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ABSTRACT An Archimedean semiprime f-algebra A for which I-A boolean AND f epsilon A for all f epsilon A is called a Stone f-algebra, where I-A is the identity operator on A. Moreover, an operator T between two Stone f-algebras A and B is... more
ABSTRACT An Archimedean semiprime f-algebra A for which I-A boolean AND f epsilon A for all f epsilon A is called a Stone f-algebra, where I-A is the identity operator on A. Moreover, an operator T between two Stone f-algebras A and B is said to be contractive if f epsilon A and 0 <= f <= I-A imply 0 <= Tf <= I-B. The set kappa (A, B) of all positive contractive operators from A into B is a convex set. This paper characterizes extreme points in kappa (A, B). In this regard, we prove that T epsilon kappa (A, B) is extreme if and only if T is an algebra homomorphism. Furthermore, we show that, T epsilon kappa (A, B) is extreme if and only if T is a Stone operator, meaning that, T (I-A boolean AND f)= I-B boolean AND Tf for all f epsilon A.
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Recently, Wickstead investigated the long-standing problem of adding an identity to a non-unital Banach lattice algebra. In this regard, he proved that, in the category whose objects are unital Banach lattice algebras and morphisms are... more
Recently, Wickstead investigated the long-standing problem of adding an identity to a non-unital Banach lattice algebra. In this regard, he proved that, in the category whose objects are unital Banach lattice algebras and morphisms are identity preserving algebra and lattice homomorphisms, there is no reflection for $$c_{0}$$ in which $$c_{0}$$ can be embedded as an algebra and order ideal. The main purpose of this note is to describe an alternative category in which a suitable reflection of $$c_{0}$$ can be located, producing a satisfactory lattice unitization of $$c_{0}$$ with $$c_{0}$$ as an algebra and order ideal.
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We introduce and study the notion of orthosymmetric Archimedean-valued vector lattices as a generalization of finite-dimensional Euclidean inner spaces. A special attention has been paid to linear operators on these spaces.
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... of algebraic \linebreak[1] order bounded disjointness preserving operators} \author{Karim Boulabiar} \address{IPEST, Universit\'{e} de Carthage, BP 51, 2070-La Marsa, Tunisia} \email{karim.boulabiar@ipest ... bibitem {BBH}K.... more
... of algebraic \linebreak[1] order bounded disjointness preserving operators} \author{Karim Boulabiar} \address{IPEST, Universit\'{e} de Carthage, BP 51, 2070-La Marsa, Tunisia} \email{karim.boulabiar@ipest ... bibitem {BBH}K. Boulabiar, G. Buskes, M. Henriksen, A ...
summary:Let $A$ be a uniformly complete almost $f$-algebra and a natural number $p\in\{3,4,\dots \}$. Then $\Pi_{p}(A)= \{a_{1}\dots a_{p}; a_{k}\in A, k=1,\dots ,p\}$ is a uniformly complete semiprime $f$-algebra under the ordering and... more
summary:Let $A$ be a uniformly complete almost $f$-algebra and a natural number $p\in\{3,4,\dots \}$. Then $\Pi_{p}(A)= \{a_{1}\dots a_{p}; a_{k}\in A, k=1,\dots ,p\}$ is a uniformly complete semiprime $f$-algebra under the ordering and multiplication inherited from $A$ with $\Sigma_{p}(A)=\{a^{p}; 0\leq a\in A\}$ as positive cone
summary:This work provides an evaluating complete description of positive homomorphisms on an arbitrary algebra of real-valued functions
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Abstract. Let Abeauniformlycompletealmost f-algebraandanaturalnumber p ∈ {3,4,...}. Then Πp(A) = {a1... ap; ak ∈ A, k = 1,..., p} is a uniformly completesemiprime f-algebraundertheorderingandmultiplicationinheritedfrom Awith Σp(A)={a p; 0... more
Abstract. Let Abeauniformlycompletealmost f-algebraandanaturalnumber p ∈ {3,4,...}. Then Πp(A) = {a1... ap; ak ∈ A, k = 1,..., p} is a uniformly completesemiprime f-algebraundertheorderingandmultiplicationinheritedfrom Awith Σp(A)={a p; 0 ≤ a ∈ A}aspositivecone.
Let $L$ be a (non necessarily unital) truncated vector lattice of real-valued functions on a nonempty set $X$. A nonzero linear functional $\psi$ on $L$ is called a truncation homomorphism if it preserves truncation, i.e.,% \[ \psi\left(... more
Let $L$ be a (non necessarily unital) truncated vector lattice of real-valued functions on a nonempty set $X$. A nonzero linear functional $\psi$ on $L$ is called a truncation homomorphism if it preserves truncation, i.e.,% \[ \psi\left( f\wedge\mathbf{1}_{X}\right) =\min\left\{ \psi\left( f\right) ,1\right\} \text{ for all }f\in L. \] We prove that a linear functional $\psi$ on $L$ is a truncation homomorphism if and only if $\psi$ is a lattice homomorphism and% \[ \sup\left\{ \psi\left( f\right) :f\leq\mathbf{1}_{X}\right\} =1. \] This allows us to prove different evaluating characterizations of truncation homomorphisms. In this regard, a special attention is paid to the continuous case and various results from the existing literature are generalized.
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This paper is mainly concerned with a representation theorem for almost /-algebras.
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An order bounded disjointness preserving operator T T on an Archimedean vector lattice is algebraic if and only if the restriction of T n ! T^{n!} to the vector sublattice generated by the range of T m T^{m} is strongly diagonal, where n... more
An order bounded disjointness preserving operator T T on an Archimedean vector lattice is algebraic if and only if the restriction of T n ! T^{n!} to the vector sublattice generated by the range of T m T^{m} is strongly diagonal, where n n is the degree of the minimal polynomial of T T and m m is its ‘valuation’.
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ABSTRACT Let A be an Archimedean f-algebra with unit element e, B be an Archimedean semiprime f-algebra and T:A→B be a lattice (or Riesz) homomorphism. The main purpose of this paper is to show, in a straightforward and elementary manner,... more
ABSTRACT Let A be an Archimedean f-algebra with unit element e, B be an Archimedean semiprime f-algebra and T:A→B be a lattice (or Riesz) homomorphism. The main purpose of this paper is to show, in a straightforward and elementary manner, that the range R(T) of T is an f-subalgebra of B if and only if Te is idempotent in B.
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We constructively prove (i.e., in ZF set theory) a decomposition theorem for certain order bounded disjointness preserving operators between any two Riesz spaces, real or complex, in terms of the absolute value of another order bounded... more
We constructively prove (i.e., in ZF set theory) a decomposition theorem for certain order bounded disjointness preserving operators between any two Riesz spaces, real or complex, in terms of the absolute value of another order bounded disjointness preserving operator. In this way, we constructively generalize results by Abramovich, Arensen and Kitover (1992), Grobler and Huijsmans (1997), Hart (1985), Kutateladze, and Meyer-Nieberg (1991).
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Let G be an abelian ℓ-group with a strong order unit u > 0. We call G u-clean after Hager, Kimber, and McGovern if every element of G can be written as a sum of a strong order unit of G and a u-component of G. We prove that G is... more
Let G be an abelian ℓ-group with a strong order unit u > 0. We call G u-clean after Hager, Kimber, and McGovern if every element of G can be written as a sum of a strong order unit of G and a u-component of G. We prove that G is u-clean if and only if u-components of G can be lifted modulo any ℓ-ideal of G. Moreover, we introduce a notion of u-suitable ℓ-groups (as a natural analogue of the corresponding notion in Ring Theory) and we prove that the ℓ-group G is u-clean when and only when it is u-suitable. Also, we show that if E is a vector lattice, then E is u-clean if and only if the space of all u-step functions of E is u-uniformly dense in E. As applications, we will generalize a result by Banaschewski on maximal ℓ-ideals of an archimedean bounded f-algebras to the non-archimedean case. We also extend a result by Miers on polynomially ideal C(X)-type algebras to the more general setting of bounded f-algebras.
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Abstract Let C (X) denote the lattice-ordered algebra of all real-valued continuous functions on a topological space X. This paper discusses in Zermelo-Fraenkel Set Theory the equivalence on C (X) between algebra homomorphisms, lattice... more
Abstract Let C (X) denote the lattice-ordered algebra of all real-valued continuous functions on a topological space X. This paper discusses in Zermelo-Fraenkel Set Theory the equivalence on C (X) between algebra homomorphisms, lattice homo- morphisms, and point evaluations.
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Abstract There is a mistake in the proof of Lemma 4.2 of Ben Amor et al. (2014), and the corrected proof is provided in this Corrigendum note.
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AbstractRecently, Ball defined a truncated $${\ell}$$ℓ-group to be an $${\ell}$$ℓ-group G along with a truncation. We constructively prove that if G is a truncated $${\ell}$$ℓ-group, then the direct sum $${G \oplus \mathbb{Q}}$$G⊕Q is... more
AbstractRecently, Ball defined a truncated $${\ell}$$ℓ-group to be an $${\ell}$$ℓ-group G along with a truncation. We constructively prove that if G is a truncated $${\ell}$$ℓ-group, then the direct sum $${G \oplus \mathbb{Q}}$$G⊕Q is equipped with a structure of an $${\ell}$$ℓ-group with weak unit the rational number 1. As a simple consequence, we get a description of the truncated $${\ell}$$ℓ- group obtained by Ball via representation theory. On the other hand, we derive some characterizations of truncation morphisms as defined by Ball himself. In particular, we show that the group homomorphism $${f : G \rightarrow H}$$f:G→H is a truncation morphism if and only its natural extension $${f^*}$$f∗ from $${G \oplus \mathbb{Q}}$$G⊕Q into $${H \oplus \mathbb{Q}}$$H⊕Q is an $${\ell}$$ℓ-homomorphism.
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Let A be an Archimedean Riesz algebra with a positive unit element e. An element f ∈ A is said to be algebraic if P (f) = 0 for some non-zero polynomial P with real coefficients. Moreover, f is called an e-step function in A if there... more
Let A be an Archimedean Riesz algebra with a positive unit element e. An element f ∈ A is said to be algebraic if P (f) = 0 for some non-zero polynomial P with real coefficients. Moreover, f is called an e-step function in A if there exist pairwise disjoint components p1, . . . , pn of e and real numbers α1, . . . , αn such that
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By a composition operator between two f-algebras we mean a positive algebra homomorphism. This paper intends to give a systematic study of such operators. A particular attention is paid to their connection with separating regular... more
By a composition operator between two f-algebras we mean a positive algebra homomorphism. This paper intends to give a systematic study of such operators. A particular attention is paid to their connection with separating regular operators as well as to their global behavior in the module of regular operators. The paper ends with some open problems.
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A F-algebra is an Archimedean lattice ordered algebra with a weak order unit. Let be A and B be F -algebras and let T be a separating linear map from A into B, that is, T is a linear map such that T(f)T(g) = 0 in B whenever fg = 0 in A.... more
A F-algebra is an Archimedean lattice ordered algebra with a weak order unit. Let be A and B be F -algebras and let T be a separating linear map from A into B, that is, T is a linear map such that T(f)T(g) = 0 in B whenever fg = 0 in A. It is proven by an order theoretical and purely algebraic method that there exist a 'weight' element w in B and a positive algebra homomorphism S from A into the maximal ring of quotients Q(B) of B such that T(f) = wS(f) holds for all f in A. Both real and complex cases are considered. This result generalizes the following theorem proved by W. Arendt in his paper [Spectral properties of Lamperti operators, Indiana Univ. J. Math., 32 (1983), 199-215]. Let C(X) and C(Y) be the F-algebras of all scalar-valued continuous functions on compact Hausdorff topological spaces X and Y, respectively. Then for every separating linear map T from C(X) into C(Y) there exist a 'weight' function w in C(Y) and a function h from Y into X (continuous on the cozero set of w) such that T(f)(y) = w(y)f(h(y)) holds for all f in C(X) and y in Y.
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La notion de groupe a ete introduite pour la premiere fois au debut du dix-neuvieme siecle. A cette epoque, elle intervient dans les travaux d’Evariste Galois sur les equations algebriques, sous forme de groupes de permutations des... more
La notion de groupe a ete introduite pour la premiere fois au debut du dix-neuvieme siecle. A cette epoque, elle intervient dans les travaux d’Evariste Galois sur les equations algebriques, sous forme de groupes de permutations des racines de ces equations. Presque au meme moment, les groupes commencent a jouer un role en geometrie, notamment, des groupes symetriques de polygone et de polyedres reguliers. C’est a partir de cette double origine,algebrique et geometrique, qu’a ete concue, vers la fin du dix- neuvieme siecle, la notion abstraite de groupe et que, petit a petit, a ete construite la theorie de groupes. Dans la theorie de groupe, une place importante a ete accordee a l’etude de la structure des groupes finis, compte tenu des nombreuses interpretations concretes qui peuvent en etre donnees. C’est precisement dans ce carde que se place ce cours d’algebre, dans lequel ont ete traites les groupes monogenes, les groupes symetriques et la notion d’un groupe operant sur un ensemble.
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... This completes the proof. ✷ Page 4. 388 KARIM BOULABIAR Notice that, as f-rings are almost f-rings, Proposition 1 above holds also for Archimedean f -rings. ... The next example shows that Proposition 1 fails to be true in an... more
... This completes the proof. ✷ Page 4. 388 KARIM BOULABIAR Notice that, as f-rings are almost f-rings, Proposition 1 above holds also for Archimedean f -rings. ... The next example shows that Proposition 1 fails to be true in an arbitrary Archi-medean l-ring. ...
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We wrote a survey [18] on lattice ordered algebras five years ago. Why do we return to f-algebras once more? We hasten to say that there is only little overlap between the current paper and that previous survey.We have three purposes for... more
We wrote a survey [18] on lattice ordered algebras five years ago. Why do we return to f-algebras once more? We hasten to say that there is only little overlap between the current paper and that previous survey.We have three purposes for the present paper. In our previous survey we remarked that one aspect that we did not discuss, while
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ABSTRACT Let G be an archimedean \({\ell}\) -group. By an f-representation of G we mean an orthomorphism-valued group homomorphism S on G for which (Sf)g = (Sg)f for all \({f, g \in G}\) . We prove that the set \({\mathfrak{Rep}(G)}\) of... more
ABSTRACT Let G be an archimedean \({\ell}\) -group. By an f-representation of G we mean an orthomorphism-valued group homomorphism S on G for which (Sf)g = (Sg)f for all \({f, g \in G}\) . We prove that the set \({\mathfrak{Rep}(G)}\) of all f-representations in G is an archimedean \({\ell}\) -group with respect to pointwise addition and ordering. Furthermore, we define an orthoproduct on G to be a bilinear map on G which is an orthomorphism in each variable separately. It turns out that the set \({\mathfrak{Opro}(G)}\) is an archimedean \({\ell}\) -group G with the set \({\mathfrak{Mult}(G)}\) of f-multiplications in G as a positive cone. Moreover, we show that \({\mathfrak{Opro}(G)}\) and \({\mathfrak{Rep}(G)}\) are isomorphic as \({\ell}\) -groups. In spite of that, we get a representation theorem for f-multiplications in an \({\ell}\) -subgroup of an archimedean f-ring R with unit element. This allows us to find an example of an archimedean \({\ell}\) -group with no nontrivial structure of an f-ring and another which cannot be a reduced f-ring.
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In this paper we study properties of bilinear maps of order bounded variation. Theorems of preservation of properties in passage to the triadjoint and the tensor product are presented.
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... preserving) H(1) = 1. K. Boulabiar (B) Département de Mathématiques, Faculté des Sciences de Tunis, Université Tunis-El Manar, 2092 Tunis, Tunisia e-mail: karim.boulabiar@ipest.rnu.tn F. Gdara Département de Mathématiques ...