# Non-Leibniz algebras with logarithms do not have the trigonometric identity

Banach Center Publications (2000)

- Volume: 53, Issue: 1, page 177-189
- ISSN: 0137-6934

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topPrzeworska-Rolewicz, D.. "Non-Leibniz algebras with logarithms do not have the trigonometric identity." Banach Center Publications 53.1 (2000): 177-189. <http://eudml.org/doc/209072>.

@article{Przeworska2000,

abstract = {Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. $[C(x)]^2 + [S(x)]^2 = e$ whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold.},

author = {Przeworska-Rolewicz, D.},

journal = {Banach Center Publications},

keywords = {sine mapping; algebra with unit; algebraic analysis; logarithmic mapping; Leibniz condition; cosine mapping; trigonometric identity},

language = {eng},

number = {1},

pages = {177-189},

title = {Non-Leibniz algebras with logarithms do not have the trigonometric identity},

url = {http://eudml.org/doc/209072},

volume = {53},

year = {2000},

}

TY - JOUR

AU - Przeworska-Rolewicz, D.

TI - Non-Leibniz algebras with logarithms do not have the trigonometric identity

JO - Banach Center Publications

PY - 2000

VL - 53

IS - 1

SP - 177

EP - 189

AB - Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. $[C(x)]^2 + [S(x)]^2 = e$ whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold.

LA - eng

KW - sine mapping; algebra with unit; algebraic analysis; logarithmic mapping; Leibniz condition; cosine mapping; trigonometric identity

UR - http://eudml.org/doc/209072

ER -

## References

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- PR[3] D. Przeworska-Rolewicz, Logarithms and Antilogarithms. An Algebraic Analysis Approach, With Appendix by Z. Binderman. Kluwer Academic Publishers, Dordrecht, 1998.
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