A New Approach to Slice Analysis Via Slice Topology

In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function.


Introduction
Quaternions are a kind of hypercomplex numbers first described by Hamilton in 1843.With the development of the theory of holomorphic functions in complex analysis and its generalizations to higher dimensional cases, similar theories for quaternions were established.The most well-known function theory in quaternionic analysis, was initiated by Gr.Moisil and R. Fueter [20] who, with his school, greatly contributed to the development of the theory.The holomorphy in quaternionic analysis is defined via the so-called Cauchy-Riemann-Fueter equation and has been widely studied, see e.g.[10,31,32,39].
Since the class of functions considered by Moisil and Fueter does not contain neither the identity function nor any other monomial function f (q) = q n , C. G. Cullen [15] considered another class, using the notion of intrinsic functions introduced by Rinehart in [36].Inspired by Cullen's approach, in [25] Gentili and Struppa started the study of a new theory of quaternionic functions which are holomorphic in a suitable sense.This theory includes convergent power series of the form n∈N q n a n and it is nowadays known as slice quaternionic analysis; it has been widely studied in the past fifteen years, see [8,12,21] and also the books [4,13,23].Slice analysis has been extended to octonions [26], Clifford algebras [11] and real alternative * -algebra [27,30].The richness of slice analysis in one variable makes it natural to look for generalization to several variables and in fact the quaternonic case is considered in [14], the case of Clifford algebras in [28], the one of octonions in [35] and finally, the real alternative * -algebras case in [29].The most general setting of Euclidean spaces is treated in [18] and in this paper.
A fundamental role in slice quaternionic analysis is played by the so-called representation formula, see [7,9].The formula allows to extend many known results in complex analysis to slice quaternionic analysis, e.g. the quaternionic power series expansion [22,38], geometric function theory [33,34,40,41], results in quaternionic Schur analysis [3] and in quaternionic operator theory [2,5,6].However, the representation formula just works on axially symmetric s-domains (see [7] for the definition).This causes difficulties while considering more general, non necessarily axially symmetric, domains.
In the recent paper [17], we generalize the representation formula to non-axially symmetric sets.To this end, a crucial tool is the use of a suitable topology on H, finer than the Euclidean one, called slice topology and denoted by τ s .In this paper we deepen the study of the slice topology also proving that it is not metrizable.
One can consider functions defined on open sets in this topology, hence enlarging the class of slice regular functions.It is important to note that, in this framework, the conditions under which analytic continuation for slice regular functions is possible are weaker than the conditions in use with the Euclidean topology.This fact makes it easier the study of analogues of the Riemann domains (Riemann domains over (H, τ s )) and generalized manifolds over (H, τ s ).In the slice topology the class of 'domains of holomorphy' includes the axially symmetric open sets and also the so-called hyper-σ-polydiscs that we shall introduce in Section 7.
We then show how the notion of slice regularity can be given for functions with values in a finitedimensional left alternative algebra, see Section 4, and also for functions with values in the Euclidean space R 2n .It is important to note that slice regularity in this paper is meant in the original sense of Gentili and Struppa, see [25] (we shall refer to it as the weak slice regularity) and not in the sense of Ghiloni and Perotti [29] (regularity in their sense may also be called strong slice regularity).Moreover, we also consider the several variables case thus giving rise to results of a more general validity than those one available in the literature.Our approach is rather new and it is spread in various papers, see [17][18][19], and the main purpose of this work is to give a unified overview of the main ideas, complemented with some other new results and examples.
The paper is organized in seven sections, besides this introduction.In Section 2 we introduce the slice topology in the quaternionic case and we prove some of its properties among which the non-metrizability.In Section 3 we discuss the definition of quaternionic slice regular functions, some main properties and examples.In Section 4 we consider the generalization to the case of functions with values in a left alternative algebra and defined on the quadratic cone whereas in Section 5 we come to the case of functions with values in a real Euclidean space, even dimensional, and defined on the so-called slicecones.In this extremely general setting we can prove, in Section 6, an extension result.The domains of slice regularity are discussed in Section 7, while a Taylor formula is proved in Section 8.

Slice topology
In this section, we define the so called slice topology τ s on H, originally introduced in [17], we state some of its properties and we discuss some examples.We shall show, in particular, that slice topology has some special features near the real axis and this fact has important consequences when considering connectedness.
We recall that the algebra of real quaternions H consists of elements of the form q = x 0 +x 1 i+x 2 j +x 3 k where the imaginary units i, j, k satisfy The set of imaginary units in H is defined by The slice topology that we introduce below is designed around the notion of slice regularity, see Definition 3.1, and is based on the following: The following result is easily proved with classical arguments: A similar terminology will be used for all the other notions in the slice topology.We made an exception and we do not use the term slice-domain to denote a domain in the slice topology, since this notion is already used in the literature to denote something different.In fact a set Ω in H is called classical slice domain, in short s-domain, if it is a domain in the Euclidean topology such that Ω R := Ω ∩ R = ∅, and Ω I is a domain in C I for any I ∈ S.
It is immediate from the definition that for any I ∈ S, the subspace topology τ s (C I \R) of τ s (H) coincides the Euclidean topology τ (C I \R).However, τ s is quite different from Euclidean topology near R as Example 3.2 in [17] shows.
The peculiarity of the topology τ s near R appears, in particular, in the notion of connectedness, so we need to introduce another useful notion.
For any q ∈ H\R, there is r ∈ R + such that the ball B I (q, r) in C I does not intersect R. Note that for such r > 0 B I (q, r) ∩ R = ∅ is connected in R. It is then clear that the slice topology has a basis of real-connected sets at any point outside R, and the following result implies that slice topology has a real-connected basis also near R.
Proposition 2.5.For any slice-open set Ω in H and q ∈ Ω, there is a real-connected st-domain U ⊂ Ω containing q.
It is useful to explain that such a U can be constructed as the slice-connected component of the set Here when q ∈ R, we take A to be the connected component of Ω R containing q in R and when q ∈ R we set A := ∅.Now we describe slice-connectedness of real-connected slice-domains by using suitable slice-paths.
One can prove, see [17,Proposition 3.6], that any path on a slice is a slice-path.Moreover we have, see [17,Proposition 3.8]: Proposition 2.7.For each real-connected st-domain U , the following assertions hold: (ii) If U R = ∅, then for each q ∈ U and x ∈ U R , there is a path on a slice from q to x.
(iii) U I is a domain in C I for each I ∈ S.
By Proposition 2.5 and 2.7 (iv), we deduce the next important result: Proposition 2.8.The topological space (H, τ s ) is connected, local path-connected and path-connected.
Corollary 2.9.A set Ω ⊂ H is an st-domain if Ω R = ∅ and Ω I is a domain in C I for any I ∈ S.
By Corollary 2.9, any s-domain is an st-domain.Therefore the notion of st-domain is a generalization of the notion of s-domain.However, the converse is not true since not every st-domain Ω is an s-domain, not even when Ω is a domain in H, as Example 3.13 in [17] shows.In that example we consider the set which is a domain in τ (H).Let us fix I ∈ S and let J ∈ S be such that J⊥I.Then is not connected in τ (C J ), and as a consequence, Ω is not an s-domain.However, Ω is slice-connected, because any point in Ω can be connected to 0 or 6 by a path in a slice, and 0 can be connected to 6 by a path in C I .So Ω is an st-domain since Ω J is open in C J for any J ∈ S. We note that in the quaternionic case, besides the Euclidean and the slice topology, we can also consider the topology introduced in [22] and based on the so-called σ-distance which is defined by The σ-balls centered at q ∈ H, namely the balls according to the σ-distance, are defined by: Σ(p, r) := {q ∈ H : σ(p, q) < r}.
The slice topology is finer than the topology τ σ induced by the σ-distance.In fact, the relations between the topologies τ, τ σ , τ s is described in the next result: Proposition 2.10.The topologies τ, τ σ , τ s are such that τ τ σ τ s .
Proof.The inclusion τ τ σ is immediate.It is also immediate to check that τ σ ⊂ τ s .To prove that the second inclusion is strict, we need to show a set Ω ∈ τ s \τ σ .To construct such a set we fix I ∈ S and we consider the slice-open set Ω in H defined by where and dist(J, C I ) is the Euclidean distance in H from J to C I .The set Ω is evidently a slice-open set.Since H\Ω ⊃ C J \Ω J , we have where dist CJ (0, C J \Ω J ) is the Euclidean distance in C J between 0 and C J \Ω J .Since 2) and (2.3), we have σ(0, H\Ω) = 0. Hence, 0 is not an interior point in Ω in the topology τ σ .We deduce that Ω is not open in τ σ .
Another peculiarity of the slice topology is described in the next proposition: Proposition 2.11.The topology τ s is not a metrizable topology.
Proof.Suppose that τ s is induced by a metric d s .Then the set O := {q ∈ H : d s (q, 0) < 1} is a slice-open set in τ s .For any ǫ > 0 we introduce the set (2.4) S ǫ := {I ∈ S : B I (0, ǫ) ⊂ O}, and we denote by |S ǫ | its cardinality.We claim that |S ǫ | is finite.Assume the contrary.Then there are J 1 , ..., J k , ... ∈ S ǫ with J ı = ±J  .Set The set is a slice-open set.However Since the cardinality of S 1 k is finite, we deduce that S is a countable set, which is absurd.Hence τ s cannot be induced by any metric.

Main results and examples
The definition of slice regular functions recalled below is well known since [24], but the novelty here is that we work with the slice topology and the functions are defined on slice-open sets in H.We present some results below, whose proof is given in [17], and we discuss some examples that further clarify how our approach with the slice topology gives a richer function theory and allows more general situations. where The following result is known as Splitting Lemma and it is based on writing the values of a quaternionic function by means of two complex-valued functions, and for this reason the result holds also in this framework: Lemma 3.2.(Splitting Lemma) Let Ω ∈ τ s (H).A function f : Ω → H is slice regular, if and only if for all I, J ∈ S with I⊥J, there are two C I -valued holomorphic functions F, G : Ω I → C I such that f I = F + GJ.
A result which is classical for holomorphic functions is the identity principle.This result holds also for slice regular functions defined on domains in the Euclidean topology, but when considering the slice topology in H, its proof is more delicate.We recall its statement here since it is crucial to prove various results: Theorem 3.3.(Identity Principle) Let Ω be an st-domain, namely a domain in τ s (H), and let f, g : Ω → H be slice regular.If f and g coincide on a subset of Ω I with an accumulation point in Ω I for some I ∈ S, then f = g on Ω.
Another crucial result for slice regular functions is the so-called extension formula, see [7,Theorem 4.2], which is used to prove the general representation formula [7,Theorem 3.2] in the class of axially symmetric slice domains.In [17], we have extended this result to a more general setting and we proved the following: Moreover, for each domain W in τ s (H), f | W is slice and the unique slice regular extension on W of f | W ∩V + , where We note that if we consider a disc B I (q, r) ⊂ C I , I ∈ S, with center q ∈ C I and radius r ∈ R + , and a holomorphic function f : B I (q, r) → H, then f can be uniquely extended to be a slice regular function on the σ-ball Σ(q, r).
Another approach to slice regular functions makes use of the notion of slice functions.To recall this notion, we first introduce a notation: for any Ω ⊂ H, define a set in C by for each x + yI ∈ Ω with x, y ∈ R and I ∈ S. We call F is a stem function of f .
For any I ∈ S, there is an isomorphism (i) f is a path-slice function.
(ii) For each γ ∈ P(C, Ω), there is an element (iv) For each γ ∈ P(C, Ω) and I, J, K ∈ S(γ, Ω) The above definition is indeed a generalization of the notion of slice functions, and in fact in [17] we proved: Proposition 3.8.Every slice function defined on a subset of H is path-slice.
The class of path-slice functions also contains the class of slice regular functions: Theorem 3.9.Every slice regular function defined on an open set in τ s (H) is path-slice.
Note that a slice regular function is not necessarily a slice function, unless one adds hypothesis on the open set of definition, for example that it is an axially symmetric s-domain.
A cornerstone of the future development of slice quaternionic analysis on slice-open sets is the following result, originally proved in [17]: Theorem 3.10.(Representation Formula) Let Ω ∈ τ s (H) and f : Ω → H be slice regular.For each γ ∈ P(C, Ω) and I, J, K ∈ S(γ, Ω) with J = K, we have Another important concept is the one of domain of holomorphy for slice regular functions, namely the notion of domain of slice regularity.(i (ii) Ω 2 is slice-connected and not contained in Ω.
(iii) For any slice regular function f on Ω, there is a slice regular function Here the situation is different from complex analysis, since not every slice-open set is a domain of slice regularity.It is true however that the σ-balls and axially symmetric slice-open sets are particular domains of slice regularity, see Proposition 9.5 and Proposition 9.6 in [17].Below we illustrate an example: The ray starts from i 2 to ∞ and the angle between the ray and the positive real axis is π 4 + s π 2 .Given a continuous function ϕ : S → [0, 1], we define a continuous function F : S × [0, 1) → H by setting The complement of the image of F is denoted by Fix J ∈ S. We now define We proved in [17] that Ω ϕ is open in the Euclidean topology if ϕ is continuous.However it may be not open if ϕ is not continuous.For example, let ϕ : S → [0, 1] be the Dirichlet type function, i.e.
Let I ∈ S with I = ±J and |I − J| ∈ Q, where J is fixed.Then ϕ(I) = 1 and We choose a sequence Notice that ϕ(K ℓ ) = 0 and ).Notice that for each t ∈ (0, 1], we have It is clear that P I • γ 0 (t) is not an interior point of Ω in the Euclidean topology and Ω is not open in the Euclidean topology.
We now consider the function where z is a complex variable.This function admits a unique holomorphic extension Ψ s on The function Ψ ϕ : Ω ϕ → H defined by for y ≥ 0, is the unique slice regular extension of Ψ to Ω ϕ .In particular, Moreover, Ψ ϕ is a slice regular function defined on a non-open set and it fails to be extended sliceregularly to a larger slice-open set (or open set) in H.Note that Ω ϕ is open in τ σ .
Example 3.14.Using the notations of the previous example, we now construct an example of a set Ω ϕ of the form given in (3.2), such that Ω ϕ is not in τ s while it is in τ σ .We take a new curve γ s , s ∈ (0, 1] which consist of three segments.We define ), (a ray from i − 1).
Consider ϕ : S → [0, 1], defined by Finally, we point out that by modifying the functions γ s and Ψ, one can construct many similar functions which are defined on sets in τ s \τ σ .

Slice topology in cones on real alternative algebras
In [27], Ghiloni and Perotti introduced slice regular functions with values in a real alternative algebra, finite dimensional and with a fixed anti-involution, using stem functions.Stem functions were used in the literature also in relation with the Fueter mapping theorem, see [16,20,36,37].The slice regular functions in [27] coincide with the class of slice regular functions over the quaternions [24] and with the class of slice monogenic functions with values in a Clifford algebra, see [11], on some special open sets.
In this section, following [18], we introduce a class of slice regular functions on a finite-dimensional real alternative algebra A following the original idea of Gentili and Struppa, i.e. following Definition 3.1.Thus the functions slice regular in this sense do not coincide with those ones studied in [27] and for this reason we sometimes called them weak slice regular and those ones in [27] strong slice regular.
We recall that an algebra over the real numbers is said to be alternative if for any pair of elements x, y in the algebra x(xy) = x 2 y, (xy)y = xy 2 .It is immediate that every associative algebra is alternative.The converse does not hold, however every alternative algebra is power associative.
A real algebra A is called left alternative, if for any x, y ∈ A, x(xy) = (xx)y.
(See [1] for the notion of right alternative algebra, which can obviously adapted to our case).
From now on we shall assume the following: We then define a map L : where L a : A → A is the right linear map given by the left multiplication by a L a : x → ax.
Note that for any I ∈ S A , we have 4.1.Slice topology.In this subsection, we discuss the slice topology whose definition, originally given over the quaternions, can be extended to a suitable set Q A that we call the quadratic cone in the real alternative algebra A: Definition 4.3.We call τ s (Q A ) the slice topology on Q A .Open sets, connected sets and paths in the slice topology are called slice-open sets, slice-connected sets and slice-paths.
Since A is finite-dimensional real vector space of dimension, say, there is a Euclidean topology on A identified with Euclidean space R m .Since A admits a complex structure, m is an even number and so m = 2n.The cone Q A , as a subset of A, also has a Euclidean topology (i.e. the subspace topology induced by A).
As in the quaternionic case, we do not use the terminology slice-domain to denote a domain in the slice topology, and we will use instead the term slice topology-domain, in short, st-domain.
The slice topology on A has similar properties of the slice topology in H and in particular: We remark that the slice topology τ s is not always strictly finer than the Euclidean topology τ as the following simple example shows.The terminology and the results in Section 3 can be stated in this more general setting, so we do not repeat them and we refer the reader to [18].We only mention the following results: Proposition 4.7.The topological space (Q A , τ s ) is connected, local path-connected and path-connected.
The notion of st-domain is also a generalization of the notion of s-domain.Let A be a left alternative algebra and I ∈ S A .The set {θ 1 , ..., θ n } ⊂ A is called an I-basis, if is a real basis of A. Since L J (see (4.1)) is a complex structure on A, there is a J-basis for all J ∈ S A .
Various results that we have stated in the quaternionic case, e.g. the Splitting Lemma and the Identity Principle, hold also in this more general case.
In the sequel, we need the following notations and definitions: for any I ∈ S A we can define an isomorphism We have the following results: Theorem 4.10.Let Ω ∈ τ s (Q A ) and f : Ω → A be slice regular.Then f is path-slice.
4.3.Slice monogenic functions.We now consider the special case in which A = R n , the real Clifford algebra over n imaginary units e 1 , ..., e n satisfying e ı e  + e  e ı = −2δ ı, , where We recall that the slice regular functions over a real Clifford algebra are also called slice monogenic functions, and they were firstly introduced in [11].
Example 4.12.In [11], the class of slice monogenic functions is defined on where The cone R n+1 is a n + 1-dimensional real vector space, i.e.
However, as observed in [27], slice monogenic functions can be defined on a larger set, namely the quadratic cone of R n : where The slice monogenic functions in [27] are defined on symmetric open sets in Q Rn .Since a Clifford algebra is a special case of an alternative algebra, we can consider the more general case of slice monogenic functions defined on a slice-open set in Q Rn where the definition of slice monogenic is given by adapting Definition 4.9 to the present case.Thus all the results in Section 4 are valid in this case.

Weak slice regular functions over slice-cones
Some results in [18] are given in a greater generality: in fact the weak slice regular functions can be considered in the case of functions which are R 2n -valued and defined on open sets in the topological space (W d C , τ s ), where W d C is a suitable weak slice cone in [End(R 2n )] d .In [18] we proved various results for these functions, among which a representation formula, and we recall some of them in this section.
We denote by C n the set of complex structures on R 2n , i.e.
Definition 5.4.For each J ∈ End R 2n k×ℓ , define by J + the unique matrix in End R 2n ℓ×k that satisfies the Moore-Penrose conditions: (i) JJ + J = J.
Here J * is the adjoint matrix of J, i.e. the unique matrix such that x, Jy = J * x, y , ∀ x ∈ R 2n k , y ∈ R 2n ℓ .
We call J + the Moore-Penrose inverse of J.

DJ
and we call it the J-slice inverse of ζ(J), where The results below were originally proved in [18], Section 5: . . .
We now present the so-called path-representation formula, see Theorem 6.1 in [18], which is a crucial result in this function theory.
Theorem 5.7.(Path-representation Formula) Let Ω be a slice-open set in In another words, if J is not a slice-solution, we can take a non-zero element Then for each K ∈ C ker (J), we have (1, K)a = 0.
Proposition 5.11.Let C ′ ⊂ C. Then there is at least one slice-solution J ∈ (C ′ ) k of C ′ for some k ∈ N + .
As a consequence we deduce the following: where f • γ J is defined by (5.5).
Next definition gives the terminology to mention a result, originally proved in [19, Corollary 6.9], which shows that also in this more general setting a slice regular function is path-slice.

Extension Theorem
In this section, we give an extension theorem for slice regular functions.This result provides a tool for extending slice regular functions to larger definition domains.We also prove a general Path-representation Formula.
For each Ω ⊂ W d C , we define C is called axially symmetric, if for each x + yI ∈ Ω, then x + yC ⊂ Ω. Proposition 6.4.Let Ω be an axially symmetric slice-domain in Proof.The proof follows the reasoning in [17,Remark 7.7].Proposition 6.5.Let Ω ⊂ W d C be an axially symmetric slice-domain and f : Ω → R 2n be path-slice (or slice regular).Then f is a slice function.
Proof.Since slice regular function are path-slice, we can assume that f is path-slice.
For each z = x + yi ∈ Ω s .We choose a fixed complex structure K z ∈ C. By definition, the point q := x + yK z belongs to Ω. Since Ω Kz is a domain in C d Kz and Ω R = ∅.We can choose a fixed path As Ω is an axially symmetric slice-domain, it follows from definition that C(Ω, It is clear that f is a slice function with its stem function F : Ω s → (R 2n ) 2×1 defined by For any J = (J 1 , ..., J k ) T ∈ C k , we set Convention.For each J ∈ C k * .We fix a slice-half subset C + J of C with respect to J. Associated with J ∈ C k * and U ∈ τ [J], we introduce the following sets: where U * R is the union of the connected components of U * C intersecting R d .Finally, we define By definition, it is easy to check that U * ∆ is an axially symmetric slice-open set and Otherwise, J is a slice-solution of C, then Proof.(6.6), (6.7) and (6.Theorem 6.9.(Extension Theorem) J = (J 1 , ..., J k ) T ∈ C k * , and can be extended to a slice regular function f : U ∽ ∆ → R 2n .Proof.According to Lemma 5.6, for each I ∈ C ker (J), we can define a holomorphic function g[I] : Note that K is a slice-solution of C, and then C ker (K) = C. Again by Lemma 5.6, for each I ∈ C, we can define a holomorphic function Moreover, by definition and (5.4), for each L 1 ∈ C ker (J) and L 2 ∈ C, Otherwise, J is a slice-solution.We define a function f : Similarly, f is slice regular, and

Domains of slice regularity
In this section, we consider domains of slice regularity for slice regular functions.These are the counterparts in this framework of the domains of holomorphy in several complex variables.Using the methods in [18, Section 7] we generalize some results proved in [17,Section 9] from the case of quaternions to a general case of LSCS algebras, which includes the case of alternative algebras.For example, for several variables, an axially symmetric slice-open set Ω is a domain of slice regularity if and only if one of its slice Ω I , I ∈ C a domain of holomorphy in C d I .We also define a generalization of σ-balls, called hyperσ-polydiscs and we give a property of domains of slice regularity, see Proposition 7.12.This proposition extends [17, (ii) Ω 2 is slice-connected and not contained in Ω. (iii) For any slice regular function f on Ω, there is a slice regular function f on Ω 2 such that f = f in Ω 1 .Moreover, if there are slice-open sets Ω, Ω 1 , Ω 2 satisfying (i)-(iii), then we call (Ω, Ω 1 , Ω 2 ) a slice-triple.
In a similar way, we give the following definition: and not contained in Ω I .(iii) For any slice regular function f on Ω, there is a holomorphic function f :  ) is an I-triple.Proof."⇒" Let Ω be a domain of slice regularity and let us suppose, by absurd, that there is an I-triple Let f : Ω → R 2n be a slice regular function.By Extension Theorem 6.9, where we take J := (I) and slice-connected and not contained in Ω.Moreover, for any slice regular function f : Ω → R 2n , there is a slice regular function f We conclude that (Ω, W 1 , W 2 ) is a slice-triple, and Ω is not a domain of slice regularity, a contradiction."⇐" Now we prove the converse, i.e. a slice-open set Ω is a domain of slice regularity if for each ) is an I-triple.So we suppose that Ω is not a domain of slice regularity.Then there are slice-open sets W 1 , W 2 such that (Ω, W 1 , W 2 ) is a slice-triple.Let U be a slice connected component of Ω ∩ W 2 with U ∩ W 1 = ∅.By the Identity Principle 5.3, (Ω, U, W 2 ) is also a slice-triple.
We claim that for any I ∈ C, U I is a union of some connected components of Ω I ∩ (W 2 ) I in C d I .(This follows from the general fact that if Σ be a slice-open set and U be a slice-connected component of Σ.Then for each I ∈ C, U I is a union of some connected components of Σ I ).Then for any I ∈ C, By Lemma 7.3, (W 2 ) J ∩∂ J U J = ∅ for some J ∈ C. Let p ∈ (W 2 ) J ∩∂ J U J .By (7.1), we have p ∈ ∂ J Ω J , and then p / ∈ Ω J .Let W 3 be the connected component of (W 2 ) J containing p in C d J , and J and not contained in Ω J (by p ∈ W 3 and p / ∈ Ω J ). (iii) Since (Ω, U, W 2 ) is a slice-triple, for any slice regular function f on Ω, there is a slice regular function ) is a J-triple, a contradiction and the assertion follows.
1 ∈ R 2n is fixed in (5.1).Using the fact that Ω is axially symmetric, and the Extension Theorem 6.9 where we set J = (I), U = (Ω), we deduce that any holomorphic function f := F θ I 1 : Ω I → C I θ I 1 ⊂ R 2n can extend to a slice regular function f defined on Ω = U ∽ ∆ (see Example 6.7), where F : Ω I → C I is a holomorphic function in several complex analysis.
Since (Ω, V 1 , V 2 ) is an I-triple, the function f | V1 = f | V1 can extend to a holomorphic function f : According to Splitting Lemma 5.2 and Identity Principle in several complex analysis, we have f = F θ I 1 for some holomorphic functions F : In summary, for any holomorphic function F : Ω I → C I , there is a holomorphic function F : ) is a triple for several complex analysis, and then Ω I is not a domain of holomorphy in C d I ."⇒" On the contrary, suppose that Ω is an axially symmetric slice-open set such that for some I ∈ C Ω I is not a domain of holomorphy in d I .Then there are domains ) is a triple for several complex analysis, i.e. the following (i), (ii) and (iii) hold.Then we deduce that (iv) also holds. (i (iv) For any slice regular function f : Ω → R 2n , it follows from Splitting Lemma 5.2, Identity Principle in several complex analysis and (iii) that there is a holomorphic function ) is an I-triple.And then by Proposition 7.4, Ω is not a domain of slice regularity.
Definition 7.6.J = (J 1 , ..., J k ) T ∈ C k .We say that J is a hyper-solution of C. If J is not a slice-solution of C and for each I ∈ C\C ker (J), (J 1 , ..., J k , I) T is a slice-solution of C. Then for each K ∈ C A , (K) is a hyper-solution of C A .Definition 7.8.Let q = x + yJ 1 ∈ W d C , J = (J 1 , J 2 , ..., J m ) T be a hyper-solution of C and r ∈ R d + .We call hyper-σ-polydisc with hyper-solution J, center q and radius r.Here P (z, r) is a polydisc in C d with center z = x + yi and radius r.It is easy to check from (1, I 1 )a = 0 that a 1 , a 2 = 0. Consider the function g 1 : Σ(p, r, I) ∩ C d I1 → R 2n defined by We know that from Splitting Lemma 5.2 (by taking ξ 1 = a 2 ) and classical complex analysis arguments, we have that g 1 does not extend to a holomorphic function near any point of the boundary of Σ I1 (p, r, I) := Σ(p, r, I) ∩ C I1 .
Obviously, there are two function F, G : D(x 0 + y 0 i, r) → R such that . By (7.2), we have Similarly, we can define holomorphic function g ℓ : [Σ(p, r, I)] I ℓ → R 2n , ℓ = 1, ..., m by Then we can take a function h : ) is also a K 0 -triple.By the same method for (7.1) we have p, r, I)) .If K 0 ∈ ±C ker (I), then the holomorphic function f K0 : Σ K0 (p, r) → R 2n can extend to a holomorphic function near a point of the boundary of Σ K0 (p, r, I), a contradiction.
Otherwise, K 0 / ∈ ±C ker (I).Take z There is r ′ ∈ R + such that Proof.We shall prove this by contradiction.Suppose that γ I ⊂ Ω for some I ∈ C ker (J).By Lemma 5.5, there is a domain Since I ∈ C ker (J), it follows by Extension Lemma 5.6 that there is a holomorphic function g For each fixed w ∈ U R , it follows from Splitting Lemma 5.2 that the Taylor series of holomorphic functions g[I] I and f I at the point w are the same.Therefore there is sufficiently small r ∈ R + such that B I (w, r) ⊂ Ψ I i (U ) ∩ Ω I and g It is easy to check that (Ω, B I (w, r), Ψ I i (U )) is an I-triple.It implies by Proposition 7.4 that Ω is not a domain of slice regularity, a contradiction.Therefore γ I ⊂ Ω for each I ∈ C ker (J).

Taylor series
When dealing with slice regular functions an important concept is that one of slice derivative.In this section, we define this concept in the very general case of functions with values in the Euclidean space R 2n and then we prove that there is a Taylor series for weak slice regular functions over slice-cones.8.1.Slice derivatives.In this subsection, we generalize the slice derivative to weak slice regular functions over slice-cones.We use the notations and definitions given in Section 5.
We now introduce some terminology: the real part and imaginary parts of W d C are defined by For
Let ℓ ∈ {1, ..., d} and I ∈ C. Since f (β) is weak slice regular, f (β) I is holomorphic.By induction hypothesis and recalling the notation (8.1), the function is holomorphic.Since the choice of I is arbitrary, it follows that f (β+θ ℓ ) is weak slice regular and (8.2) holds when α = β + θ ℓ .By induction, for any α ∈ N n , f (α) is weak slice regular and (8.2) holds.Hence, we can write P (z, r) instead of P I (z, r), without ambiguity.We call P (z, r) the σ-polydisc with center z and radius r.
For simplicity, below we shall write (q−q 0 ) * α instead of (id−p) * α (q) and (q−p) * α a instead of [(q−p) * α ](a).Since the choice of z and K is arbitrary, f K is holomorphic and then f is weak slice regular.

Lemma 2 . 2 .Definition 2 . 3 .
The family τ s (H) := {Ω ⊂ H : Ω is slice-open} defines a topology of H.We call τ s (H) the slice topology on H. Open sets, connected sets and paths in the slice topology are called slice-open sets, slice-connected sets and slice-paths.A domain in the slice topology is called slice topology-domain or, in short, st-domain.

Definition 3 .
11.A slice-open set Ω ⊂ H is called a domain of slice regularity if there are no slice-open sets Ω 1 and Ω 2 in H with the following properties.

Example 3 . 12 .
The σ-ball Σ( I 2 , 1) is a domain of slice regularity for any I ∈ S. The function f : Σ( I 2 , 1) → H defined by f (q) = n∈N (q − I/2) * 2 n , does not extend to a slice regular function near any point of the boundary in any slice C J , J ∈ S.This example shows that there exist slice regular functions defined on a slice-open set which is not open in the Euclidean topology but it is open in the τ σ topology.Moreover, there are examples of functions defined on slice-open sets which are neither open in the Euclidean topology nor in τ σ , as the Example 3.14 shows.The example is obtained by further elaborating Example 3.13 which comes from ideas in [17], Section 8. Example 3.13.We consider a ray γ s so that the point 0 is not an interior point in Ω ϕ in the topology τ σ .The function Ψ ϕ defined above is a slice regular function defined in a non-open set and it could not be extended slice regularly to a larger slice-open set (or open set or open set in τ σ ) in H.
Example 4.6.C is an algebra satisfing Assumption 4.1.The slice topology and Euclidean topology coincide on C.

4. 2 .
Main results in left alternative algebra.The slice regularity given in Definition 3.1 can be extended to the case of a left alternative algebra satisfying Assumption 4.1, and in this subsection we state some main results about this class of functions.Definition 4.9.Let Ω ∈ τ s (Q A ).A function f : Ω → A is called slice regular if and only if for each I ∈ S A , f I := f | ΩI is (left I-)holomorphic, i.e. f I is real differentiable and For each path γ : [0, 1] → C, we define its corresponding path in C d I by γ I := P I • γ.Finally, for any Ω ⊂ Q A and γ ∈ P(C) we set P(C, Ω) := {δ ∈ P(C) : ∃ I ∈ S A , s.t.δ I ⊂ Ω} and S A (γ, Ω) := {I ∈ S A : γ I ⊂ Ω}.Mimicking Definition 3.6, given Ω ⊂ Q A , we say that a function f : Ω → A is path-slice if there is a function F : P(C, Ω) → A 2×1 such that f • γ I (1) = (1, L I )F (γ), for each γ ∈ P(C, Ω) and I ∈ S A (γ, Ω).

C
n := T ∈ End R 2n : T 2 = −1 , where the identity map id R 2n on R 2n is denoted by 1.Let C ⊂ C n with C = −C.We call W d C := I∈C C d I the d-dimensional weak slice-cone of C, where C d I := (R + RI) d .The slice topology on W d C is defined by τ s W d C := Ω ⊂ W d C : Ω I ∈ τ (C d I ), ∀ I ∈ C , where Ω I := Ω ∩ C d I .Convention: Let Ω ⊂ W d C .Denote by τ s (Ω) the subspace topology induced by τ s W d C .Open sets, domains, connected sets and paths in τ s (Ω) are called slice-open sets, slice-domains, slice-connected sets and slice-paths in Ω, respectively.We write x + yI ∈ Ω short for x + yI ∈ Ω with x, y ∈ R d and I ∈ C. Definition 5.1.Let Ω ∈ τ s (W d C ).A function f : Ω → R 2n is called weak slice regular if and only if for each I ∈ C, f I := f | ΩI is (left I-)holomorphic, i.e. f I is real differentiable and for each ℓ = 1, 2, ..., d, I (x + yI) = 0, on Ω I .For any I ∈ C, the set {ξ 1 , ..., ξ n } ⊂ R 2n is called an I-basis of R 2n if {ξ 1 , ..., ξ n , I(ξ 1 ), ..., I(ξ n )} is a basis of R 2n as a real vector space.Lemma 5.2.(Splitting Lemma) Let Ω ∈ τ s W d C .A function f : Ω → R 2n is weak slice regular if and only if for any I ∈ C and I-basis {ξ 1 , ...., ξ n }, there are n holomorphic functions F 1 , ..., F n : Ω I ⊂ C d I → C I , such that f I = n ℓ=1 (F ℓ ξ ℓ ).Theorem 5.3.(Identity Principle) Let Ω be a slice-domain in W d C and f, g : Ω → R 2n be weak slice regular.Then the following statements holds.(i) If f = g on a non-empty open subset U of Ω R , then f = g on Ω. (ii) If f = g on a non-empty open subset U of Ω I for some I ∈ C, then f = g on Ω.For any I ∈ C, let us choose a fixed I-basis of R 2n which is denoted by (5.1) θ I := θ I 1 , ..., θ I n , and let us consider the 2 n × 2 n real matrix D I given by (5.2) Let γ : [0, 1] → C d and I ∈ C. We define the image path of γ in C d I byγ I := Ψ I i • γ, where Ψ I i : C d → C d I , x + yi → x + yI is an isomorphism.As we already did in the preceding sections, we define:P(C d ) := {γ : [0, 1] → C d , γ is a path s.t.γ(0) ∈ R d }; for any Ω ⊂ W dC and for an arbitrary, but fixed γ ∈ P C d we define P C d , Ω := δ ∈ P C d : ∃ I ∈ C, s.t.Ran(δ I ) ⊂ Ω ; C(Ω, γ) := I ∈ C : Ran(γ I ) ⊂ Ω , where Ran(•) denotes the range.Let J = (J 1 , ..., J k ) T ∈ C k , Ω ⊂ W d C and γ ∈ P(C d , Ω).We define (5.3) C ker (J) := I ∈ C : ker(1, I) ⊃ ker[ζ(J)] = k ℓ=1 ker(1, J ℓ ) , and C(Ω, γ, J) := C(Ω, γ) ∩ C ker (J), where ker(•) is the kernel of a map, and ker(1, J ℓ ) stands for ker((1, J ℓ )).

Remark 5 . 10 .ker( 1 ,
Note that for any I ∈ C, It is easy to check that, by definition (see (5.3)), J ∈ C k is a slice-solution of C if and only if k ℓ=1 J ℓ ) = {0}.
8) hold directly by definition.Note that (U ℓ \R d ) ∪ (U ∆ ) R (if I = ±J ℓ ), Ψ I i (U C ) and Ψ I i (U * C ) are open in C d I for each I ∈ C. Therefore by (6.7) and (6.8), U ∽ ∆ ∩ C d I is open in C d I for each I ∈ C. By definition, U ∽ ∆ is a slice-open set.

Proposition 9. 7 ]
Definition 7.1.A slice-open set Ω ⊂ W d C is called a domain of slice regularity if there are no slice-open sets Ω 1 and Ω 2 in W d C with the following properties.

Lemma 7 . 3 .
Let U be a non-empty slice-open set and Ω be an slice-domain with U Ω. Then Ω∩∂ I U I = ∅ for some I ∈ C, where ∂ I U I is the boundary of U I in C d I .Proof.This proof is similar with the proof of [17, Theorem 9.3].Suppose that Ω ∩ ∂ I U I = ∅ for each I ∈ C. Since Ω I and

Proposition 7 . 5 .
Any axially symmetric slice-open set Ω is a domain of slice regularity, if and only if Ω I is a domain of holomorphy in C d I for some I ∈ C. Proof."⇐" Suppose that an axially symmetric slice-open set Ω is not a domain of slice regularity.By Proposition 7.4, there is an I

8. 2 .
Taylor series.In this subsection, we shall prove a Taylor expansion on σ-polydiscs for weak slice regular functions.To this end, we consider I ∈ C and r = (r 1 , ..., r d ) ∈ R d + = (0, +∞] d .For any z = (z 1 , ..., z d ) ∈ C d I , we denote the polydisc with center z and radius r by P I (z, r) := w = (w 1 , ..., w d ) ∈ C d I : |z ı − w ı | ≤ r ı , ı = 1, ..., n , and we set P I (z, r) := {x + yJ ∈ W d C : J ∈ C, and x ± yI ∈ P I (z, r)} P I (z, r).It is easy to check that if z ∈ C d I ∩ C d J , for some I, J ∈ C, then P I (z, r) = P J (z, r).

where s = 1 2
[f (x + yI) + f (x − yI)] , r = − 1 2 [f (x + yI) − f (x − yI)].It iseasy to check that |r|, |s| ≤ M and |JIr| ≤ |J||Ir| ≤ |J||I||r| ≤ |J||I|M, so (8.3) holds.Theorem 8.8.(Taylor series) Let Ω ∈ τ s W d so 0 is not an interior point of U in the slice topology, and U is not slice-open, which is a contradiction.Hence |S ǫ | is finite.On the other hand, for each I ∈ S, dist CI (0, H\O I ) > 0, since O I is open in C I .Then I ∈ S 1 n for some n ∈ N + and S = Definition 3.1.Let Ω be a slice-open set in H.A function f : Ω → H is called (left) slice regular, if f I := f | ΩI is left holomorphic for any I ∈ S, i.e. if f is real differentiable and satisfies Hence Ω is the disjoint union of the nonempty slice-open sets Ω ∩ (W d C \U ) and Ω ∩ U .It implies that Ω is not slice-connected, a contradiction.Proposition 7.4.A slice-open set Ω ⊂ W d C is a domain of slice regularity if and only if for any I ∈ C there are no open sets U 1 and U 2 in C d I such that (Ω, U 1 , U 2