Groups – Additive Notation

Roland Coghetto

Formalized Mathematics (2015)

  • Volume: 23, Issue: 2, page 127-160
  • ISSN: 1426-2630

Abstract

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We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.

How to cite

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Roland Coghetto. "Groups – Additive Notation." Formalized Mathematics 23.2 (2015): 127-160. <http://eudml.org/doc/271755>.

@article{RolandCoghetto2015,
abstract = {We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.},
author = {Roland Coghetto},
journal = {Formalized Mathematics},
keywords = {additive group; subgroup; Lagrange theorem; conjugation; normal subgroup; index; additive topological group; basis; neighborhood; additive abelian group; Z-module; additive groups; normal subgroups; additive topological groups; bases; neighborhoods; additive Abelian groups; -modules; formalization},
language = {eng},
number = {2},
pages = {127-160},
title = {Groups – Additive Notation},
url = {http://eudml.org/doc/271755},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Roland Coghetto
TI - Groups – Additive Notation
JO - Formalized Mathematics
PY - 2015
VL - 23
IS - 2
SP - 127
EP - 160
AB - We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.
LA - eng
KW - additive group; subgroup; Lagrange theorem; conjugation; normal subgroup; index; additive topological group; basis; neighborhood; additive abelian group; Z-module; additive groups; normal subgroups; additive topological groups; bases; neighborhoods; additive Abelian groups; -modules; formalization
UR - http://eudml.org/doc/271755
ER -

References

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