Structure of unital 3-fields
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Structure of unital 3-fields Steven Duplij · Wend Werner
Received: 14 January 2019 / Accepted: 27 July 2020 © The Author(s) 2020
Abstract We investigate fields in which addition requires three summands. These ternary fields are shown to be isomorphic to the set of invertible elements in a local ring R having Z=2Z as a residual field. One of the important technical ingredients is to intrinsically characterize the maximal ideal of R. We include a number of illustrative examples and prove that the structure of a finite 3-field is not connected to any binary field.
1 Introduction Most of us seem to be biologically biased towards thinking that it always requires two in order to generate a third. In mathematics or physics, however, this idea does not seem to rest on a sound foundation: The theory of symmetric spaces, for example, is nicely described in terms of Lie or Jordan triple systems ([5, 20]; see e.g. [2] for a recent development), and in physics, higher Lie algebras have come into focus in [18] (for later development see e.g. [9, 15]) and were e.g. applied to the theory of M2-branes in [1]. Ternary Hopf algebras were introduced and investigated in [11]. In order to illustrate why oftentimes (but not here), ternary algebraic structure does not introduce new aspects, let us digress somewhat and have a closer look at a simple example, commutative ternary groups. S. Duplij Center for Information Technology (WWU IT), Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany W. Werner () Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany E-Mail: [email protected]
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S. Duplij, W. Werner
If G is a set and m W G 3 ! G is a mapping, what properties should m have in order that G be called a (commutative) ternary group with multiplication given by m? Here is a list.
Associativity, as always, should mean that there is no need for brackets when we multiply several times in a row. The ternary multiplication is commutative, iff it is invariant under any permutation of the factors Finally, every element g 2 G has a (ternary) inverse g (the quer element) so that for all g0 2 G g0 gg D g0 :
This is all that is needed in the way of a definition. Note that there is no neutral element, which nonetheless can be defined by saying e 2 G is neutral, iff eeg D g, for all g 2 G. The reason that we did not include the existence of a neutral element e in the definition is that as soon as there is one, G equipped with the product g h WD geh; becomes a binary group (with neutral element e and inverses g 1 D g) such that for all g1;2;3 2 G g1 g2 g3 D g1 g2 g3 : So here, G is trivial in that its ternary product directly comes from a binary one. 1 This, however always is the case, albeit for a different reason. A more general example of this kind is the following. Pick a binary commutative group G, fix g0 2 G, and let m.g1 ; g2 ; g3 / D g0 g1 g2 g3 Then .G; m/ is a ternary group with g D g01 g 1 for each g 2 G. Also, .G; m/ is unital iff g0 D w 2 f
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