Nice local global fields. IV.
Nichtaxiomatisierbarkeit von Satzmengen durch Ausdrücke spezieller Gestalt
Niven’s Theorem
This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].
Non additive ordinal relations representable by lower or upper probabilities
We characterize (in terms of necessary and sufficient conditions) binary relations representable by a lower probability. Such relations can be non- additive (as the relations representable by a probability) and also not “partially monotone” (as the relations representable by a belief function). Moreover we characterize relations representable by upper probabilities and those representable by plausibility. In fact the conditions characterizing these relations are not immediately deducible by means...
Non-axiomatizability of real general affine geometry
Nondeterminism and fully abstract models
Non-finitizability of a weak second-order theory
Normal forms in partial modal logic
A "partial" generalization of Fine's definition [Fin] of normal forms in normal minimal modal logic is given. This means quick access to complete axiomatizations and decidability proofs for partial modal logic [Thi].
Normal forms in the typed -calculus with tuple types
Normal Subgroup of Product of Groups
In [6] it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all j ∈ I, the group G = Πi∈IGi has a normal subgroup isomorphic to Gj. Moreover, we show some relations between a family of groups and its direct product.
Normalisation of the theory of Cartesian closed categories and conservativity of extensions of
Normalisation of the Theory T of Cartesian Closed Categories and Conservativity of Extensions T[x] of T
Using an inductive definition of normal terms of the theory of Cartesian Closed Categories with a given graph of distinguished morphisms, we give a reduction free proof of the decidability of this theory. This inductive definition enables us to show via functional completeness that extensions of such a theory by new constants (“indeterminates”) are conservative.
Notational and Logical Completeness in three-Valued Logic
Note on "construction of uninorms on bounded lattices"
In this note, we point out that Theorem 3.1 as well as Theorem 3.5 in G. D. Çaylı and F. Karaçal (Kybernetika 53 (2017), 394-417) contains a superfluous condition. We have also generalized them by using closure (interior, resp.) operators.
Notes on a class of tiling problems
Notes on locally internal uninorm on bounded lattices
In the study, we introduce the definition of a locally internal uninorm on an arbitrary bounded lattice . We examine some properties of an idempotent and locally internal uninorm on an arbitrary bounded latice , and investigate relationship between these operators. Moreover, some illustrative examples are added to show the connection between idempotent and locally internal uninorm.
Null events and stochastical independence
In this paper we point out the lack of the classical definitions of stochastical independence (particularly with respect to events of 0 and 1 probability) and then we propose a definition that agrees with all the classical ones when the probabilities of the relevant events are both different from 0 and 1, but that is able to focus the actual stochastical independence also in these extreme cases. Therefore this definition avoids inconsistencies such as the possibility that an event can be at the...
O dynamické logice
O intuicionismu
Object-Free Definition of Categories
Category theory was formalized in Mizar with two different approaches [7], [18] that correspond to those most commonly used [16], [5]. Since there is a one-to-one correspondence between objects and identity morphisms, some authors have used an approach that does not refer to objects as elements of the theory, and are usually indicated as object-free category [1] or as arrowsonly category [16]. In this article is proposed a new definition of an object-free category, introducing the two properties:...