2021
DOI: 10.1007/s00025-021-01548-6
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Around the Sine Addition Law and d’Alembert’s Equation on Semigroups

Abstract: We treat several related trigonometric functional equations on semigroups. First we make a small improvement to the description of solutions of the sine addition law f (xy) = f (x)g(y) + g(x)f (y) for f, g : S → C, where S is a semigroup. Next we solve the partial Pexiderizationwhere σ, τ are involutive automorphisms of M . The solutions of the latter two equations were given by Ng, Zhao, and Lin assuming that M is a group, then by the author assuming that the monoid M is either regular or generated by its squ… Show more

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Cited by 16 publications
(6 citation statements)
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“…Since f and g are even, we get that h * = h and then χ * = χ. This occurs in case (7). Subcase B: g • = 0 and f • = 0.…”
Section: The Main Resultsmentioning
confidence: 94%
See 2 more Smart Citations
“…Since f and g are even, we get that h * = h and then χ * = χ. This occurs in case (7). Subcase B: g • = 0 and f • = 0.…”
Section: The Main Resultsmentioning
confidence: 94%
“…Also Stetkaer [11,Theorem 6.1] gives a description of the solution of (1.1) with σ = id on a general semigroup in terms of the solutions of (1.3) with σ = id. The most recent result about (1.3) with σ = id was obtained by Ebanks [5,Theorem 2.1] and [7,Theorem 3.1] on semigroups.…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…The next result is essentially [3, Theorem 3.2] (with a small technical improvement in part (c) as stated in [4]).…”
Section: Preliminary Resultsmentioning
confidence: 99%
“…Note that it is possible for some values of f P to be chosen arbitrarily (see [3,Remark 3.2] for details). Note also that solution class (iii) does not arise on monoids, semigroups generated by their squares, or regular semigroups.…”
Section: Notation Terminology and Preliminariesmentioning
confidence: 99%