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Abstract Motivated by recent work on the use of topological methods to study collections of rings between an integral domain and its quotient field, we examine spaces of subrings of a commutative ring, endowed with the Zariski or patch topologies. We introduce three notions to study such a space : patch bundles, patch presheaves and patch algebras. When is compact and Hausdorff, patch bundles give a way to approximate with topologically more tractable spaces, namely Stone spaces. Patch presheaves encode the space into stalks of a presheaf of rings over a Boolean algebra, thus giving a more geometrical setting for studying . To both objects, a patch bundle and a patch presheaf, we associate what we call a patch algebra, a commutative ring that efficiently realizes the rings in as factor rings, or even localizations, and whose structure reflects various properties of the rings in . 

Abstract An irreducible complete atomicomlof infinite height cannot be algebraic and have the covering property. However, modest departure from these conditions allows infinite-height examples. We use an extension of Kalmbach’s construction to the setting of infinite chains to provide an example of an infinite-height, irreducible, algebraicomlwith the 2-covering property, and Keller’s construction provides an example of an infinite-height, irreducible, completeomlthat has the covering property and is completely hereditarily atomic. Completely hereditarily atomicomlsgeneralize algebraicomls suitably to quantum predicate logic. 

Abstract On relational structures and on polymodal logics, we describe operations which preserve local tabularity. This provides new sufficient semantic and axiomatic conditions for local tabularity of a modal logic. The main results are the following. We show that local tabularity does not depend on reflexivity. Namely, given a class$$\mathcal {F}$$of frames, consider the class$$\mathcal {F}^{\mathrm {r}}$$of frames, where the reflexive closure operation was applied to each relation in every frame in$$\mathcal {F}$$. We show that if the logic of$$\mathcal {F}^{\mathrm {r}}$$is locally tabular, then the logic of$$\mathcal {F}$$is locally tabular as well. Then we consider the operation of sum on Kripke frames, where a family of frames-summands is indexed by elements of another frame. We show that if both the logic of indices and the logic of summands are locally tabular, then the logic of corresponding sums is also locally tabular. Finally, using the previous theorem, we describe an operation on logics that preserves local tabularity: we provide a set of formulas such that the extension of the fusion of two canonical locally tabular logics with these formulas is locally tabular. 

We show that the variety of monadic ortholattices is closed under MacNeille and canonical completions. In each case, the completion of L is obtained by forming an associated dual space X that is a monadic orthoframe. This is a set with an orthogonality relation and an additional binary relation satisfying certain conditions. For the MacNeille completion, X is formed from the non-zero elements of L, and for the canonical completion, X is formed from the proper filters of L. The corresponding completion of L is then obtained as the ortholattice of bi-orthogonally closed subsets of X with an additional operation defined through the binary relation of X. With the introduction of a suitable topology on an orthoframe, as was done by Goldblatt and Bimb´o, we obtain a dual adjunction between the categories of monadic ortholattices and monadic orthospaces. A restriction of this dual adjunction provides a dual equivalence.