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Dedekind cuts in C(X)

Nicolae Dăneţ (2011)

Banach Center Publications

The aim of this paper is to show that every Hausdorff continuous interval-valued function on a completely regular topological space X corresponds to a Dedekind cut in C(X) and conversely.

Denotational aspects of untyped normalization by evaluation

Andrzej Filinski, Henning Korsholm Rohde (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We show that the standard normalization-by-evaluation construction for the simply-typed λ β η -calculus has a natural counterpart for the untyped λ β -calculus, with the central type-indexed logical relation replaced by a “recursively defined” invariant relation, in the style of Pitts. In fact, the construction can be seen as generalizing a computational-adequacy argument for an untyped, call-by-name language to normalization instead of evaluation.In the untyped setting, not all terms have normal forms,...

Denotational aspects of untyped normalization by evaluation

Andrzej Filinski, Henning Korsholm Rohde (2010)

RAIRO - Theoretical Informatics and Applications

We show that the standard normalization-by-evaluation construction for the simply-typed λβη-calculus has a natural counterpart for the untyped λβ-calculus, with the central type-indexed logical relation replaced by a “recursively defined” invariant relation, in the style of Pitts. In fact, the construction can be seen as generalizing a computational-adequacy argument for an untyped, call-by-name language to normalization instead of evaluation.In the untyped setting, not all terms have normal...

Derivations and Translations on Trellises

Shashirekha B. Rai, S. Parameshwara Bhatta (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

G. Szász, J. Szendrei, K. Iseki and J. Nieminen have made an extensive study of derivations and translations on lattices. In this paper, the concepts of meet-translations and derivations have been studied in trellises (also called weakly associative lattices or WA-lattices) and several results in lattices are extended to trellises. The main theorem of this paper, namely, that every derivatrion of a trellis is a meet-translation, is proved without using associativity and it generalizes a well-known...

Determinant of Some Matrices of Field Elements

Yatsuka Nakamura (2006)

Formalized Mathematics

Here, we present determinants of some square matrices of field elements. First, the determinat of 2 * 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that the determinant of a matrix is the same as the determinant of its transpose.

Direct product decompositions of infinitely distributive lattices

Ján Jakubík (2000)

Mathematica Bohemica

Let α be an infinite cardinal. Let 𝒯 α be the class of all lattices which are conditionally α -complete and infinitely distributive. We denote by 𝒯 σ ' the class of all lattices X such that X is infinitely distributive, σ -complete and has the least element. In this paper we deal with direct factors of lattices belonging to 𝒯 α . As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class 𝒯 σ ' .

Direct summands of Goldie extending elements in modular lattices

Rupal Shroff (2022)

Mathematica Bohemica

In this paper some results on direct summands of Goldie extending elements are studied in a modular lattice. An element a of a lattice L with 0 is said to be a Goldie extending element if and only if for every b ≤ a there exists a direct summand c of a such that b ∧ c is essential in both b and c . Some characterizations of decomposition of a Goldie extending element in a modular lattice are obtained.

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