On the Properties of the Möbius Function

Self Cite

Summary The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the paper, the notion of a pseudocomplement in a lattice is formally introduced in Mizar, and based on this we define the notion of the skeleton and the set of dense elements in a pseudocomplemented lattice, giving the meet-decomposition of arbitrary element of a lattice as the infimum of two elements: one belonging to the skeleton, and the other which is dense. The core of the paper is of course the idea of Stone identity $$a^* \sqcup a^{**} = {\rm{T}},$$ which is fundamental for us: Stone lattices are those lattices L, which are distributive, bounded, and satisfy Stone identity for all elements a ∈ L. Stone algebras were introduced by Grätzer and Schmidt in [18]. Of course, the pseudocomplement is unique (if exists), so in a pseudcomplemented lattice we defined a * as the Mizar functor (unary operation mapping every element to its pseudocomplement). In Section 2 we prove formally a collection of ordinary properties of pseudocomplemented lattices. All Boolean lattices are Stone, and a natural example of the lattice which is Stone, but not Boolean, is the lattice of all natural divisors of p 2 for arbitrary prime number p (Section 6). At the end we formalize the notion of the Stone lattice B [2] (of pairs of elements a, b of B such that a ⩽ b) constructed as a sublattice of B 2, where B is arbitrary Boolean algebra (and we describe skeleton and the set of dense elements in such lattices). In a natural way, we deal with Cartesian product of pseudocomplemented lattices. Our formalization was inspired by [17], and is an important step in formalizing Jouni Järvinen Lattice theory for rough sets [19], so it follows rather the latter paper. We deal essentially with Section 4.3, pages 423–426. The description of handling complemented structures in Mizar [6] can be found in [12]. The current article together with [15] establishes the formal background for algebraic structures which are important for [10], [16] by means of mechanisms of merging theories as described in [11].

mentioning

confidence: 99%

Stone Lattices

Grabowski

2015

Self Cite

“…

In the article the formal characterization of square-free numbers is shown; in this manner the paper is the continuation of [19]. Essentially, we prepared some lemmas for convenient work with numbers (including the proof that the sequence of prime reciprocals diverges [1]) according to [18] which were absent in the Mizar Mathematical Library.

…”

mentioning

confidence: 99%

“…The notation and terminology used in this paper have been introduced in the following articles: [8], [2], [3], [30], [34], [6], [9], [16], [10], [11], [39], [27], [31], [42], [36], [19], [4], [23], [15], [26], [5], [12], [22], [37], [17], [20], [7], [41], [13], [25], [33], [32], [38], [40], [21], and [14].…”

mentioning

confidence: 99%

On Square-Free Numbers

Grabowski¹

2013

Self Cite

Summary In the article the formal characterization of square-free numbers is shown; in this manner the paper is the continuation of [19]. Essentially, we prepared some lemmas for convenient work with numbers (including the proof that the sequence of prime reciprocals diverges [1]) according to [18] which were absent in the Mizar Mathematical Library. Some of them were expressed in terms of clusters’ registrations, enabling automatization machinery available in the Mizar system. Our main result of the article is in the final section; we proved that the lattice of positive divisors of a positive integer n is Boolean if and only if n is square-free.

“…The articles [14], [38], [28], [32], [39], [11], [40], [13], [33], [12], [5], [4], [2], [6], [10], [37], [36], [25], [3], [15], [19], [35], [24], [30], [18], [34], [16], [9], [22], [21], [41], [17], [20], [7], [31], [29], [8], [23], and [27] provide the notation and terminology for this paper.…”

mentioning

confidence: 99%

The Perfect Number Theorem and Wilson's Theorem

Riccardi¹

2009

Summary. This article formalizes proofs of some elementary theorems of number theory (see [1,26]): Wilson's theorem (that n is prime iff n > 1 and (n − 1)! ∼ = −1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler's sum of divisors function φ, proves that φ is multiplicative and that PreliminariesWe adopt the following convention: k, n, m, l, p denote natural numbers and n 0 , m 0 denote non zero natural numbers.We now state several propositions: (1) 2 n+1 < 2 n+2 − 1.