Author: "Picado, Jorge" - Searchworks@Jio Institute Digital Library Search Results

Author: Picado, Jorge, Pultr, Aleš, Santana, Ana Paula, Series Editor, Neves, Júlio S., Series Editor, Viana, Marcelo, Series Editor, Serra de Oliveira, Maria Paula, Series Editor, Loja Fernandes, Rui, Series Editor, Clementino, Maria Manuel, editor, Facchini, Alberto, editor, and Gran, Marino, editor
Published: 2021
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Author: Erné, Marcel and Picado, Jorge
Subjects: Mathematics - Category Theory, Mathematics - Rings and Algebras, 06F05, 08A30, 03G05, 16D30, 20M12, 20N02, 16N80, 17A65
Abstract: A classical tensor product $A \,\otimes\, B$ of complete lattices $A$ and $B$, consisting of all down-sets in $A \times B$ that are join-closed in either coordinate, is isomorphic to the complete lattice $Gal(A,B)$ of Galois maps from $A$ to $B$, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (separately continuous maps or maps preserving restricted joins in the two components) into complete lattices. The appropriate ingredient for quantale constructions is here distributivity at the bottom, a generalization of pseudo\-complementedness. We show that the truncated tensor product of a complete lattice $B$ with itself becomes a quantale with the closure of the relation product as multiplication iff $B$ is pseudocomplemented, and the tensor product has a unit element iff $B$ is atomistic. The pseudocomplemented complete lattices form a semicategory in which the hom-set between two objects is their tensor product. The largest subcategory of that semicategory has as objects the atomic boolean complete lattices, which is equivalent to the category of sets and relations. More general results are obtained for closure spaces and posets.
Published: 2016

Author: Andrew Moshier, M., primary, Picado, Jorge, additional, and Pultr, Aleš, additional
Published: 2023
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Author: Gutiérrez García, Javier, Mozo Carollo, Imanol, Picado, Jorge, and Walters-Wayland, Joanne
Published: 2019
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Author: Arrieta, Igor, Picado, Jorge, and Pultr, Ales
Subjects: AXIOMS, TOPOLOGY
Abstract: A locale (frame) L has a largest spatial sublocale generated by the primes (spectrum points), the spatial part SpL. In this paper we discuss some of the properties of the embeddings SpL = L. First we analyze the behaviour of the spatial parts in the assembly: the points of L and of S(L)op (~= the congruence frame) are in a natural one-one correspondence while the topologies of SpL and Sp(S(L)op) differ. Then we concentrate on some special types of embeddings of SpL into L, namely in the questions when SpL is complemented, closed, or open. While in the first part L was general, here we need some restrictions (weak separation axioms) to obtain suitable formulas. [ABSTRACT FROM AUTHOR]
Published: 2024
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