elements.

Examples 4.2–4.6 are available to show that all elements separated in G(R^{2}) are also separated in the semigroup G(E). To show this we consider g(x) = f_{j}(ξ_{1}(x)), j = 1, ...4, and g(x) = g_{1}(ξ_{1}(x), ξ_{2}(x)), where ξ_{1} and ξ_{2} are two linearly independent continuous linear forms on E. However, we need to separate also lmc and lmcl as in Example 4.7. This can be done using g(x) = g_{2}(η(x), ξ_{1}(x)), where η is a discontinuous linear form on E, thus η ∈ E^{∗} ∖ E′. Such a form always exists, at least if we admit the existence of Hamel bases. Define first θ(p_{j}) = 1, and then extend θ to the union of all E_{j} by linearity. Then, using a Hamel basis, θ can be extended to a linear form η on all of E. Since η(p_{j}) = θ(p_{j}) = 1 and ∥p_{j}∥ = 1/j, η is discontinuous.

With all these examples in mind we can now finish the proof. Examples 4.2–4.5 show that the number of different elements is at least that stated in the case (A_{15}). Combined with Example 4.6 they show the same thing in case (A_{16}). With Example 4.8 we can complete the proof of (C), since G(E) can never have more than eighteen elements. We shall now see that there are not more elements in the other cases.

Proposition 4.9. Let E be a topological vector space of finite dimension. Then lmcl(f) = mlc(f) for all functions f: E → [−∞, +∞]. If the topology is separated, then lmc(f) = lmcl(f) = mlc(f) for all functions. More generally, the last conclusion holds whenever E is a topological space such that all linear forms are continuous.

Lemma 4.10. If E is a finite-dimensional topological vector space and if cl(f) > −∞, then lc(f) > −∞. If we assume the topology to be separated, then c(f) > −∞ implies that also lc(f) > −∞. The last conclusion holds whenever E is a topological vector space such that all linear forms are continuous.

Proof. It is classical that a convex function on a vector space which does not take the value −∞ admits an affine minorant. In case E^{∗} = E′, this affine function is continuous, so it is also a minorant of lc(f).

In the general case we know that cl(f) > −∞, so, again, there is an affine function φ such that cl(f) ⩾ φ, hence lc(f) = lcl(f) ⩾ l(φ). The affine function is given by φ(x) = φ(0) + ξ(x), where ξ is a linear form on E.

If cl(f) is identically +∞, then also lc(f) is identically +∞, and so does not take the value −∞.

If, on the other hand, there exists a point a such that cl(f)(a) < +∞, then we use the fact that cl(f) is constant along the cosets of \overline{\{0\}}, meaning that +∞ > cl(f)(a) = cl(f)(a + x) ⩾ φ(a + x) = φ(0) + ξ(a + x) for all x ∈ \overline{\{0\}}. Hence ξ is constant on the cosets of \overline{\{0\}}, implying that it is continuous for the given topology on E. Thus cl(f) ⩾ φ implies that lc(f) = lcl(f) ⩾ l(φ) = φ, so that lc(f) cannot take the value −∞. This proves the lemma. Proof of Proposition 4.9. Assume first that cl(f) takes the value −∞. Then mcl(f) is identically −∞, and since mcl(f) ⩾ lmcl(f) ⩾ mlc(f), all three functions are equal to minus infinity. Next, if cl(f) does not take the value −∞, then neither does lc(f) (Lemma 4.10), so mcl(f) = cl(f) and mlc(f) = lc(f) and we obtain lmcl(f) = lcl(f) = lc(f) = mlc(f) as required.

In case E^{∗} = E′, we assume first that c(f) takes the value −∞. Then mc(f) is identically minus infinity, as well as lmc(f), lmcl(f), and mlc(f), so the three functions mentioned in the statement of the proposition are all equal. On the other hand, if c(f) > −∞, then by Lemma 4.10, lc(f) > −∞ so that mlc(f) = lc(f). We always have mlc(f) ⩽ lmcl(f) ⩽ lmc(f) ⩽ lc(f), so if mlc(f) = lc(f), then all elements in this chain are equal. This proves Proposition 4.9.

Proposition 4.11. If E = R with the usual topology, then lmc(f) = mcl(f) = lmcl(f) = mlc(f) for all functions f: R → [−∞, +∞].

Lemma 4.12. If E = R with the usual topology, then c(f) > −∞ implies that lc(f) = cl(f).

Proof. If f is any function of one real variable such that c(f) > −∞, then c(f) is finite, hence continuous, in an open interval I and equal to plus infinity in the complement of its closure. We therefore have lc(f) = c(f) in I ∪ ∁\overline{I}, as well as c(f) ⩾ cl(f) ⩾ lc(f) everywhere, which implies that lc(f) = cl(f) in I ∪ ∁\overline{I}. The endpoints of I remain to be studied. For definiteness, let I = ]0, b[ and assume that lc(f)(0) < cl(f)(0), say lc(f)(0) = 0 < 2ε = cl(f)(0) ⩽ l(f)(0). Then there exists a positive number δ such that f(x) ⩾ ε for all x with |x| ⩽ δ. We may of course take δ < b. Define an auxiliary function g to be equal to +∞ for x < 0, to take the value ε at the origin and the value min(c(f)(δ), ε) at the point δ, to be affine in the interval [0, δ], and finally to be equal to (x/δ) min(c(f)(δ), ε) for x > δ. The function g is convex and a minorant of f, so we get g = c(g) ⩽ c(f). Then ε = lc(g)(0) ⩽ lc(f)(0), contradicting the assumption that lc(f) = 0. Proof of Proposition 4.11. Comparing with Proposition 4.9, we see that it only remains to be proved that mcl(f) is equal to the three other functions. If cl(f) assumes the value −∞, then mcl(f) is −∞ identically, so we are done. If, on the other hand, cl(f) > −∞, then by Lemma 4.12, lc(f) = cl(f), so mcl(f) = mlc(f) also in this case.

Corollary 4.13. If E is a topological vector space such that \overline{\{0\}} has codimension one, then mcl = lmcl = 0.

Proof. For any function f on E we define g = l(f) and note that it is constant on the cosets of \overline{\{0\}}, hence a function of one variable. We apply Proposition 4.11 to it and obtain lmc(g) = mcl(g) = lmcl(g) = mlc(g). Inserting the definition of g we get lmcl(f) = mcl(f) = mlc(f) as claimed.

We can now finish the proof of Theorem 4.1. If E = {0}, all elements in G(E) are equal to the identity, so (A_{1}) follows. On the basis of the Examples 4.2–4.5 and Proposition 4.11 we can finish the proof in case (A_{15}). Similarly, case (A_{16}) follows from Examples 4.2–4.6 and Proposition 4.9.

If E is equipped with the chaotic topology, then bla(f) is a constant function for all elements a, b ∈ G(E); moreover, it is the same constant. Therefore G(E)(f) cannot contain more than six different functions, viz. f, c(f), m(f), cm(f), mc(f), and l(f). On the other hand, if E is nonzero, then Examples 4.3–4.5, where we now use the chaotic topology, show that 1, c, m, cm, mc, and l are actually different. So (B_{6}) is proved.

Examples 4.2–4.5 and the function g_{2} constructed in Example 4.7 show that there are at least sixteen elements in case (B_{16})—however, not the same sixteen elements as in case (A_{16}). Corollary 4.13 then shows that the number cannot exceed sixteen. If the codimension of \overline{\{0\}} is at least two we may use the function g_{2} constructed in Example 4.7 together with all the functions in Examples 4.2–4.6 to show that the number is at least seventeen; Proposition 4.9 then shows that there are not more that seventeen elements in case (B_{17}). Finally, as already remarked, case (C) is dealt with by Examples 4.2–4.5, 4.7, and 4.8, since they show that there are at least eighteen elements in G(E) in that case.

5. The multiplicative structure of the semigroup

The multiplicative structure of the semigroup 𝒢 is as shown in Table 1.

The semigroup is highly noncommutative. To turn that statement into something quantitative we may argue as follows. We must have ab = ba if a or b are equal, or if one of them is equal to 0 or 1. There are 84 such pairs; they commute automatically. The remaining pairs a, b with b following a in the list are 120 in number; of those there are 90 pairs such that ab ≠ ba. In other words, out of the pairs which do not commute automatically, 75 per cent do not commute. In Table 2 the 30 commuting pairs which do not commute automatically are marked by an equality sign.

Another measure of noncommutativity is the order of the abelization 𝒢_{0} (eight) compared with that of 𝒢 (eighteen).

6. The order structure of the semigroup

For every vector space E we may define an order in the semigroup G(E) by declaring that a ⩽ b for two elements a, b of G(E) if a(f)(x) ⩾ b(f)(x) for all functions f: E → [−∞, +∞] and all points x ∈ E. (The order between functions is that of inclusion between their epigraphs.) Since G(E) is isomorphic to 𝒢 for large spaces E, we also get an order in 𝒢.

Thus we see that 1 ⩽ c, l, m ⩽ 0, and that cl ⩽ lc, cm ⩽ mc, and lm ⩽ ml. Multiplication preserves order in the sense that a ⩽ a′, b ⩽ b′ implies ab ⩽ a′b′.

The order structure of 𝒢 can be read off from its Hasse diagram, which is shown in Figure 1. Of course the geometric aspect of the arrangement is not uniquely determined by the order. In the presentation we have chosen, the semigroup consists basically of two cubes which share a common vertex. The upper cube hangs from the vertex 1, and its bottom vertex clm is the top vertex of the lower cube, which, however, is not complete in that there is no edge from clm to mc. In addition to the vertices of the two cubes, there is one elements between cl and lcm, viz. lc, and one between lm and cml, viz. ml. Finally, the zero element mlc = 0 is attached below the bottom vertex lmcl of the lower cube.

The upper cube realizes the abelization 𝒢_{0} of 𝒢 mentioned at the end of Section 3. The whole semigroup possesses a 2-symmetry: it is equal to its mirror image in

the plane determined by 1, 0 and l. Mathematically the symmetry is achieved by interchanging c and m and then reversing the order of all products (the conditions (3.1) and (3.2) are invariant under this change). The upper cube in Figure 1 possesses of course more symmetry, but this does not correspond to any symmetry of the semigroup, only of its abelization 𝒢_{0}.

We emphasize that there is no order relation (no edge) between clm and mc. This follows from Example 4.2, where the function f_{1} shows that we do not have clm ⩾ mc, and Example 4.4, where the function f_{3} shows that we do not have clm ⩽ mc. As a consequence, G(R) is not a lattice. The same is then true of 𝒢. In fact, both lmc and mcl are minimal majorants of the set {clm, mc}, so there is no smallest majorant of that set. Similarly, the infimum of lmc and mcl does not exist; mc and clm are both maximal minorants of the set {lmc, mcl}.

In Table 3 the infimum and the supremum (when they exist) of two elements of the semigroup are listed. Only the upper triangle, corresponding to the case where a precedes b in the listing we have chosen, is entered. When a ⩽ b in the partial order of the semigroup, we have of course inf(a, b) = a and sup(a, b) = b; in these cases nothing is written. A star ⋆ indicates that the supremum or infimum does not exist.

7. Comparison between the supremum and the product of two elements

With the order we have defined in 𝒢, it is clear that a, b ⩽ ab, thus that sup(a, b) ⩽ ab when the supremum exists. By symmetry, sup(a, b) ⩽ ba, so that

(7.1)  sup(a, b) ⩽ inf(ab, ba)

whenever the supremum and infimum exist. In Table 4 we list, for each pair a, b with a preceding b, sup(a, b) and inf(ab, ba) (a nonexisting supremum or infimum is indicated by a star). Let us agree to call a pair regular if we have equality in (7.1). Every pair (a, b) with a or b equal to 0 or 1 is regular. A pair (a, a) is regular if and only if a is idempotent.

There are 136 pairs (a, b) with a preceding b or a equal to b but with none of them equal to 0 or 1. Of these, there are 90 regular pairs, thus satisfying sup(a, b) = inf(ab, ba), marked by an equality sign in Table 4; 41 irregular pairs with sup(a, b) ≠ inf(ab, ba); four pairs where neither sup(a, b) nor inf(ab, ba) exists; and one pair where sup(a, b) exists but inf(ab, ba) does not exist.

References

Birkhoff, Garrett 1948 Lattice Theory. Revised edition. New York: American Mathematical Society. xiii + 283 pp.

Everett, C. J.

1944  Closure operators and Galois theory in lattices. Trans. Amer. Math. Soc. 55, 514–525.

16 C. O. Kiselman

Hewitt, Edwin, and Zuckerman, Herbert S.

1955  Finite dimensional convolution algebras. Acta Math. 93, 67–119.

Kuroš, A. G.

1962  Lekcii po obščej algebre. Moscow: F.M. 396 pp.

Ore, Oystein

1944  Galois connexions. Trans. Amer. Math. Soc. 55, 493–513.

A semigroup of operators in convexity theory 17

1  c  l  m cl cm lm clm lc mc ml lcm cml lcml lmc mcl lmcl 0
c  c  cl cm cl cm clm clm lc mc cml lcm cml lcml lmc mcl lmcl 0
l  lc  l  lm lc lcm lm lcm lc lmc ml lcm lcml lcml lmc lmcl lmcl 0
m mc ml m mcl mc ml mcl 0 mc ml 0 mcl 0  0 mcl 0 0
cl lc cl clm lc lcm clm lcm lc lmc cml lcm lcml lcml lmc lmcl lmcl 0
cm mc cml cm mcl mc cml mcl 0 mc cml 0 mcl 0  0 mcl 0 0
lm lmc ml lm lmcl lmc ml lmcl 0 lmc ml 0 lmcl 0  0 lmcl 0 0
clm lmc cml clm lmcl lmc cml lmcl 0 lmc cml 0 lmcl 0  0 lmcl 0 0
lc lc lc lcm lc lcm lcm lcm lc lmc lcml lcm lcml lcml lmc lmcl lmcl 0
mc mc mcl mc mcl mc mcl mcl 0 mc mcl 0 mcl 0  0 mcl 0 0
ml 0 ml ml 0  0 ml 0 0 0 ml 0  0  0  0  0  0 0
lcm lmc lcml lcm lmcl lmc lcml lmcl 0 lmc lcml 0 lmcl 0  0 lmcl 0 0
cml 0 cml cml 0  0 cml 0 0 0 cml 0  0  0  0  0  0 0
lcml 0 lcml lcml 0  0 lcml 0 0 0 lcml 0  0  0  0  0  0 0
lmc lmc lmcl lmc lmcl lmc lmcl lmcl 0 lmc lmcl 0 lmcl 0  0 lmcl 0 0
mcl 0 mcl mcl 0  0 mcl 0 0 0 mcl 0  0  0  0  0  0 0
lmcl 0 lmcl lmcl 0  0 lmcl 0 0 0 lmcl 0  0  0  0  0  0 0
0  0  0  0  0  0  0  0 0 0  0  0  0  0  0  0  0 0

Table 1. The multiplication table of the semigroup.

18 C. O. Kiselman

1 c l m cl cm lm clm lc mc ml lcm cml lcml lmc mcl lmcl 0
c c cl cm cl cm clm clm lc mc cml lcm cml lcml lmc mcl lmcl 0
c lc mc lc mc lmc lmc = = 0 lmc 0 0 = 0 0 0
l  l lm lc lcm lm lcm lc lmc ml lcm lcml lcml lmc lmcl lmcl 0
l ml cl cml ml cml = mcl = lcml cml = lmcl mcl = 0
m m mcl mc ml mcl 0 mc ml 0 mcl 0 0 mcl 0 0
m clm cm lm clm lcm = = lcm cml lcml lmc = lmcl 0
cl  lc lcm clm lcm lc lmc cml lcm lcml lcml lmc lmcl lmcl 0
lc mcl lmcl lmcl = mcl 0 lmcl 0  0 lmcl 0  0 0
cm  mc cml mcl 0 mc cml 0 mcl 0  0 mcl 0 0
mc lmc lmc lcm = 0 lmc 0  = lmc 0  = 0
lm  ml lmcl 0 lmc ml 0 lmcl 0  0 lmcl 0 0
ml cml lcm mcl = lcml cml lcml lmcl mcl lmcl 0
clm  lmcl 0 lmc cml 0 lmcl 0  0 lmcl 0 0
lmcl lcm mcl 0 lmcl 0  = lmcl 0  = 0
lc  lc lmc lcml lcm lcml lcml lmc lmcl lmcl 0
lc 0  0  0  0  0  0  0  0 0
mc  mc mcl 0 mcl 0  0 mcl 0 0
mc 0 lmc 0  = lmc 0  = 0
ml  ml 0  0  0  0  0  0 0
ml lcml cml lcml lmcl mcl lmcl 0
lcm  0 lmcl 0  0 lmcl 0 0
0  0  = =  0  = 0
cml  0  0  0  0  0 0
0  = lmcl = = 0
lcml 0 0 0 0 0
0  = = = 0
lmc  0 lmcl 0 0
0  0  = 0
mcl  0  0 0
0  = 0
lmcl  0 0
0 0
0  0

0

Table 2. Listing ab and ba for comparison.

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•_{lmc}  •_{lcml}  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_{0}

Figure 1. The Hasse diagram of the semigroup.

20 C. O. Kiselman

1 c l m cl cm lm clm lc mc ml lcm cml lcml lmc mcl lmcl 0
c  1 1  1  1
cl cm  clm  cml
l  1  1  1 lm  clm  ⋆
m  1  1
clm  lcm
cl  c  l  c  l
clm clm  ⋆ cml
cm  m  c  m
clm  lcm  cml
lm  l m lcm ⋆
clm  cl cm lm
lcm ⋆ cml
lc  c  l  cl  cl
lmc lcml  lcml  lmcl
mc  m cm cm cm
mcl lmc mcl lmcl
ml lm lm
lcml  lmcl
lcm clm clm
lcml  lmcl
cml  clm

lmcl

lcml  lcm cml
lmcl lmcl
lmc  ⋆ lmcl

mcl lmcl 0

Table 3. The infimum and supremum (when they exist) of two elements of the semigroup.

A semigroup of operators in convexity theory 21

1 c l m cl cm lm clm lc mc ml lcm cml lcml lmc mcl lmcl 0
c  c cl cm cl cm clm clm lc mc cml lcm cml lcml lmc mcl lmcl 0
= = = = = = = = = = = = = = = = 0
l  l lm cl clm lm clm lc ⋆ ml lcm cml lcml lmc mcl lmcl 0
= = = = = = = ⋆ = = = = = = = 0
m  m clm cm lm clm lcm mc ml lcm cml lcml lmc mcl lmcl 0
= = = = = = = = = = = = = = 0
cl  cl clm clm clm lc ⋆ cml lcm cml lcml lmc mcl lmcl 0
lc = = lcm = ⋆ = = lcml = = lmcl = 0
cm  cm clm clm lcm mc mcl lcm cml lcml lmc mcl lmcl 0
mc = ⋆ = = cml lmc mcl 0  = =  0 0
lm  lm clm lcm ⋆ ml lcm cml lcml lmc mcl lmcl 0
ml cml = ⋆ = lcml = = lmcl = = 0
clm  clm lcm ⋆ cml lcm cml lcml lmc mcl lmcl 0
lmcl = ⋆ = lmcl lmcl 0 lmcl lmcl 0 0
lc  lc lmc lcml lcm lcml lcml lmc lmcl lmcl 0
= = = = = = = = = 0
mc  mc mcl lmc mcl lmcl lmc mcl lmcl 0
= = = =  0  = =  0 0
ml  ml lcml cml lcml lmcl mcl lmcl 0
= = = = = = = 0
lcm  lcm lcml lcml lmc lmcl lmcl 0
0 lmcl 0  0  =  0 0
cml  cml lcml lmcl mcl lmcl 0
0  0  =  0  0 0
lcml  lcml lmcl lmcl lmcl 0
0  0  0  0 0
lmc  lmc lmcl lmcl 0
0  =  0 0
mcl  mcl lmcl 0
0  0 0
lmcl  lmcl 0
0 0
0  0

0

Table 4. Comparison between sup(a, b) and inf(ab, ba).

Author’s address: Uppsala University, Department of Mathematics,

P. O. Box 480, SE-751 06 Uppsala, Sweden.

Telephone: +46 18 4713216 (office); +46 18 300708 (home) Fax: +46 18 4713201

Electronic mail: kiselman@math.uu.se  URL: http://www.math.uu.se/~kiselman