Atomic operators in vector lattices

In this paper we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator $T$ from a vector lattice $E$ to a vector lattice $F$ is atomic if there exists a Boolean homomorphism $\Phi$ from the Boolean algebra $\mathfrak{B}(E)$ of all order projections on $E$ to $\mathfrak{B}(F)$ such that $T\pi=\Phi(\pi)T$ for every order projection $\pi\in\mathfrak{B}(E)$. We show that the set of all atomic operators defined on a vector lattice $E$ with the principal projection property and taking values in a Dedekind complete vector lattice $F$, is a band in the vector lattice of all regular orthogonally additive operators from $E$ to $F$. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space.


Introduction and preliminaries
Local operators and, more generally, atomic operators in classical function spaces find numerous applications in control theory, the theory of dynamical systems and the theory of partial differential equations (see [6,19,23]). The concept of a local operator was in the context of vector lattices first introduced in [22]. It is an abstract form of the well known property of a nonlinear superposition operator and can be stated in the following form: the value of the image function on a certain set depends only on the values of the preimage function on the same set. In this article we analyse the notion of an atomic operator in the framework of the theory of vector lattices and orthogonally additive operators. Today the theory of orthogonally additive operators in vector lattices is an active area in functional analysis; see for instance [1,2,7,8,11,10,13,17,16,14,18,25].
Abstract results of this theory can be applied to the theory of nonlinear integral operators [12,21], and there are connections with problems of convex geometry [24].
Let us introduce some basic facts concerning vector lattices and orthogonally additive operators. We assume that the reader is acquainted with the theory of vector lattices and Boolean algebras. For the standard information we refer to [3,4,9]. All vector lattices below are assumed to be Archimedean.
Let E be a vector lattice. A net (x α ) α∈Λ in E order converges to an element −→ x) if there exists a net (u α ) α∈Λ in E + such that u α ↓ 0 and |x α − x| ≤ u α for all α ∈ Λ satisfying α ≥ α 0 for some α 0 ∈ Λ. Two elements x, y of the vector lattice E are disjoint (notation x⊥y), if |x| ∧ |y| = 0. The sum x + y of two disjoint elements x and y is denoted by x ⊔ y. The equality x = n i=1 x i means that x = n i=1 x i and x i ⊥x j if i = j. An element y of E is called a fragment of an element x ∈ E, provided y⊥(x − y). The notation y ⊑ x means that y is a fragment of x. If E is a vector lattice and x ∈ E then we denote by F x the set of all fragments of x. A positive, linear projection π : E → E is said to be an order projection if 0 ≤ π ≤ Id, where Id is the identity operator on E. The set of all order projections on E is denoted by B(E). The set B(E) is ordered by π ≤ ρ :⇔ π • ρ = π, and it is a Boolean algebra with respect to the Boolean operations: π ∧ ρ := π • ρ; π ∨ ρ := π + ρ − π • ρ; π = Id − π.
An element x of a vector lattice E is called a projection element if the band generated by x is a projection band, and then we denote by π x the order projection onto the band generated by x. A vector lattice E is said to have the principal projection property if every element of E is a projection element. For example, every σ-Dedekind complete vector lattice has the principal projection property.
A (possibly nonlinear) operator T : E → F from a vector lattice E into a real vector space is called orthogonally additive if T (x + y) = T (x) + T (y) for every disjoint elements x, y ∈ E. It is clear that if T is orthogonally additive, then T (0) = 0. The set of all orthogonally additive operators from E into F , denoted by OA(E, F ), is a real vector space for the natural linear operations.
An operator T : E → F between two vector lattices E and F is said to be: • order bounded, if T maps order bounded sets in E to order bounded sets in F .
for two positive, orthogonally additive operators T 1 and T 2 from E to F .
An orthogonally additive, order bounded operator T : E → F is called an abstract Urysohn operator. This class of operators was introduced and studied in 1990 by Mazón and Segura de León [12]. We notice that the order boundedness is a restrictive condition for orthogonally additive operator. Indeed, every operator T : R → R satisfying T (0) = 0 is orthogonally additive, but not every operator of this form is order bounded. Consider, for instance, the positive function T defined by The notion of a laterally-to-order bounded operator was introduced in [15]. It is obviously weaker than the notion of order bounded operator. An orthogonally additive, laterally-to-order bounded operator T : E → F is also called a Popov operator.
We denote by OA + (E, F ) the set of all positive, orthogonally additive operators from E to F (so that OA(E, F ) becomes an ordered vector space with this cone), by OA r (E, F ) := OA + (E, F ) − OA + (E, F ) the regular, orthogonally additive operators, and by P(E, F ) the laterally-to-order bounded, orthogonally additive operators from E to F . Also OA r (E, F ) and P(E, F ) are ordered vector spaces. In general, OA r (E, F ) = P(E, F ) (see [15]), but for a Dedekind complete vector lattice F we have the following strong properties of OA r (E, F ) and P(E, F ). . Let E and F be vector lattices, and assume that F is Dedekind complete. Then OA r (E, F ) = P(E, F ), and OA r (E, F ) is a Dedekind complete vector lattice. Moreover, for every S, T ∈ OA r (E, F ) and every x ∈ E, (1) (T ∨ S)(x) = sup{T y + Sz : x = y ⊔ z}; (2) (T ∧ S)(x) = inf{T y + Sz : x = y ⊔ z}; (3) (T ) + (x) = sup{T y : y ⊑ x};

Basic properties of atomic operators
In this section we introduce a new subclass of orthogonally additive operators, namely the class of atomic operators, and show that under some assumptions on the vector lattices E and F the set of all atomic operators from E to F subordinate to a Boolean homomorphism Φ : B(E) → B(F ) is a band in the vector lattice OA r (E, F ). We further obtain a formula for the order projection onto this band.
Let us first recall the definition of Boolean homomorphisms. Let A, B be Boolean algebras. A map Φ : A → B is called a Boolean homomorphism, if the following conditions hold: for every family (resp. countable family) (x λ ) λ∈Λ of elements of A then Φ is said to be an order continuous (resp. a sequentially order continuous) Boolean homomorphism.
Let E and F be vector lattices and Φ be a Boolean homomorphism from B(E) to B(F ). A map T : E → F is said to be an atomic operator subordinate to Φ, or briefly atomic operator, if T π = Φ(π)T for every order projection π ∈ B(E). The set of all atomic operators from E to F subordinate to Φ is denoted by Φ(E, F ).
We remark that the class of atomic operators was first introduced in [1]. It is easy to verify that Φ(E, F ) is a vector space. Indeed, let λ ∈ R, T , S ∈ Φ(E, F ), x ∈ E and π ∈ B(E). Then Φ(π)λT (x) = λΦ(π)T (x) = λT π(x) and  In particular, if E has the principal projection property, then every linear orthomorphism T : E → E (a band preserving, order bounded operator) is atomic with respect to the identity homomorphism.
For every subset A ⊆ Z one can define an order projection π A which corresponds in fact to the multiplication by the characteristic function 1 A . This gives a one-to-one correspondence between the Boolean algebra P(Z) of all subsets of Z and the Boolean algebra of order projections on E. With this identification and for fixed k ∈ Z, if we define the shift Boolean homomorphism Φ k : The following is an example of a nonlinear atomic operator. (C 1 ) N(·, r) is measurable for all r ∈ R; (C 2 ) N(s, ·) is continuous on R for ν-almost all s ∈ B.
If the function satisfies only the conditions (C 1 ) and (C 2 ), then we call it a Carathéodory function. Given a Caratheodory function N : B × R → R, one defines the superposition operator T N : L 0 (ν) → L 0 (ν) by We note that a superposition operator is in the literature also known as Nemytskii operator. The theory of these operators is widely represented in the literature (see [5]).
Proof. Let Ξ 0 = {D ∈ Ξ : ν(D) = 0} be the set of all ν-null sets. Then Ξ 0 is an ideal in the Boolean algebra Ξ. We let Ξ ′ := Ξ/Ξ 0 be the factor algebra. It is well known that the Boolean algebra B(L 0 (ν)) of all order projections on L 0 (ν) is isomorphic to the Boolean algebra Ξ ′ . In fact, to every (equivalence class of a) measurable subset D ∈ Ξ ′ there corresponds an order projection π D which is in fact the multiplication by the characteristic function 1 D , and vice versa; see [3, Section 1.6]. Now we show that First, for every s ∈ D, Second, for every s ∈ B \ D, by condition (C 0 ), Hence, for every f ∈ L 0 (ν) and π = π D ∈ B(L 0 (ν)), and the assertion is proved.
The following lemma shows that on Banach lattices with the principal projection property every atomic operator is regular orthogonally additive.
Lemma 2.5. Let E be a vector lattice with the principal projection property, F be a vector lattice, Φ be a Boolean homomorphism from B(E) to B(F ) and T ∈ Φ(E, F ). Then T is orthogonally additive, laterally-to-order bounded (that is, T ∈ P(E, F )) and disjointness preserving.
Hence, T is orthogonally additive.
We next show that T is disjointness preserving. Let x, y ∈ E be disjoint elements. Then the order projections π x , π y are disjoint elements in the Boolean algebra B(E), and hence Φ(π x ) and Φ(π y ) are disjoint elements in the Boolean algebra B(F ). Since Now fix x ∈ E and assume that y ∈ F x . Then by definition y ⊥ (x − y), and therefore, since T is disjointness preserving, T y ⊥ T (x − y). Since T is orthogonally additive, this implies T (F x ) ⊆ F T x , and hence |T y| ≤ |T x|. Hence T is a laterally-to-order bounded.
It is worth to notice that without any assumption on the vector lattices E and F the space P(E, F ) is not a vector lattice and we say nothing about the order structure of the space of laterally-to-order bounded orthogonally additive operators. Nevertheless the next lemma shows that Φ(E, F ) is a vector lattice if E has the principal projection property. Lemma 2.6. Let E be a vector lattice with the principal projection property, F be a vector lattice, and Φ be a Boolean homomorphism from B(E) to B(F ). Then For every x ∈ E, We show that R is orthogonally additive. Indeed, let x, y ∈ E with x ⊥ y. By Lemma 2.5, T is disjointness preserving. Then Finally, we show that R is an atomic operator. Let π ∈ B(E) and x ∈ E. Then and the proof is complete.
The following theorem is the first main result of this section.
Theorem 2.7. Let E be a vector lattice with the principal projection property, F be a Dedekind complete vector lattice and Φ be a Boolean homomorphism from Proof. By Theorem 1.1, OA r (E, F ) is a vector lattice, and by Lemmas 2.5 and 2.6 and Theorem 1.
We remark that if x = y ⊔ z, then y = π y x = π y y, z = π z Passing to the supremum in the left-hand side of the above inequality over all y, z ∈ F x such that x = y ⊔ z yields to and it follows that (T ∨ S)(x) = T x ∨ Sx. Now it is easy to deduce formulas for the infimum, module, positive and negative parts of operators.
Let E be a vector lattice with the principal projection property and F be a Dedekind complete vector lattice. Then OA r (E, F ) is Dedekind complete by Theorem 1.1, and therefore, by a theorem of F. Riesz, every band is a projection band. By Theorem 2.7, every positive orthogonally additive operator T : E → F thus has a unique decomposition T = T 1 + T 2 with 0 ≤ T 1 ∈ Φ(E, F ) and T 2 ∈ Φ(E, F ) ⊥ . The next theorem, which is the second main result of this section, gives a description of the band projection onto Φ(E, F ).
By D 0 (E), or D 0 for short, we denote the set of all finite partitions of the identity operator Id, that is Theorem 2.8. Let E be a vector lattice with the principal projection property and F be a Dedekind complete vector lattice and T ∈ OA + (E, F ). Then the component Proof. For any T ∈ OA + (E, F ), set Clearly A(T ) is a downward directed set of positive orthogonally additive operators and taking into account the Dedekind completeness of the vector lattice OA r (E, F ) we deduce that there exists R(T ) := inf A(T ). We verify the following properties for every T ∈ OA + (E, F ): The relation (1) is obvious. In order to prove (2), we show that R is additive on the positive cone. If T 1 , T 2 ∈ OA + (E, F ), then for arbitrary (π i ), (π j ) ∈ D 0 we have where (π k ) ∈ D 0 is finer than (π i ) and (π j ). Taking the infimum, we obtain Let us prove the equivalence in (3). Assume that 0 ≤ T ∈ Φ(E, F ). We notice that for every (π i ) ∈ D 0 we have that Φ(π i ) = Id. Thus Passing to infimum on the left hand side of the above equality over all (π i ) ∈ D 0 we get that R(T ) = T . On the other hand, assume that R(T ) = T . We show that T ∈ Φ(E, F ). Indeed, fix π ∈ B(E). Then T ≤ Φ(π)T π + Φ(Id − π)T (Id − π). It follows that Φ(π)T ≤ Φ(π)T π and therefore Φ(π)T = T π.
It remains to verify the equality (4). Suppose that W = R(T ), with T ∈ OA + (E, F ). For every ρ ∈ B(E), we may write Thus, W ρ = Φ(ρ)W for every ρ ∈ B(E). By the equivalence (3) which is established above, we obtain W = R(W ).
We remark that a similar theorem for orthomorphisms was proved in [20].

Atomic operators in spaces of measurable functions
In this section we investigate atomic operators in spaces of real valued, measurable functions and get an analytic representation for this class of operators.
Let (B, Ξ, ν) be a σ-finite measure space. Choose an equivalent finite measure λ on Ξ such that ν and λ have the same sets of measure 0. As in Example 2.3 above, we denote by L 0 (B, Ξ, ν) (or L 0 (ν) for brevity) the set of all real valued, measurable functions on B. More precisely, L 0 (ν) consists of equivalence classes of such functions, where as usual two functions f and g are said to be equivalent if they coincide almost everywhere on B; note that L 0 (ν) and L 0 (λ) coincide. The vector space L 0 (ν) with the metric ρ L 0 , defined by becomes a complete metric space, and the convergence with respect to the metric ρ L 0 is equivalent to the convergence in measure, meaning here the measure λ. Recall that (f n ) converges to f in measure (notation f n λ → f ; see [3, Theorem 1.82]), if, for every δ > 0, lim n→∞ λ({|f n − f | > δ}) = 0. More precisely, the convergence in measure is characterised by the following statement. Choose an equivalent finite measure λ on Ξ such that ν and λ have the same sets of measure 0. For a sequence (f n ) ⊆ L 0 (ν) and and element f ∈ L 0 (ν) the following equivalent: (1) f n λ → f ; (2) every subsequence of (f n ) has a subsequence that converges pointwise νalmost everywhere to f ; We say that a function N : B × R → R is a superpositionally measurable function, or briefly that it is sup-measurable, if is measurable for every f ∈ L 0 (ν). We call the function N an S-function, if it is sup-measurable and if This normalisation condition already appeared in Example 2.3, where we also defined K-functions and Caratheodory functions. Note that every sup-measurable function N satisfies automatically the condition (C 1 ) from Example 2.3, that is, N(·, r) is measurable for all r ∈ R. Indeed, it suffices to identify r ∈ R with the corresponding constant function r1 B . It is well known that every Carathéodory function N is sup-measurable; see for instance [5,Chapter 1.4]. Sup-measurability of a function N : B × R → R is the weakest condition under which the superposition operator T N given by is well-defined on L 0 (ν).
Given two sup-measurable functions N, K : B × R → R, we write N K if, for every f ∈ L 0 (ν), N(·, f (·)) ≤ K(·, f (·)) ν-almost everywhere on B. We say that N and K are sup-equivalent (notation N ≃ K) if both N K and K N.
Let N : B × R → R be an K-function, and let T N be the associated superposition operator on L 0 (ν). Then, by Example 2.3, T N is atomic with respect to the identity Boolean homomorphism. In addition, since L 0 (ν) is Dedekind complete and by Lemma 2.5 and Theorem 1.1, T N is regular orthogonally additive and disjointness preserving, but actually these two properties can easily be verified directly for a superposition operator. The main result of this section shows that in L 0 (ν), and up to Boolean homomorphisms, all sequentially order continuous, atomic operators are superposition operators. Before stating the precise statement, let us recall a definition and introduce an operator.
We recall that an orthogonally additive operator T : E → F is sequentially order continuous, if for every order convergent sequence (x n ) n ⊆ E with x n (o) −→ x the sequence (T x n ) n ⊆ F is order convergent to T x. Now, let (A, Σ, µ) be a second σ-finite measure space. Recall that for every measurable set A ′ ∈ Σ the multiplication operator π A ′ associated to the multiplication by the characteristic function 1 A ′ is an order projection on L 0 (µ). In fact, every order projection is of this form, and when we consider the factor Boolean algebra Σ ′ = Σ/Σ 0 as in Example 2.3 (factorization by the sets of µ-measure zero), then we obtain a one-to-one correspondence, that is, the Boolean algebras Σ ′ and B(L 0 (µ)) are isomorphic. Now let Φ : B(L 0 (µ)) → B(L 0 (ν)) be a Boolean homomorphism. We identify it with a Boolean homomorphism Φ : Σ ′ → Ξ ′ , and we define an associated, linear shift operator S Φ : L 0 (µ) → L 0 (ν) in the following way. First, for every simple we set The function S Φ f is a simple function and therefore measurable, and its definition does not depend on the representation of f . Note in this context that since Φ is a Boolean homomorphism, then the Φ(A i ) are mutually disjoint, too. Second, for every positive, measurable function f ∈ L 0 (µ) + there exists an increasing sequence (f n ) of positive, simple functions such that f = sup n f n . One can easily show that the sequence (Sf n ) is order bounded in L 0 (ν). We then put This definition of S Φ f does not depend on the choice of the approximating sequence (f n ). Finally, for arbitrary f ∈ L 0 (µ) we set The operator S Φ thus defined is a linear, positive operator from L 0 (µ) into L 0 (ν). Now let Φ be, in addition, a Boolean isomorphism. Then Φ −1 : Ξ ′ → Σ ′ is a Boolean isomorphism, too, and it follows from the definition that S Φ is invertible and S −1 Φ = S Φ −1 . In particular, S −1 Φ is linear and positive, too. We show that S Φ is a sequentially order continuous operator. Assume, on the contrary, that S Φ is not sequentially order continuous. Then there exists a sequence (f n ) in L 0 (µ) such that f n ↓ 0 and S Φ f n ↓ 0. Passing to an appropriate subsequence, we can find g > 0 such that S Φ f n ≥ g for every n ∈ N. Applying S −1 Φ to this inequality yields f n ≥ S −1 Φ g for every n ∈ N. Since S −1 Φ g > 0, this yields a contradiction. The next theorem is the main result of the section. Let (A, Σ, µ) and (B, Ξ, ν) be σ-finite measure spaces, Φ : Σ ′ → Ξ ′ be a Boolean isomorphism, and T : L 0 (µ) → L 0 (ν) be a regular orthogonally additive operator. Then the following statements are equivalent: (1) T is a continuous (with respect to the metric ̺ L 0 , atomic operator subor- T N is the superposition operator associated with N and S Φ is the shift operator associated with Φ, that is, Proof. (1) ⇒ (2). Let T : L 0 (µ) → L 0 (ν) be a continuous, atomic operator subordinate to Φ. Then we define a functionN : We note thatN (·, 0) = T (0) = 0 and thereforeN (·, 0) = 0 ν-almost everywhere. Moreover,N(·, r) is Ξ-measurable for every r ∈ R. Now, take a simple function where the A i are mutually disjoint measurable subsets of A and In other words, when we define the operator TN : L 0 (ν) → L 0 (ν), TN := T • S −1 Φ , then TN f =N(·, f (·)) for every finite step function f , so that on the space of finite step functions, the operator TN acts like a superposition operator. At the same time, TN is defined everywhere on L 0 (ν) and it is continuous with respect to the metric ̺ L 0 by assumption on T and by sequential order continuity of S −1 Φ which implies continuity with respect to ̺ L 0 .
It is, however, not clear whetherN is sup-measurable. If it was sup-measurable, then we could invoke [5,Theorem 1.4] in order to show thatN is sup-equivalent to a Caratheodory function N. For the construction of a Caratheodory function associated with TN , we proceed as in the proof of [5,Lemma 1.7], that is, by regularisation and approximation.
We may for our purposes without loss of generality assume that (B, Ξ, ν) is a finite measure space. In fact, if this measure space was only σ-finite, then we could replace the measure ν by an equivalent finite measure λ, as in the definition of the metric ̺ L 0 . Define, for every k ∈ N, the function t k : Ξ × L 0 (ν) → R by and then for every k ∈ N and every λ > 0 the regularized function t k,λ : Ξ × L 0 (ν) → R by Then, for every k ∈ N, D ∈ Ξ, f ∈ L 0 (ν), r,r ∈ R, where in the third line we identify a real number r with the corresponding constant function r 1 B . In order to see that t k,λ (D, f ) ≤ t k (D, f ) (see (3.2)), it suffices simply to take g = f in the definition of t k,λ . Similarly, from the definition one sees that t k,λ increasing in λ > 0 (see (3.2)). The property (3.3) follows from the order continuity of TN . Finally, in order to prove (3.4), fix f ,f ∈ L 0 (ν). By definition, for every g ∈ L 0 (ν), Moreover, for every ε > 0 there exists g ε ∈ L 0 (ν) such that When we substract both inequalities and take g = g ε in the first inequality, then we obtain or, when f = r andf =r are constant functions, Since this inequality holds for arbitrary ε > 0, and by changing the roles of r and r, one obtains (3.4). One easily shows that for every k ∈ N, λ > 0 and r ∈ R the function t k,λ (·, r) is a measure on (B, Ξ). By the inequality (3.4), this measure is absolutely continuous with respect to ν. By the Radon-Nikodym theorem, for every k ∈ N, λ ∈ N and r ∈ Q, there exist densities N k,λ (·, r) such that t k,λ (D, r) = D N k,λ (x, r) dν(x).
Since we are only dealing with a countable set of parameters k, λ and r, by the definition of t k,λ , and by the properties (3.2) and (3.4), there exists a set D 0 ∈ Ξ of ν-measure zero such that, for every k ∈ N, λ ∈ N, r,r ∈ Q and x ∈ B \ D 0 , From the last inequality it follows that, for every k ∈ N, λ ∈ N and every x ∈ B \ D 0 , the function N k,λ (x, ·) uniquely extends to a Lipschitz continuous function on R, which we still denote by N k,λ (x, ·). In particular, the functions N k,λ are Caratheodory functions (more precisely, they are K-functions), and the associated superposition operators are continuous on L 0 (ν) by Example 2.3. Set As a pointwise supremum of (Lipschitz) continuous functions, for every x ∈ B\D 0 , the function N k (x, ·) is lower semicontinuous. By (3.6), for every x ∈ B \ D 0 , r ∈ R, By [5,Theorem 1.1], N k is a so-called Shragin function. By Lebesgue's dominated convergence theorem, for every k ∈ N, every D ∈ Ξ and every r ∈ R, As a consequence, there exists a set D 1 ∈ Ξ of ν-measure zero, such that D 1 ⊇ D 0 and, for every x ∈ B \ D 1 and every r ∈ Q, (3.8) In particular, the superposition operator associated with the Shragin function N k and the superposition operator associated with the function (−k) ∨N ∧ k coincide on the space of rational step functions (step functions taking values in Q). The latter operator, however, uniquely extends to a continuous operator on L 0 (ν). By [5,Theorem 1.3], the function N k already is a Caratheodory function.
It remains now to let k → ∞, and to note that N k (x, ·) is uniquely determined on [−k, k] ∩ Q by (3.8), independently of k ∈ N, in order to obtain a Caratheodory function N : B × R → R such that the associated superposition operator T N coincides with TN on the space of rational step functions. Hence, by continuity, T N = TN everywhere on L 0 (ν). Since T N 0 = 0, N is in fact a K-function, and we have proved one implication.
(2) ⇒ (1). Assume that there exists a K-function N : B × R → R such that for any f ∈ L 0 (ν) that is, T = T N • S Φ . Since any superposition operator associated with a Caratheodory function is order continuous, and since S Φ is order continuous, then T : L 0 (µ) → L 0 (ν) is order continuous. By Lemma 2.4, the superposition operator T N is an atomic operator subordinate to the identity homomorphism. By construction, the shift operator S Φ is an atomic operator subordinate to Φ. It follows easily that the composition T = T N • S Φ is an atomic operator subordinate to Φ. The proof is finished.
4. An extension of positive atomic operators and laterally continuous orthogonally additive operators In this section we show that any atomic operator is laterally-to-order continuous. We also prove that an atomic, orthogonally additive map defined on a lateral ideal can be extended to an atomic orthogonally additive operator defined on the whole space.
Let E, F be vector lattices. A net (x α ) α∈Λ ⊆ E is said to be laterally convergent Proof. Take a laterally convergent net (x λ ) λ∈Λ with x λ lat −→ x. Denote by π λ , ρ λ and π the order projections onto the bands {x λ } ⊥⊥ , {x − x λ } ⊥⊥ and {x} ⊥⊥ , respectively. Since the elements x−x λ and x λ are disjoint for any λ ∈ Λ it follows that Moreover, the net (π λ ) order converges to π and the net (ρ λ ) order converges to 0 B(E) in the Boolean algebra B(E). Taking into account that Φ is an order continuous homomorphism of Boolean algebras we deduce that the net Φ(ρ λ ) converges to 0 B(F ) in the Boolean algebra B(F ). Hence, and this completes the proof.
A subset D of a vector lattice E is said to be a lateral ideal if the following conditions hold: (1) if x ∈ D and y ∈ F x , then y ∈ D; (2) if x, y ∈ D and x⊥y, then x + y ∈ D.
Example 4.2. Let E be a vector lattice. Then any order ideal in E is a lateral ideal.  Let E, F be vector lattices, D be a lateral ideal in E. A map T : D → F is said to be • orthogonally additive, if T (x + y) = T x + T y for every disjoint elements x, y ∈ D; • positive, if T x ≥ 0 for every x ∈ D; • atomic, if T π = Φ(π)T for every order projection π ∈ B(E) .
with the interpretation sup ∅ = 0, is positive, orthogonally additive and laterallyto-order continuous, that is, T D ∈ P + (E, F ). Moreover, T D x = T x for every x ∈ D.
The operator T D ∈ P + (E, F ) is called the minimal extension of the positive, orthogonally additive operator T : D → F .
We recall the following auxiliary result. . Let E be a vector lattice. Then the relation ⊑ is a partial order on E. Moreover for every x ∈ E the set F x , partially ordered by ⊑, is a Boolean algebra with the least element 0, maximal element x, and the Boolean operations z ∪ y := (z + ∨ y + ) − (z − ∨ y − ), z ∩ y := (z + ∧ y + ) − (z − ∧ y − ), The next theorem is the main result of this section. It shows that the minimal extension of an atomic orthogonally additive map is an atomic operator as well. Proof. By Theorem 4.5, the operator T D is well defined and T D ∈ OA + (E, F ). We show that T D is an atomic operator. Take an order projection π ∈ B(E) and x ∈ E. First we show that D := {T y : y ∈ F x ∩ D} is an upward directed set. Indeed, take y, z ∈ D. By Lemma 4.6 there exists u ∈ F x such that u = z ∩ y. Then y ′ := y − u ∈ F x ∩ D and y ′ ⊥ z. Hence v := y ′ + z ∈ F x ∩ D, y ⊑ v and z ⊑ v. Thus, for every y, z ∈ F x ∩ D there exists v. Taking into account that the relation z ⊑ x implies that T z ≤ T x we deduce that for every y, z ∈ F x ∩ D there exists v ∈ F x ∩ D such that T y ≤ T v and T z ≤ T v. Now, T D π(x) = sup{T π(y) : y ∈ F x ∩ D} = sup{Φ(π)T (y) : y ∈ F x ∩ D}.
Taking into account that the set {T (y) : y ∈ F x ∩ D} is upward directed and that Φ(π) is an order continuous positive linear operator for any order projection π ∈ B(E) we get T D π(x) = sup{Φ(π)T (y) : y ∈ F x ∩ D}