A Levi–Civita Equation on Monoids, Two Ways
We consider the Levi–Civita equation f(xy)=g1(x)h1(y)+g2(x)h2(y)f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right) for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many...
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| Format: | Article |
| Language: | English |
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Sciendo
2022-09-01
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| Series: | Annales Mathematicae Silesianae |
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| Online Access: | https://doi.org/10.2478/amsil-2022-0009 |
| _version_ | 1797989245114122240 |
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| author | Ebanks Bruce |
| author_facet | Ebanks Bruce |
| author_sort | Ebanks Bruce |
| collection | DOAJ |
| description | We consider the Levi–Civita equation
f(xy)=g1(x)h1(y)+g2(x)h2(y)f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right)
for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids. |
| first_indexed | 2024-04-11T08:15:58Z |
| format | Article |
| id | doaj.art-78349633eed748e5bdb8fd45eb36d8c1 |
| institution | Directory Open Access Journal |
| issn | 2391-4238 |
| language | English |
| last_indexed | 2024-04-11T08:15:58Z |
| publishDate | 2022-09-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Annales Mathematicae Silesianae |
| spelling | doaj.art-78349633eed748e5bdb8fd45eb36d8c12022-12-22T04:35:09ZengSciendoAnnales Mathematicae Silesianae2391-42382022-09-0136215116610.2478/amsil-2022-0009A Levi–Civita Equation on Monoids, Two WaysEbanks Bruce0Department of Mathematics, University of Louisville, Louisville, Kentucky 40292 and Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USAWe consider the Levi–Civita equation f(xy)=g1(x)h1(y)+g2(x)h2(y)f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right) for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.https://doi.org/10.2478/amsil-2022-0009levi–civita equationsine addition formulacosine addition formulasemigroupmonoidexponential function39b3239b52 |
| spellingShingle | Ebanks Bruce A Levi–Civita Equation on Monoids, Two Ways Annales Mathematicae Silesianae levi–civita equation sine addition formula cosine addition formula semigroup monoid exponential function 39b32 39b52 |
| title | A Levi–Civita Equation on Monoids, Two Ways |
| title_full | A Levi–Civita Equation on Monoids, Two Ways |
| title_fullStr | A Levi–Civita Equation on Monoids, Two Ways |
| title_full_unstemmed | A Levi–Civita Equation on Monoids, Two Ways |
| title_short | A Levi–Civita Equation on Monoids, Two Ways |
| title_sort | levi civita equation on monoids two ways |
| topic | levi–civita equation sine addition formula cosine addition formula semigroup monoid exponential function 39b32 39b52 |
| url | https://doi.org/10.2478/amsil-2022-0009 |
| work_keys_str_mv | AT ebanksbruce alevicivitaequationonmonoidstwoways AT ebanksbruce levicivitaequationonmonoidstwoways |