W is the category of archimedean `-groups with distinguished weak order unit. For G ∈W, we have t... more W is the category of archimedean `-groups with distinguished weak order unit. For G ∈W, we have the contravariantly functorial Yosida space YG. For an embedding G ≤ H, the resulting YG← YH is surjective; when this is one-to-one, we write “YH = YG”. This is the case with the divisible hull G ≤ dG, where, always, YdG = YG; however for the vector lattice hull G ≤ vG, we frequently have YvG 6= YG. Theorem. A compact space X is quasi-F if and only if: ∀G ∈W with YG = X , also YvG = X . (“quasi-F” means each dense cozero set is C∗-embedded.)
An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We s... more An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-algebras. We outline the general theory, and then apply it to three particular MV-convergences and their corresponding Cauchy completions. The Cauchy completion arising from order convergence coincides with the Dedekind-MacNeille completion of an MV-algebra. The Cauchy completion arising from polar convergence allows a tidy proof of the existence and uniqueness of the lateral completion of an MV-algebra. And the Cauchy completion arising from α-convergence gives rise to the cut completion of an MV-algebra.
summary:We describe the extension of the multiplication on a not-necessarily-discrete topological... more summary:We describe the extension of the multiplication on a not-necessarily-discrete topological monoid to its flow compactification. We offer two applications. The first is a nondiscrete version of Hindman’s Theorem, and the second is a characterization of the projective minimal and elementary flows in terms of idempotents of the flow compactification of the monoid
In analysis, truncation is the operation of replacing a nonnegative real-valued function a (x) by... more In analysis, truncation is the operation of replacing a nonnegative real-valued function a (x) by its pointwise meet a (x) ∧ 1 with the constant $1$ function. A vector lattice A is said to be closed under truncation if a ∧ 1 ∈ A for all a ∈ A+. Note that A need notcontain 1 itself.Truncation is fundamental to analysis. To give only one example, Lebesgue integration generalizes beautifully to any vector lattice of real-valued functions on a set X, provided the vector lattice is closed under truncation. But vector lattices lacking this property may have integrals which cannot be represented by any measure on X. Nevertheless, when the integral is formulated in a context broader than RX, for example in pointfree analysis, the question oftruncation inevitably arises.What is truncation, or more properly, what are its essential properties? In this paper we answer this question by providing the appropriate axiomatization, and then go on to present several representation theorems. The first ...
Abstract. We show that the 2-crown is not coproductive, which is to say that the class of those b... more Abstract. We show that the 2-crown is not coproductive, which is to say that the class of those bounded distributive lattices whose Priestley spaces lack any copy of the 2-crown is not productive. We do this by rst exhibiting a general construction to handle questions of this sort. We then use a particular instance of this constrution, along with some of the combinatorial features of projective planes, to show that the 2-crown is not coproductive. 1.
We prove that, roughly speaking, the •-completion G i " is the least extension of a lattice ... more We prove that, roughly speaking, the •-completion G i " is the least extension of a lattice ordered group (&group) G in which G is large and which is order closed in every extension i which it (G ia) is large. Numerous related completion results are proved: in addition, we obtain fairly detailed structural descriptions ofG i • in the cases when G is completely distributive, archimedean or strongly projectable. The study of •-congergence was begun by Papangelou [27], carried forward in Ellis ' thesis [22], and culminated in a paper of Madell [25]. Ball and Davis [10] used the general Cauchy completion techniques of [2] to prove the existence and uniqueness of the c•'-completion G i". In addition, many of these ideas have been developed in the broader context of distribu-tive lattices in [6]. This paper, whose purpose is to explicate the structure of G i•, is motivated by the opinion that •-completeness is a natural and important lattice property with interesting c...
For Tychonof\text{}f $X$ and $\alpha$ an infinite cardinal, let $\alpha \operatorname{def} X := $... more For Tychonof\text{}f $X$ and $\alpha$ an infinite cardinal, let $\alpha \operatorname{def} X := $ the minimum number of $\alpha $\,cozero-sets of the Cech-Stone compactification which intersect to $X$ (generalizing $\Bbb R$-defect), and let $\operatorname{rt} X := \min _\alpha \max (\alpha , \alpha \operatorname{def} X)$. Give $C(X)$ the compact-open topology. It is shown that $\tau C(X)\leq n\chi C(X) \leq \operatorname{rt}X=\max (L(X),L(X) \operatorname{def} X)$, where: $\tau$ is tightness; $n\chi$ is the network character; $L(X)$ is the Lindel"{o}f number. For example, it follows that, for $X$ Cech-complete, $\tau C(X)=L(X)$. The (apparently new) cardinal functions $n\chi C$ and $\operatorname{rt}$ are compared with several others.
We show that prohibiting a combinatorial tree in the Priestley duals deter- mines an axiomatizabl... more We show that prohibiting a combinatorial tree in the Priestley duals deter- mines an axiomatizable class of distributive lattices. On the other hand, prohibiting n-crowns withn ≥ 3 does not. Given what is known about the diamond, this is an- other strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.
The normal, or Dedekind-MacNeille, completion δ(L) of a distributive lattice L need not be distri... more The normal, or Dedekind-MacNeille, completion δ(L) of a distributive lattice L need not be distributive. However, δ(L) does contain a largest distributive sublattice β(L) containing L, and δ(L) is distributive if and only if β(L) is complete if and only if δ(L) = β(L). In light of these facts, it may come as a surprise to learn that β(L) was developed (in [1]) for reasons having nothing to do with distributivity. In fact, the cuts of β(L) can be readily identified as those having the property we here term exactness. This provides a useful criterion for testing whether the normal completion of a given lattice is distributive. We illustrate the utility of this criterion by providing a simple demonstration that the normal completion of a Heyting algebra is distributive. We prove these facts by simple arguments from first princples, and then bring out the geometry of the situation by developing the construct in Priestley spaces. While the elements of L appear as clopen up-sets of the (o...
Les auteurs presentent une formule du premier ordre caracterisant les treillis distributifs L don... more Les auteurs presentent une formule du premier ordre caracterisant les treillis distributifs L dont les espaces de Priestley P(L) ne contiennent aucune copie d'une foret finie T. Pour des algebres de Heyting L le fait qu'il n'y ait pas d'ordre fini T dans P(L) est caracterise par des equations ssi T est un arbre. Ils donnent une condition qui caracterise les treillis distributifs dont les espaces de Priestley ne contiennent aucune copie d'une foret finie avec un seul point additionnel a la base.
W is the category of archimedean `-groups with distinguished weak order unit. For G ∈W, we have t... more W is the category of archimedean `-groups with distinguished weak order unit. For G ∈W, we have the contravariantly functorial Yosida space YG. For an embedding G ≤ H, the resulting YG← YH is surjective; when this is one-to-one, we write “YH = YG”. This is the case with the divisible hull G ≤ dG, where, always, YdG = YG; however for the vector lattice hull G ≤ vG, we frequently have YvG 6= YG. Theorem. A compact space X is quasi-F if and only if: ∀G ∈W with YG = X , also YvG = X . (“quasi-F” means each dense cozero set is C∗-embedded.)
An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We s... more An MV-convergence is a convergence on an MV-algebra which renders the operations continuous. We show that such convergences on a given MV-algebra A are exactly the restrictions of the bounded -convergences on the abelian -group in which A appears as the unit interval. Thus the theory of -convergence and Cauchy structures transfers to MV-algebras. We outline the general theory, and then apply it to three particular MV-convergences and their corresponding Cauchy completions. The Cauchy completion arising from order convergence coincides with the Dedekind-MacNeille completion of an MV-algebra. The Cauchy completion arising from polar convergence allows a tidy proof of the existence and uniqueness of the lateral completion of an MV-algebra. And the Cauchy completion arising from α-convergence gives rise to the cut completion of an MV-algebra.
summary:We describe the extension of the multiplication on a not-necessarily-discrete topological... more summary:We describe the extension of the multiplication on a not-necessarily-discrete topological monoid to its flow compactification. We offer two applications. The first is a nondiscrete version of Hindman’s Theorem, and the second is a characterization of the projective minimal and elementary flows in terms of idempotents of the flow compactification of the monoid
In analysis, truncation is the operation of replacing a nonnegative real-valued function a (x) by... more In analysis, truncation is the operation of replacing a nonnegative real-valued function a (x) by its pointwise meet a (x) ∧ 1 with the constant $1$ function. A vector lattice A is said to be closed under truncation if a ∧ 1 ∈ A for all a ∈ A+. Note that A need notcontain 1 itself.Truncation is fundamental to analysis. To give only one example, Lebesgue integration generalizes beautifully to any vector lattice of real-valued functions on a set X, provided the vector lattice is closed under truncation. But vector lattices lacking this property may have integrals which cannot be represented by any measure on X. Nevertheless, when the integral is formulated in a context broader than RX, for example in pointfree analysis, the question oftruncation inevitably arises.What is truncation, or more properly, what are its essential properties? In this paper we answer this question by providing the appropriate axiomatization, and then go on to present several representation theorems. The first ...
Abstract. We show that the 2-crown is not coproductive, which is to say that the class of those b... more Abstract. We show that the 2-crown is not coproductive, which is to say that the class of those bounded distributive lattices whose Priestley spaces lack any copy of the 2-crown is not productive. We do this by rst exhibiting a general construction to handle questions of this sort. We then use a particular instance of this constrution, along with some of the combinatorial features of projective planes, to show that the 2-crown is not coproductive. 1.
We prove that, roughly speaking, the •-completion G i " is the least extension of a lattice ... more We prove that, roughly speaking, the •-completion G i " is the least extension of a lattice ordered group (&group) G in which G is large and which is order closed in every extension i which it (G ia) is large. Numerous related completion results are proved: in addition, we obtain fairly detailed structural descriptions ofG i • in the cases when G is completely distributive, archimedean or strongly projectable. The study of •-congergence was begun by Papangelou [27], carried forward in Ellis ' thesis [22], and culminated in a paper of Madell [25]. Ball and Davis [10] used the general Cauchy completion techniques of [2] to prove the existence and uniqueness of the c•'-completion G i". In addition, many of these ideas have been developed in the broader context of distribu-tive lattices in [6]. This paper, whose purpose is to explicate the structure of G i•, is motivated by the opinion that •-completeness is a natural and important lattice property with interesting c...
For Tychonof\text{}f $X$ and $\alpha$ an infinite cardinal, let $\alpha \operatorname{def} X := $... more For Tychonof\text{}f $X$ and $\alpha$ an infinite cardinal, let $\alpha \operatorname{def} X := $ the minimum number of $\alpha $\,cozero-sets of the Cech-Stone compactification which intersect to $X$ (generalizing $\Bbb R$-defect), and let $\operatorname{rt} X := \min _\alpha \max (\alpha , \alpha \operatorname{def} X)$. Give $C(X)$ the compact-open topology. It is shown that $\tau C(X)\leq n\chi C(X) \leq \operatorname{rt}X=\max (L(X),L(X) \operatorname{def} X)$, where: $\tau$ is tightness; $n\chi$ is the network character; $L(X)$ is the Lindel"{o}f number. For example, it follows that, for $X$ Cech-complete, $\tau C(X)=L(X)$. The (apparently new) cardinal functions $n\chi C$ and $\operatorname{rt}$ are compared with several others.
We show that prohibiting a combinatorial tree in the Priestley duals deter- mines an axiomatizabl... more We show that prohibiting a combinatorial tree in the Priestley duals deter- mines an axiomatizable class of distributive lattices. On the other hand, prohibiting n-crowns withn ≥ 3 does not. Given what is known about the diamond, this is an- other strong indication that this fact characterizes combinatorial trees. We also discuss varieties of 2-Heyting algebras in this context.
The normal, or Dedekind-MacNeille, completion δ(L) of a distributive lattice L need not be distri... more The normal, or Dedekind-MacNeille, completion δ(L) of a distributive lattice L need not be distributive. However, δ(L) does contain a largest distributive sublattice β(L) containing L, and δ(L) is distributive if and only if β(L) is complete if and only if δ(L) = β(L). In light of these facts, it may come as a surprise to learn that β(L) was developed (in [1]) for reasons having nothing to do with distributivity. In fact, the cuts of β(L) can be readily identified as those having the property we here term exactness. This provides a useful criterion for testing whether the normal completion of a given lattice is distributive. We illustrate the utility of this criterion by providing a simple demonstration that the normal completion of a Heyting algebra is distributive. We prove these facts by simple arguments from first princples, and then bring out the geometry of the situation by developing the construct in Priestley spaces. While the elements of L appear as clopen up-sets of the (o...
Les auteurs presentent une formule du premier ordre caracterisant les treillis distributifs L don... more Les auteurs presentent une formule du premier ordre caracterisant les treillis distributifs L dont les espaces de Priestley P(L) ne contiennent aucune copie d'une foret finie T. Pour des algebres de Heyting L le fait qu'il n'y ait pas d'ordre fini T dans P(L) est caracterise par des equations ssi T est un arbre. Ils donnent une condition qui caracterise les treillis distributifs dont les espaces de Priestley ne contiennent aucune copie d'une foret finie avec un seul point additionnel a la base.
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