Internal monoids and groups in the category of commutative cancellative medial magmas

Fatelo, J. P.; Martins-Ferreira, Nelson

doi:10.4171/pm/1986

“…This leads naturally to the study of which elements e ∈ X can be chosen as the origin of a module. In [5] this approach was used for the study of internal monoids in the category of midpoint algebras.…”

Section: Comparison With R-modulesmentioning

confidence: 99%

Affine mobi spaces

Fatelo,

Martins-Ferreira

2021

Preprint

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The category of mobi algebras has been introduced as a model to the unit interval of real numbers. The notion of mobi space over a mobi algebra has been proposed as a model for spaces with geodesic paths. In this paper we analyse the particular case of affine mobi spaces and show that there is an isomorphism of categories between R-modules and pointed affine mobi spaces over a mobi algebra R as soon as R is a unitary ring in which 2 is an invertible element.

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“…The first results concerning this investigation were presented in [6] and [7]. In [8], the first version of this text, mobility algebras (or mobi algebras) and mobility affine spaces are considered in more details.…”

Section: Introductionmentioning

confidence: 99%

Mobility spaces and geodesics for the n-sphere

Fatelo¹,

Martins-Ferreira²

2021

Preprint

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We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. We show that there is a strong connection between modules over a ring and affine mobility spaces over a mobility algebra. However, geodesics in general fail to be affine thus giving rise to the new algebraic structure of mobility space. We show that the so called formula for spherical linear interpolation, which gives geodesics on the n-sphere, is an example of a mobility space over the unit interval mobility algebra.

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“…First results concerning this investigation were presented in [5] where a binary operation, obtained by fixing t to a value that positions the particle at half way from x to y, is studied. The whole movement of a particle on a geodesic path is captured when the variable t is allowed to range over a set of values, of which the unit interval is the most natural choice.…”

Section: Introductionmentioning

confidence: 99%

“…Section 3 is totally devoted to the definition of mobi space and a short list of its properties that will be needed in this paper. Further studies are postponed to a future work, namely the transformations between such structures, and the fact that the category of mobi spaces is a weakly Mal'tsev category [5,12,13]. Or its comparison with the work of A. Kock on affine connections with neighbouring relations [8,9,10] and its relation to the work of Buseman [1,2] on spaces with unique geodesic paths.…”

Section: Introductionmentioning

confidence: 99%

Mobility spaces and their geodesic paths

Fatelo,

Martins-Ferreira

2020

Preprint

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We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. Mobility algebras consist on a set A together with three constants and a ternary operation. In the case of the closed unit interval A = [0, 1], the three constants are 0, 1 and 1/2 while the ternary operation is p(x, y, z) = x − yx + yz. A mobility space is a set X together with a map q : X × A × X → X with the meaning that q(x, t, y) indicates the position of a particle moving from point x to point y at the instant t ∈ A, along a geodesic path within the space X. A mobility space is thus defined with respect to a mobility algebra, in the same way as a module is defined over a ring. We introduce the axioms for mobility spaces, investigate the main properties and give examples. We also establish the connection between the algebraic context and the one of spaces with geodesic paths. The connection with affine spaces is briefly mentioned.

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Internal monoids and groups in the category of commutative cancellative medial magmas

Cited by 7 publications

References 20 publications

Affine mobi spaces

Affine mobi spaces

Mobility spaces and geodesics for the n-sphere

Mobility spaces and their geodesic paths

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