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Keywords = point functor

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32 pages, 463 KiB  
Article
The Calculus of Names—The Legacy of Jan Łukasiewicz
by Andrzej Pietruszczak
Axioms 2025, 14(3), 160; https://doi.org/10.3390/axioms14030160 - 23 Feb 2025
Viewed by 280
Abstract
With his research on Aristotle’s syllogistic, Jan Łukasiewicz initiates the branch of logic known as the calculus of names. This field deals with axiomatic systems that analyse various fragments of the logic of names, i.e., that branch of logic that studies various forms [...] Read more.
With his research on Aristotle’s syllogistic, Jan Łukasiewicz initiates the branch of logic known as the calculus of names. This field deals with axiomatic systems that analyse various fragments of the logic of names, i.e., that branch of logic that studies various forms of names and functors acting on them, as well as logical relationships between sentences in which these names and functors occur. In this work, we want not only to present the genesis of the calculus of names and its first system created by Łukasiewicz, but we also want to deliver systems that extend the first. In this work, we will also show that, from the point of view of modern logic, Łukasiewicz’s approach to the syllogistic is not the only possible one. However, this does not diminish Łukasiewicz’s role in the study of syllogism. We believe that the calculus of names is undoubtedly the legacy of Łukasiewicz. Full article
(This article belongs to the Section Logic)
38 pages, 1204 KiB  
Article
Frames of Group Sets and Their Application in Bundle Theory
by Eric J. Pap and Holger Waalkens
Mathematics 2024, 12(13), 2135; https://doi.org/10.3390/math12132135 - 7 Jul 2024
Viewed by 1581
Abstract
We study fiber bundles where the fibers are not a group G but a free G-space with disjoint orbits. The fibers are then not torsors but disjoint unions of these; hence, we like to call them semi-torsors. Bundles of semi-torsors naturally generalize [...] Read more.
We study fiber bundles where the fibers are not a group G but a free G-space with disjoint orbits. The fibers are then not torsors but disjoint unions of these; hence, we like to call them semi-torsors. Bundles of semi-torsors naturally generalize principal bundles, and we call these semi-principal bundles. These bundles admit parallel transport in the same way that principal bundles do. The main difference is that lifts may end up in another group orbit, meaning that the change cannot be described by group translations alone. The study of such effects is facilitated by defining the notion of a basis of a G-set, in analogy with a basis of a vector space. The basis elements serve as reference points for the orbits so that parallel transport amounts to reordering the basis elements and scaling them with the appropriate group elements. These two symmetries of the bases are described by a wreath product group. The notion of basis also leads to a frame bundle, which is principal and so allows for a conventional treatment. In fact, the frame bundle functor is found to be a retraction from the semi-principal bundles to the principal bundles. The theory presented provides a mathematical framework for a unified description of geometric phases and exceptional points in adiabatic quantum mechanics. Full article
(This article belongs to the Section B: Geometry and Topology)
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7 pages, 230 KiB  
Article
The Zahl-Anzahl Distinction in Gottlob Frege: Arithmetic of Natural Numbers with Anzahl as a Primitive Term
by Eugeniusz Wojciechowski
Axioms 2020, 9(1), 6; https://doi.org/10.3390/axioms9010006 - 31 Dec 2019
Cited by 1 | Viewed by 2728
Abstract
The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the [...] Read more.
The starting point is Peano’s expression of the axiomatics of natural numbers in the framework of Leśniewski’s elementary ontology. The author enriches elementary ontology with the so-called Frege’s predication scheme and goes on to propose the formulations of this axiomatic, in which the original natural number (N) term is replaced by the term Anzahl (A). The functor of the successor (S) is defined in it. Full article
(This article belongs to the Special Issue Deductive Systems in Traditional and Modern Logic)
22 pages, 393 KiB  
Article
Koszulity and Point Modules of Finitely Semi-Graded Rings and Algebras
by Oswaldo Lezama and Jaime Gomez
Symmetry 2019, 11(7), 881; https://doi.org/10.3390/sym11070881 - 4 Jul 2019
Cited by 13 | Viewed by 2540
Abstract
In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and [...] Read more.
In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non- N -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincaré series. Full article
(This article belongs to the Special Issue Mirror Symmetry and Algebraic Geometry)
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