Generalized Priestley Quasi-Orders

We introduce generalized Priestley quasi-orders and show that subalgebras of bounded distributive meet-semilattices are dually characterized by means of generalized Priestley quasi-orders. This generalizes the well-known characterization of subalgebras of bounded distributive lattices by means of Priestley quasi-orders (Adams, Algebra Univers 3:216–228, 1973; Cignoli et al., Order 8(3):299–315, 1991; Schmid, Order 19(1):11–34, 2002). We also introduce Vietoris families and prove that homomorphic images of bounded distributive meet-semilattices are dually characterized by Vietoris families. We show that this generalizes the well-known characterization (Priestley, Proc Lond Math Soc 24(3):507–530, 1972) of homomorphic images of a bounded distributive lattice by means of closed subsets of its Priestley space. We also show how to modify the notions of generalized Priestley quasi-order and Vietoris family to obtain the dual characterizations of subalgebras and homomorphic images of bounded implicative semilattices, which generalize the well-known dual characterizations of subalgebras and homomorphic images of Heyting algebras (Esakia, Sov Math Dokl 15:147–151, 1974).     

Funayama’s theorem revisited

Funayama’s theorem states that there is an embedding e of a lattice L into a complete Boolean algebra B such that e preserves all existing joins and meets in L iff L satisfies the join infinite distributive law (JID) and the meet infinite distributive law (MID). More generally, there is a lattice embedding e: L → B preserving all existing joins in L iff L satisfies (JID), and there is a lattice embedding e: L → B preserving all existing meets in L iff L satisfies (MID). Funayama’s original proof is quite involved. There are two more accessible proofs in case L is complete. One was given by Grätzer by means of free Boolean extensions and MacNeille completions, and the other by Johnstone by means of nuclei and Booleanization. We show that Grätzer’s proof has an obvious generalization to the non-complete case, and that in the complete case the complete Boolean algebras produced by Grätzer and Johnstone are isomorphic. We prove that in the non-complete case, the class of lattices satisfying (JID) properly contains the class of Heyting algebras, and we characterize lattices satisfying (JID) and (MID) by means of their Priestley duals. Utilizing duality theory, we give alternative proofs of Funayama’s theorem and of the isomorphism between the complete Boolean algebras produced by Grätzer and Johnstone. We also show that unlike Grätzer’s proof, there is no obvious way to generalize Johnstone’s proof to the non-complete case.     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

Canonical extensions of bounded archimedean vector lattices

Canonical extensions of Boolean algebras with operators were introduced in the seminal paper of Jónsson and Tarski. The two defining properties of canonical extensions are the density and compactness axioms. While the density axiom can be extended to the setting of vector lattices of continuous real-valued functions, the compactness axiom requires appropriate weakening. This provides a motivation for defining the concept of canonical extension in the category \(\varvec{ bav }\) of bounded archimedean vector lattices. We prove existence and uniqueness theorems for canonical extensions in \(\varvec{ bav }\). We show that the underlying vector lattice of the canonical extension of \(A\in \varvec{ bav }\) is isomorphic to the vector lattice of all bounded real-valued functions on the Yosida space of A, and give an intrinsic characterization of those \(B \in \varvec{ bav }\) that arise as the canonical extension of some \(A \in \varvec{ bav }\).     

A proof of the Anderson–Badawi $$ \mathbf{rad}\varvec{(I)}^{\varvec{n}} \varvec{\subseteq } {\varvec{I}}$$ formula for $$\varvec{n}$$n-absorbing ideals

In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.     

Stone duality and Gleason covers through de Vries duality

We introduce zero-dimensional de Vries algebras and show that the category of zero-dimensional de Vries algebras is dually equivalent to the category of Stone spaces. This shows that Stone duality can be obtained as a particular case of de Vries duality. We also introduce extremally disconnected de Vries algebras and show that the category of extremally disconnected de Vries algebras is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. As a result, we give a simple construction of the Gleason cover of a compact Hausdorff space by means of de Vries duality. We also discuss the insight that Stone duality provides in better understanding of de Vries duality.     

Varieties of two-dimensional cylindric algebras II

In [2] we investigated the lattice (Df₂) of all subvarieties of the variety Df₂ of two-dimensional diagonal free cylindric algebras. In the present paper we investigate the lattice (CA₂) of all subvarieties of the variety CA₂ of two-dimensional cylindric algebras. We prove that the cardinality of (CA₂) is that of the continuum, give a criterion for a subvariety of CA₂ to be locally finite, and describe the only pre locally nite subvariety of CA2. We also characterize nitely generated subvarieties of CA2 by describing all fteen pre nitely generated subvarieties of CA₂. Finally, we give a rough picture of (CA₂), and investigate algebraic properties preserved and reected by the reduct functors .     

New catalysts for Suzuki-Miyaura coupling reactions of heteroatom-substituted heteroaryl chlorides

The new air-stable PdCl2[PR2(Ph-R')]2 complexes, readily prepared from commercial reagents, exhibit unique efficiency as catalysts for the Suzuki-Miyaura coupling reactions of a variety of heteroatom-substituted heteroaryl chlorides with a diverse range of aryl/heteroaryl boronic acids. The coupling reactions catalyzed by the new complexes exhibit high product yields (88-99%) and high catalyst turnover numbers (up to 10,000 TON).     

Copper-catalyzed N-arylation of sulfonamides with aryl bromides under mild conditions

This Letter describes a copper catalyzed sulfonamide coupling reaction with aryl bromides to form N-aryl sulfonamides under mild conditions, including the first examples of Cu-catalyzed sulfonamide coupling at room temperature. The reaction protocol tolerates a broad range of substrates including a variety of primary and secondary sulfonamides and challenging heteroaryl bromides such as 2-bromothiazole.     

Preservation of Sahlqvist fixed point equations in completions of relativized fixed point Boolean algebras with operators

We define Sahlqvist fixed point equations and relativized fixed point Boolean algebras with operators (relativized fixed point BAOs). We show that every Sahlqvist fixed point equation is preserved under completions of conjugated relativized fixed point BAOs. This extends the result of Givant and Venema (1999) to the setting of relativized fixed point BAOs.