Structure of unital 3-fields. (English) Zbl 1489.12021
A set \(\mathcal R\) with two ternary operations \(\nu :{\mathcal R}^3\to \mathcal R \) and \(\mu : {\mathcal R}^3 \to \mathcal R \) is called a \((3,3)\)-ring if 1) both operations are totally associative, 2) they are connected by a ternary analog of the distributive law and 3) for all \(a,b,c \in \mathcal R\) there exists a unique solution of the equation \(\nu (a,b,x)=c\). The last condition means that \((\mathcal R, \nu)\) is a ternary group. The operations \(\nu\) and \(\mu\) are called an additive and a multiplicative operations correspondingly. If \((\mathcal R, \mu)\) is also a ternary group the \((3,3)\)-ring is called a 3-field. Ternary analogues of the concepts of a vector space and a polynomial algebra in \(n\) variables over a 3-field are also introduced. The mentioned objects are investigated and many examples are given.
Reviewer: Grigori I. Zhitomirski (Herzliyya)
MSC:
| 12K99 | Generalizations of fields |
| 11S99 | Algebraic number theory: local fields |
| 20N10 | Ternary systems (heaps, semiheaps, heapoids, etc.) |
Keywords:
ternary operations; ternary group; ternary rings; ternary fields; ternary polynomial algebrasReferences:
| [1] | Bagger, J.; Lambert, N., Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev., D77, 065008 (2008) |
| [2] | Bohle, D.; Werner, W., A K-theoretic approach to the classification of symmetric spaces, J Pure App Algebra, 219, 10, 4295-4321 (2015) · Zbl 1434.46042 · doi:10.1016/j.jpaa.2015.02.003 |
| [3] | Bosch, S.; Guentzer, U.; Remmert, R., Non-Archimedean Analysis (1984) · Zbl 0539.14017 · doi:10.1007/978-3-642-52229-1 |
| [4] | Celakoski, N., On (F,G)-rings, God Zb Med Fak Skopje, 28, 5-15 (1977) · Zbl 0448.16024 |
| [5] | Chu, C-H, Jordan structures in geometry and analysis (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1238.17001 |
| [6] | Crombez, G., The post coset theorem for (n,m)-rings, Ist. Veneto Sci. Lett. Arti, 131, 1-7 (1973) · Zbl 0338.08007 |
| [7] | Crombez, G.; Timm, J., On (n,m)-quotient rings, Abh. Math. Semin. Univ. Hamb., 37, 200-203 (1972) · Zbl 0247.08002 · doi:10.1007/BF02999696 |
| [8] | Čupona, G., On [m,n]-rings, Bull. Soc. Math. Phys. Macedoine, 16, 5-9 (1965) · Zbl 0144.25503 |
| [9] | de Azcarraga, JA; Izquierdo, JM, n-Ary algebras: a review with applications, J. Phys., A43, 293001 (2010) · Zbl 1202.81187 |
| [10] | Dörnte, W., Unterschungen über einen verallgemeinerten Gruppenbegriff, Math. Z., 29, 1-19 (1929) · JFM 54.0152.01 · doi:10.1007/BF01180515 |
| [11] | Duplij, S.; Nikitin, AG; Boyko, VM, Ternary Hopf algebras, Symmetry in nonlinear mathematical physics, 25-34 (2001), Kiev: Institute of Mathematics, Kiev |
| [12] | Elgendy, HA; Bremner, MR, Universal associative envelopes of (n+1)-dimensional n-Lie algebras, Comm. Algebra, 40, 5, 1827-1842 (2012) · Zbl 1260.17005 · doi:10.1080/00927872.2011.558549 |
| [13] | Higgins, PJ, Groups with multiple operators, Proc. Lond. Math. Soc., 3, 6, 367-416 (1956) · Zbl 0073.01704 |
| [14] | Jacobson, N., Basic Algebra II (1989), New York: Freeman and Company, New York · Zbl 0694.16001 |
| [15] | Kerner, R., Ternary algebraic structures and their applications in physics, 15 (2000), Paris: Univ. P. & M. Curie, Paris |
| [16] | Kurosh, AG, Multioperator rings and algebras, Russ. Math. Surv., 24, 1-13 (1969) · Zbl 0204.35701 · doi:10.1070/RM1969v024n01ABEH001334 |
| [17] | Leeson, JJ; Butson, AT, On the general theory of (m,n) rings, Algebra Univers, 11, 42-76 (1980) · Zbl 0461.16029 · doi:10.1007/BF02483082 |
| [18] | Nambu, Y., Generalized Hamiltonian dynamics, Phys. Rev., D7, 2405-2412 (1973) · Zbl 1027.70503 |
| [19] | Post, EL, Polyadic groups, Trans. Amer. Math. Soc., 48, 208-350 (1940) · JFM 66.0099.01 · doi:10.1090/S0002-9947-1940-0002894-7 |
| [20] | Upmeier, H., Symmetric Banach manifolds and Jordan C*-algebras (1985), Amsterdam: North-Holland Publishing, Amsterdam · Zbl 0561.46032 |
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