"new foundations must give new math"

from "The Basic Picture. Structures for constructive topology"

by Giovanni Sambin, Oxford University Press, forthcoming


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Under the patronage of

Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Italy

University of Padova, Italy

AILA (Italian Association of Logic and Applications), whose first president was Giovanni Sambin (from 1987 to 1994).

Abstracts » Toto

Paola Toto

Many-valued overlap algebras
(j.w.w. Maria Emilia Maietti and Giovanni Sambin)

In his forthcoming book Sambin introduces a new topological theory, called "The Basic Picture", that generalizes the notion of topological space and in turn its point-free version. He also introduces the notion of overlap algebra to give a purely algebraic presentation of such concepts.

The ultimate goal of our work is to generalize such topological notions in the context of many-valued sets and to test whether the original algebrization of Sambin’s topological notions can be considered also the algebrization of their many-valued version.

To include the many-valued power collection of a set as an example of overlap algebra, we need to work in the many-valued set theory whose structure of truth values is an overlap algebra itself.

After defining an evaluated set-based overlap algebra, we prove that the evaluated power collection functor is full and faithful and also dense over evaluated atomic overlap algebras, as proved by Sambin in his book.