0605674 - MÚ 2026 RIV DE eng J - Journal Article
Akbar Tabatabai, Seyed Amirhossein
On geometric implications.
Studia Logica. Roč. 113, č. 1 (2025), s. 79-108. ISSN 0039-3215. E-ISSN 1572-8730
Institutional support: RVO:67985840
Keywords : frame representation * geometricity * implications * modal algebras
OECD category: Pure mathematics
Impact factor: 0.6, year: 2023 ; AIS: 0.494, rok: 2023
Method of publishing: Open access
DOI: https://doi.org/10.1007/s11225-023-10094-x

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.
Permanent Link: https://hdl.handle.net/11104/0363327