On geometric implications
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F25%3A00605674" target="_blank" >RIV/67985840:_____/25:00605674 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s11225-023-10094-x" target="_blank" >https://doi.org/10.1007/s11225-023-10094-x</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s11225-023-10094-x" target="_blank" >10.1007/s11225-023-10094-x</a>
Alternative languages
Result language
angličtina
Original language name
On geometric implications
Original language description
It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (Implication via spacetime. In: Mathematics, logic, and their philosophies: essays in honour of Mohammad Ardeshir, pp 161–216, 2021). Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category S, provided that S has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Studia Logica
ISSN
0039-3215
e-ISSN
1572-8730
Volume of the periodical
113
Issue of the periodical within the volume
1
Country of publishing house
DE - GERMANY
Number of pages
30
Pages from-to
79-108
UT code for WoS article
001175447900003
EID of the result in the Scopus database
2-s2.0-85186948968