A complete classification of continuous fractional operations on $$\mathbb {C}$$ C
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A complete classification of continuous fractional operations on C Yuji Kobayashi1 · Sin-Ei Takahasi1 · Makoto Tsukada1
Published online: 14 August 2017 © Akadémiai Kiadó, Budapest, Hungary 2017
Abstract We completely classify continuous fractional operations on the complex number field C modulo equivalence. A continuous fraction is described by a pair of complex numbers. We prove that a continuous fraction is completely characterized by the (conjugate) ratio of two numbers describing the fraction. Furthermore, we show that the set of all the equivalence classes of continuous fractions is equipped with a natural topology and it is homeomorphic to the unit disk {z ∈ C: |z| ≤ 1}. Keywords Fractional operation · Automorphism · Homomorphism Mathematics Subject Classification Primary 46N99 · 30E99; Secondary 43A25 · 12D99
1 Introduction In [3] we study continuous fractional operations (continuous fractions for short) on the real number field R and on the complex number field C. In this paper we give a complete classification of continuous fractional operations on C.
Dedicated to Professor Emeritus Jyunji Inoue on the occasion of his 77th birthday. This research was partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science, No. 25400120.
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Sin-Ei Takahasi [email protected] Yuji Kobayashi [email protected] Makoto Tsukada [email protected]
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Department of Information Science, Toho University, Funabashi 274–8510, Japan
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A fractional operation (fraction for short) on C is a binary operation ∗ on C satisfying (a + b) ∗ c = (a ∗ c) + (b ∗ c) and (ax) ∗ (bx) = a ∗ b for all a, b, c, x ∈ C with x = 0. It is continuous, if the mapping x → x ∗ b is continuous on C for every b ∈ C. This notion was introduced in order to include zero in the denominator [1,3]. Two continuous operations ∗ and ∗ on C is equivalent, if there is a homeomorphism f : C → C such that f (z ∗ w) = f (z) ∗ f (w) holds for any z, w ∈ C. We call f a corresponding homeomorphism to the equivalence ∗∼ = ∗ . We are interested in classifying continuous fractions modulo equivalence. Let α, β ∈ C. The operation ∗(α,β) on C defined by z z α Re + β Im if w = 0 z∗w = (1.1) w w 0 if w = 0 for z, w ∈ C is a continuous fraction, and conversely any continuous fraction is given in this way [3, Theorem 2]. Let ∗(α,β) denote the operation ∗ given by (1.1). In particular, ∗(0,0) is the trivial fraction; z ∗(0,0) w = 0 for all z, w ∈ C. Clearly, the trivial fraction is not equivalent to any non-trivial fraction. In [3, Theorem 3] we prove that non-trivial fractions ∗(α,β) and ∗(α ,β ) are equivalent if αβ = βα . The converse is true in some special cases [3, Theorem 4], but the general converse problem is left unsolved. Actually, the converse is nottrue in general. For (α, β) , α .β ∈ (C × C) \ (0, 0) , we write (α, β) ∼ α , β , if αβ = βα or αβ = −βα . In the main theorem of this paper we prove that non-trivial operations ∗(α,β)
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