1 Introduction and Motivation

In order to develop further our previous idea [1]; Ławrynowicz et al. [16] of identification of two ternaries and three binaries with the help of noncommutative Galois extension applied to fractals and chaos related to Ising [2]–Onsager [3]–Zhang [4] lattices, in this considerations we start with involving the Galois extension structure of the nonion algebra and analyzing the ternary and binary structures of \({\mathfrak {su}}(3)\). Our approach is also motivated by unclear statements in the Zhang [4] paper and further treatise [5, 6]. The main results of this paper are summarized as Theorems 1–3. Next we analyze the identification of construction of the collection of two ternaries with the collection of three binaries, and observe that the approach is applicable to quarks and elementary particles including introduction of colours. Our results announced above provide an example of a mathematical procedure applicable parallelly in the physics of condensed matter and physics of elementary particles. In this place it seems naturally to quote the suprising results due to [58,59,60,61,62,63,64,65] dealing with discovery of topological phase transition and topological phases matter [57].

Finally we suggest an analysis of three quaternaries vs. four ternaries, involving duodevicenion and/or quindenion algebra. More precisely, we suggest the construction of the collection of four quaternaries and the corresponding Dirac-like operators in connection with the nonion algebra. We suggest to construct then the collection of four quaternaries and the corresponding Dirac-like operators in connection with the duodevicenion or quindenion algebra [52,53,54] and its ternary extension. The next steps suggested are noncommutative Galois extensions and a study of their basic relations and Galois extensions of ternary Clifford type [67], quaternary Galois extensions and Galois extensions of quaternary Clifford type [9], the Galois extension structure of the duodevicenion and quindenion algebras, quaternary and ternary structure of \({\mathfrak {su}}(3)\), ternary and quaternary Dirac-like operators of noncommutative Galois extensions [26, 27, 68] and, finally, identification of the constructed collection of three quaternaries with the proper collection of four ternaries.

An approach of ternary numbers, algebras, and complex analysis, coming back to geometric ideas suggested in [66].

2 The Galois Extension Structure of the Nonion Algebra

We recall the concept of nonion algebra \({\mathbb {N}}\) [52,53,54] and discuss its Galois extension structure.

Consider the following matrices

$$\begin{aligned}&Q_1=\left( \begin{array}{ccc}0&{}\quad {\mathbf {j}}&{}\quad 0\\ 0&{}\quad 0&{}\quad {\mathbf {j}}^2\\ 1&{}\quad 0&{}\quad 0\end{array}\right) ,\;\; Q_2=\left( \begin{array}{ccc}0&{}\quad {\mathbf {j}}^2&{}\quad 0\\ 0&{}\quad 0&{}\quad {\mathbf {j}}\\ 1&{}\quad 0&{}\quad 0\end{array}\right) ,\;\; Q_3=\left( \begin{array}{ccc}0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 1\\ 1&{}\quad 0&{}\quad 0\end{array}\right) ;\quad \quad \end{aligned}$$
(2.1)
$$\begin{aligned}&\overline{Q}_1=\left( \begin{array}{ccc}0&{}\quad 0&{}\quad 1\\ {\mathbf {j}}^2&{}\quad 0&{}\quad 0\\ 0&{}\quad {\mathbf {j}}&{}\quad 0\end{array}\right) ,\;\; \overline{Q}_2=\left( \begin{array}{ccc}0&{}\quad 0&{}\quad 1\\ {\mathbf {j}}&{}\quad 0&{}\quad 0\\ 0&{}\quad {\mathbf {j}}^2&{}\quad 0\end{array}\right) ,\;\; \overline{Q}_3=\left( \begin{array}{ccc}0&{}\quad 0&{}\quad 1\\ 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0\end{array}\right) ;\quad \quad \end{aligned}$$
(2.2)
$$\begin{aligned}&R_1=\left( \begin{array}{ccc}1&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 1\end{array}\right) ,\;\; R_2=\left( \begin{array}{ccc}1&{}\quad 0&{}\quad 0\\ 0&{}\quad {\mathbf {j}}&{}\quad 0\\ 0&{}\quad 0&{}\quad {\mathbf {j}}^2\end{array}\right) ,\;\; R_3=\left( \begin{array}{ccc}1&{}\quad 0&{}\quad 0\\ 0&{}\quad {\mathbf {j}}^2&{}\quad 0\\ 0&{}\quad 0&{}\quad {\mathbf {j}}\end{array}\right) . \end{aligned}$$
(2.3)

where \({\mathbf {j}}\) is one of the roots of \(z^3-1=0\) different from 1. The matrix algebra which is generated by two of the three elements (2.1) over \({\mathbb {R}}[\root 3 \of {I_3}]\) is called nonion algebra \({\mathbb {N}}\).

Consider in addition the matrices of the form

$$\begin{aligned}&T_1=I_6:=\left( \begin{array}{cc} I_3 &{}\quad 0_3 \\ 0_3 &{}\quad I_3\end{array}\right) ,\;\; T_2=\left( \begin{array}{cc} T^0_1 &{}\quad 0_3 \\ 0_3 &{}\quad T^0_1\end{array}\right) ,\;\; T_3=\left( \begin{array}{cc} T^0_2 &{}\quad 0_3 \\ 0_3 &{}\quad T^0_2\end{array}\right) , \end{aligned}$$
(2.4)
$$\begin{aligned}&T_4=\left( \begin{array}{cc} 0_3 &{}\quad I_3 \\ -I_3 &{}\quad 0_3\end{array}\right) ,\;\; T_5=\left( \begin{array}{cc} 0_3&{}\quad T^0_1 \\ -T_1^0 &{}\quad 0_3\end{array}\right) ,\;\; T_6=\left( \begin{array}{cc} 0_3&{}\quad T^0_2 \\ -T_2^0 &{}\quad 0_3\end{array}\right) ,\qquad \end{aligned}$$
(2.5)

where

$$\begin{aligned} T_1^0=\left( \begin{array}{ccc}0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 1\\ 1&{}\quad 0&{}\quad 0\end{array}\right) ,\;\; T_2^0=\left( \begin{array}{ccc}0&{}\quad 0&{}\quad 1\\ 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad 0\end{array}\right) . \end{aligned}$$
(2.6)

We recall that the nine elements (2.1)–(2.3) constitute linear basis of the nonion algebra \({\mathbb {N}}\). The matrix algebra which is generated by two of the three elements (2.5) over the real field \({\mathbb {R}}\) will be denoted by \({\mathbb {B}}^\prime \). The algebra generated by \(T_1^0\) or \(T_2^0\) is called cubic algebra, denoted by \({\mathbb {B}}\). The algebra \(\widetilde{{\mathbb {N}}}\) generated by \(T_4\) is the binary extension of \({\mathbb {N}}\),

$$\begin{aligned} \widetilde{{\mathbb {N}}}={\mathbb {N}}[\root 2 \of {I_3}]\; : \;\widetilde{{\mathbb {N}}}=\{x+yT_4 \,|\,x,y \in {\mathbb {N}}\}. \end{aligned}$$

The three elements (2.4) are a linear basis of \({\mathbb {B}}\). The six elements (2.4), (2.5) form a linear basis of \({\mathbb {B}}'\). Besides, \({\mathbb {B}}\) and \({\mathbb {B}}'\) are subalgebras of \({\mathbb {N}}\) and \(\widetilde{{\mathbb {N}}}\) respectively. We can prove the above assertions with the help of the two enclosed product tables.

Now, owing to hints [13] and motivation given in references [7, 8, 13, 15], following some earlier demands appearing in references [11, 12, 16,17,18], we can prove the following basic theorems on the Galois extensions.

Theorem 1

(1) The nonion algebra is a ternary Galois extension of the algebra \({\mathbb {B}}\,:\,{\mathbb {N}}={\mathbb {B}}[\root 3 \of {I_3}]\). The extension can be realized by \({\mathbb {B}}[ \tau ]\;(\tau ^3=I_3)\) with the choice of \(\tau = Q_i,\,\bar{Q}_i\) (\(i=1,2,3\)). (2) \(\widetilde{{\mathbb {N}}}\) is a binary extension of \({\mathbb {B}}^\prime \,:\,\widetilde{{\mathbb {N}}}={\mathbb {B}}^\prime [\root 2 \of {I_3}]\). Hence we have the following commutative diagram:

figure a

Proof will be given in the next section.

Theorem 2

(3) We have the following Galois extensions:

$$\begin{aligned} \left\{ \begin{array}{lll} {\mathbb {A}}[R_1, R_2, R_3]&{}=&{}\{xR_1+yR_2+zR_3\,|\,x,y,z\in {\mathbb {R}}[{\mathbf {j}}]\},\\ {\mathbb {A}}[R_1, Q_i, \bar{Q_i}]&{}=&{}\{xR_1+yQ_i+z\bar{Q_i}\,|\,x,y,z\in {\mathbb {R}}[{\mathbf {j}}]\},\;\;(i=1,2,3), \\ {\mathbb {A}}[R_1, \bar{Q_i}, Q_i]&{}=&{}\{xR_1+y\bar{Q_i}+zQ_i\,|\,x,y,z\in {\mathbb {R}}[{\mathbf {j}}]\},\;\;(i=1,2,3). \end{array} \right. \end{aligned}$$
(2.8)

The extension does not depend on the choice of \(\tau \) with \({\mathbb {B}}[\tau ]\;\;(\tau ^3=I_3)\): we have

$$\begin{aligned} {\mathbb {N}}={\mathbb {A}}[R_1, Q_1, \bar{Q_1}]={\mathbb {A}}[R_1, Q_2, \bar{Q_2}]={\mathbb {A}}[R_1, Q_3, \bar{Q_3}]. \end{aligned}$$
(2.9)

(4) \(Q_i\), \(\bar{Q_j}\) \((i,j=1,2,3)\) give a part of generators of the Galois group of \({\mathbb {N}}\) : \( {\mathbb {N}}={\mathbb {B}}[\root 3 \of {I_3}]\). Namely putting

$$\begin{aligned} \begin{array}{c} {\mathbb {A}}_U[R_1, R_2, R_3]=\{xR_1+yUR_2+z\bar{U}R_3\,|\,x,y,z\in {\mathbb {R}}[{\mathbf {j}}]\},\\ {where}\;\; U =Q_i, \bar{Q_i},\;(i=1,2,3), \end{array} \end{aligned}$$
(2.10)

we can obtain new Galois extensions

$$\begin{aligned} \left\{ \begin{array}{l} {\mathbb {A}}_{Q_1}[R_1, R_2, R_3]={\mathbb {A}}[R_1, Q_2, \bar{Q_2}],\;\; {\mathbb {A}}_{Q_2}[R_1, R_2, R_3]={\mathbb {A}}[R_1, Q_3, \bar{Q_3}],\\ {\mathbb {A}}_{\bar{Q_1}}[R_1, R_2, R_3]={\mathbb {A}}[R_1, Q_3, \bar{Q_3}],\;\; {\mathbb {A}}_{\bar{Q_2}}[R_1, R_2, R_3]={\mathbb {A}}[R_1, Q_1, \bar{Q_1}], \\ {\mathbb {A}}_{R_2}[R_1, Q_1, \bar{Q_1}]={\mathbb {A}}[R_1, Q_2, \bar{Q_2}],\;\; {\mathbb {A}}_{R_2}[R_1, Q_2, \bar{Q_2}]={\mathbb {A}}[R_1, Q_3, \bar{Q_3}], \\ {\mathbb {A}}_{R_2}[R_1, Q_3, \bar{Q_3}]={\mathbb {A}}[R_1, Q_1, \bar{Q_1}],\;\; {\mathbb {A}}_{\bar{Q}_1}[R_1, R_2, R_3]={\mathbb {A}}[R_1, Q_3, \bar{Q_3}], \\ {\mathbb {A}}_{\bar{Q}_2}[R_1, R_2, R_3]={\mathbb {A}}[R_1, Q_1, \bar{Q_1}], \;\; {\mathbb {A}}_{\bar{Q}_3}[R_1, R_2, R_3]={\mathbb {A}}[R_1, Q_2, \bar{Q_2}]. \end{array} \right. \quad \quad \quad \end{aligned}$$
(2.11)

(5) We have the following results for the adjoint operations:

$$\begin{aligned} \left\{ \begin{array}{llll} \mathrm {Ad}_{Q_i}R_1=R_1,&{} \mathrm {Ad}_{Q_i}R_2={\mathbf {j}}R_2,&{} \mathrm {Ad}_{Q_{i1}}R_3={\mathbf {j}}^2R_3, &{}\;(i=1,2,3),\\ \mathrm {Ad}_{Q_i}Q_1=Q_1,&{} \mathrm {Ad}_{Q_i}Q_2={\mathbf {j}}Q_2,&{} \mathrm {Ad}_{Q_{i1}}Q_3={\mathbf {j}}^2Q_3, &{}\;(i=1,2,3),\\ \mathrm {Ad}_{Q_i}\bar{Q_1}=\bar{Q_1},&{} \mathrm {Ad}_{Q_i}\bar{Q_2}={\mathbf {j}}^2\bar{Q_2},&{} \mathrm {Ad}_{Q_{i1}}\bar{Q_3}={\mathbf {j}}\bar{Q_3}, &{}\;(i=1,2,3), \end{array} \right. \quad \end{aligned}$$
(2.12)

where \({\mathbf {j}}\ne 1\), \({\mathbf {j}}^3=1\).

The proof will be given in the next section.

3 Proofs of Basic Theorems 1 and 2 [Assertions (1)–(5)]

Proof. Ad (1). We notice that \({\mathbb {B}}\) is the commutative Galois extension: \({\mathbb {B}}={\mathbb {R}}[\root 3 \of {I_3}]\).

Then we observe that, for a Clifford algebra \({\mathcal {A}}\) with generators \(T_1,T_2,\ldots ,T_n\), there is a sequence of noncommutative binary Galois extensions of \({\mathbb {R}}\) which realizes the given Clifford algebra \({\mathcal {A}}\) [19].

Indeed, let us prove the above statement by induction with respect to m, where \({\mathcal {A}}_0={\mathbb {R}}\), \({\mathcal {A}}_m={\mathcal {A}}\), and

$$\begin{aligned} T_iT_j+T_jT_i=-\delta _{ij}I_n \;\Rightarrow \; {\mathcal {A}}_k={\mathcal {A}}_{k-1}[\root 2 \of {-I_n}],\;\;k=1,2,\ldots ,m. \end{aligned}$$

Complex numbers can be obtained by commutative extensions of real numbers. Now, setting

$$\begin{aligned} {\hat{T}}_i=\left( \begin{array}{c@{\quad }c}T_{i} &{} 0 \\ 0 &{} -T_{i} \end{array} \right) ,\;\; i=1,2,\ldots ,m;\;\; {\hat{H}}_{n+1}=\left( \begin{array}{c@{\quad }c} 0 &{} I_{n} \\ -I_{n} &{} 0 \end{array} \right) , \end{aligned}$$

we get Clifford algebra which is generated by \(({\hat{T}}_1, {\hat{T}}_2, \ldots , {\hat{T}}_n, {\hat{H}}_{n+1})\) on one hand, and the right (or left) module binary extension of \({\mathcal {A}}_n\) by \({\hat{H}}_{n+1}\) on the other hand.

If we choose in (1) \(\tau = Q_i\overline{Q}_i,\) \(i=1,2,3,\) we make the Galois extension \({\mathbb {B}}[\root 3 \of {I_3}]\). Then we can see that it is identical with \({\mathbb {N}}\).

Ad (2). We notice that \({\mathbb {B}}^\prime \) is the noncommutative Galois extension of \({\mathbb {B}}\) : \({\mathbb {B}}^\prime ={\mathbb {B}}[\root 2 \of {I_3}]\), where \(\root 2 \of {I_3}=T_4\). Choosing \(\tau =T_4\), we make the Galois extension. Then we can see that it is identical with \(\widetilde{{\mathbb {N}}}\) : \(\widetilde{{\mathbb {N}}}={\mathbb {B}}^\prime [\root 2 \of {I_3}]\).

Ad (3)–(5). Clearly, for \(i=1\) we have

$$\begin{aligned} {\mathbb {R}}[\mathbf{j}, Q_1]=\{xR_1+yQ_1+zQ_{1}^{2} | x,y,z \in {\mathbb {R}}[\mathbf{j}]\}. \end{aligned}$$

From \(Q_1^2=\bar{Q_1}\) it follows that

$$\begin{aligned} {\mathbb {R}}[\mathbf{j}, Q_1]=\{xR_1+yQ_1+z\bar{Q_1} | x,y,z\in {\mathbb {R}}[\mathbf{j}]\}. \end{aligned}$$

Hence \({\mathbb {R}}[\mathbf{j}, Q_1]\) is a ternary Galois extension with the corresponding Galois group \({1, Q_1, Q_1^2}.\) Analogously we perform the reasoning for \({\mathbb {R}}[\mathbf{j}, Q_2]\) and \({\mathbb {R}}[\mathbf{j}, Q_3].\)

In general, we follow the enclosed schemes (Fig. 1) which indicate how to use in the clockwise way Tables 1 and 2. The scheme is clarified by a detailed calculation in the case expressed in the second line of the formulae (2.10).

Fig. 1
figure 1

Schemes illustrating reasoning leading to Assertions (1)–(5)

Full size image

4 Ternary and Binary Galois Extension Structures of \({\mathfrak {su}}(3)\)

In addition to the desires followed from our programme formulated in [1], the present section meets also the demands appearing in references [20,21,22,23,24,25]. We are going to discuss the structure of the Galois extension in the context of \({\mathfrak {su}}(3)\).

We recall base of \({\mathfrak {su}}(3)\) (where selected generators are proportional to the well known Gell-Mann matrices [14]):

$$\begin{aligned}&f_1=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}i&{}0\\ i&{}0&{}0\\ 0&{}0&{}0\end{array}\right) ,\;\; f_2=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}-1&{}0\\ 1&{}0&{}0\\ 0&{}0&{}0\end{array}\right) ,\;\; f_3=\left( \begin{array}{c@{\quad }c@{\quad }c}i&{}0&{}0\\ 0&{}-i&{}0\\ 0&{}0&{}0\end{array}\right) ,\;\;\\&f_4=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}0&{}i\\ 0&{}0&{}0\\ i&{}0&{}0\end{array}\right) ,\\&f_5=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}0&{}-1\\ 0&{}0&{}0\\ 1&{}0&{}0\end{array}\right) ,\;\; f_6=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}0&{}0\\ 0&{}0&{}i\\ 0&{}i&{}0\end{array}\right) ,\;\; f_7=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}0&{}0\\ 0&{}0&{}-1\\ 0&{}1&{}0\end{array}\right) ,\;\;\\&f_8=\frac{1}{\sqrt{3}} \left( \begin{array}{c@{\quad }c@{\quad }c}1&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}-2\end{array}\right) , \end{aligned}$$
Table 1 The (Q, Q̄, R)-matrices product table
Full size table
Table 2 The T-matrices product table
Full size table

and construct the three linear subspaces:

(4.1)

We notice the relation

$$\begin{aligned} f_8=\frac{{-1}}{{i} \sqrt{3}}(e'_3+e''_3), \end{aligned}$$

while \(\{ e_3, e_3^{'}, e_3^{''}\}\) is linearly dependent. Hence we can see that \((e_1,e_2,\ldots , e''_3)\) constitute the basis omitting one of \(e_3, e'_3, e''_3\).

We can prove the following basic theorem on both binary and ternary Galois extension structures on \({\mathfrak {su}}(3)\), extremely important from the point of view of the of our research programme formulated in Section 4 of [69]: identification of two ternaries with three binaries with the help of noncommutative Galois extensions; cf. [26, 27]:

Theorem 3

We have the binary and ternary extension structures on \({\mathfrak {su}}(3)\):

\(\mathbf {(6)}\) We have the following adjoint representation on \(L_i\) \((i=1,2,3)\):

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l@{\quad }l} He_1H^{-1}=-e_2 , &{}He_2H^{-1}=e_1 , &{}He_3H^{-1}=e_3 ,\\ H'e'_1H'^{-1}=-e'_2,\,&{}H'e'_2H'^{-1}=-e'_1 ,\, &{}H'e'_3H'^{-1}=e'_3 ,\\ H'e''_1H'^{-1}=e''_2 ,&{}H'e''_2H'^{-1}=e''_1 , &{}H'e''_3H'^{-1}=e''_3 ,\end{array}\right. \end{aligned}$$
(4.2)

where

$$\begin{aligned} H= \left( \begin{array}{c@{\quad }c@{\quad }c}1&{}0&{}0\\ 0&{}i&{}0\\ 0&{}0&{}1\end{array}\right) ,\;\; H'=\left( \begin{array}{c@{\quad }c@{\quad }c}1&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}i\end{array}\right) . \end{aligned}$$
(4.3)

\(\mathbf {(7)}\) We can obtain the following commutation relations:

$$\begin{aligned} \left\{ \begin{array}{l} e_1^2=e_2^2=e_3^2=-1\\ e_1e_2=-e_2e_1=e_3,\;e_2e_3=-e_3e_2=e_1,\;e_3e_1=-e_1e_3=e_2. \end{array}\right. \end{aligned}$$
(4.4)

After the central extension, we have the Clifford algebra which is isomorphic to the quaternion algebra. For the case of \(e_i'\) and \(e_i''\) \((i=1,2,3)\), we have the same assertions on \(L_i\) \((i=1,2,3)\). Hence we obtain the Dirac-like operators desired.

\(\mathbf {(8)}\) We have

$$\begin{aligned} \begin{array}{l} G_1e_kG_1^{-1}=e'_{k} \;(k=1,2,3),\;G_1e'_kG_1^{-1}=e''_k (k=1,2),\;G_1e'_3G_1^{-1}=-e''_3\\ G_1e''_kG_1^{-1}=e_k \;(k=1,3),\;G_1e''_2G_1^{-1}=-e_2, \end{array} \end{aligned}$$
(4.5)

where

$$\begin{aligned} G_1=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}1&{}0\\ 0&{}0&{}1\\ 1&{}0&{}0 \end{array} \right) . \end{aligned}$$
(4.6)

By this result, we can find a ternary commutative Galois extension. Hence we can introduce the ternary Dirac-like operators.

Proof

Basic elements of our considerations are summarized as properties (1) of su(3), (2) of su(2), and (3) of \({\mathfrak {su}}(3)\); as schematic comparison of the commutative Galois extension structures on \({\mathfrak {su}}(3)\) (Fig. 2), and as the diagram (4.7):

$$\begin{aligned} \begin{array}{rllr} {\mathrm {(1)}}&{} {\textit{su}}(3) =L_1\cup L_2\cup L_3&{}\;\;&{}{\textit{su}}(3)\\ &{}&{}&{}\downarrow \root 3 \of {I_3}\\ \mathrm {(2)}&{} L_i (i=1,2,3)\; \mathrm { is \;isomorphic \;to\; su(2) \;and\; it\; is\; a\; binary\;}&{}&{}{\mathrm {su(2)}}\\ &{}\mathrm { Galois\; extension}\; L_i=B_0[\root 2 \of {I_3}]\; \mathrm {over} \;B_0=R[e_3]&{}&{}\downarrow \root 2 \of {I_3}\\ \mathrm {(3)}&{}{{\mathfrak {su}}}(3)\;is\; a\; ternary\; Galois\; extension\; B'[\root 3 \of {I_3}]\; \mathrm {over} \;{\mathbb {B}}^\prime ={{\mathrm {su}}}(2)\;&{}\;\;&{}{{\mathfrak {su}}(3)} \;\\ \end{array}\nonumber \\ \end{aligned}$$
(4.7)
Fig. 2
figure 2

Schematic comparison of the commutative Galois extension structures on \({\mathfrak {su}}(3)\)

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All the formulae (4.2), (4.4) and (4.5) can be checked directly from the definitions concerned. A comparison of the commutative Galois extension structures on \({\mathfrak {su}}(3)\) is visualized in Fig. 2. As we can see, \(G_1\) is isomorphism of (sub)algebras induced by the fact that they are isomorphic to \({\mathfrak {su}}(2)\) and \(H_1\) is binary extension different for each (sub)algebra by the different choice of element \(\tau ,\) that \(\tau ^2=I_3\).

5 Identification of Construction of the Collection of Two Ternaries with the Collection of Three Binaries

Starting with definitions [1]

$$\begin{aligned} x_1+x_2 i=x_1+x_2\sqrt{-1}\Leftrightarrow \left( \begin{array}{c@{\quad }c} x_1 &{} x_2 \\ -x_2 &{} x_1\end{array}\right) ,\nonumber \\ y_1+\mathbf{j}y_2+\mathbf{j}^2y_3\Leftrightarrow \left( \begin{array}{c@{\quad }c@{\quad }c}y_1 &{} y_2 &{}y_3 \\ y_3 &{} y_1 &{} y_2 \\ y_2 &{} y_3 &{}y_1\end{array}\right) \; (=Y_1) \end{aligned}$$
(5.1)

and

$$\begin{aligned} Y_2=\left( \begin{array}{c@{\quad }c@{\quad }c}y_4 &{} y_5 &{}y_6 \\ y_6 &{} y_4 &{} y_5 \\ y_5 &{} y_6 &{}y_4\end{array}\right) \;({\Leftrightarrow }y_4+\mathbf{j}y_5+\mathbf{j}^2y_6) \end{aligned}$$

with \(x_1,x_2,y_1,\ldots ,y_6 \in {\mathbb {R}},\)

we observe that

$$\begin{aligned} \left( \begin{array}{c@{\quad }c@{\quad }c}X_1&{}X_2&{}X_3\\ X_3&{}X_1&{}X_2\\ X_2&{}X_3&{}X_1\end{array}\right) \Leftrightarrow \left( \begin{array}{c@{\quad }c}Y_1&{}Y_2\\ -Y_2&{}Y_1\end{array}\right) \end{aligned}$$

follows from the definitions of \(X_1,X_2,X_3\) and \(Y_1,Y_2.\)

The identification includes a correspondence of the related ternary and binary Galois structures in the sense precised in assertions of Theorem 1–3 regarding the Galois extension structures on the nonion algebra (enlightening, in particular, the passage from cubic algebra \({\mathbb {B}}\) to the nonion algebra \({\mathbb {N}}\)) and the analogous structure on \({\text {su}}(3).\)

6 An Analogue for Sects. 2–5: Two Ternaries vs. Three Binaries for Quarks and Elementary Particles

At the beginning of this and next section we shortly present the idea which is essentially contained in [11, 27] and related papers. We present it here in a slightly altered form and order for the seek of completeness.

In the procedure (4.1)–(4.4) and Fig. 6 of [1] we may replace crystallographic lattices by quarks and elementary particles with the minimal requirements for their definition at the initial stage of model [7, 8, 11, 19,20,21]. In particular, we may start with mesons and baryons as examples of binary and ternary particles (Fig. 3).

Fig. 3
figure 3

Examples of binary and ternary particles

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We know that each meson constitutes a quark and an anti-quark, and that each baryon constitutes only three quarks or anti-quarks. This generates the duality related with the ternary Pauli exclusion principles generalized by Kerner [11] and specifying in [27] by Th. 4 (Fig. 4).

Fig. 4
figure 4

Duality related with the ternary Pauli exclusion principle

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In turn we may study the quark realization of mesons and baryons by use of the binary Galois extension on \({\mathfrak {su}}(3)\) [27], Figs. 3 and 4. We come to the quark realization of mesons and baryons by use of the binary resp. ternary Galois extension of \({\mathfrak {su}}(3)\) refering to the Kobayashi–Masukawa theory; [27, 28], Figs. 8, 9 and 10.

Following the Kobayashi–Masukawa model, we consider Galois extension of In this way we obtain six kinds of quarks. We remark that, usually, the Kobayashi–Masukawa theory is attributed to their study of two kinds of quarks.

Introducing the extension by \(e_0 \,(=diag[1,1,0])\) we may identify the quarks as up-quark, down-quark, strange-quark, as follows

$$\begin{aligned} \{e_0, e_1, e_2, e_3\}\Rightarrow u, \;\; \{e'_0, e'_1, e'_2, e'_3\}\Rightarrow d,\;\; \{e''_0, e''_1, e''_2, e''_3\}\Rightarrow s. \end{aligned}$$
(6.1)

Further, using the conjugate elements of (6.1)

$$\begin{aligned} \{\bar{e}_0, \bar{e}_1, \bar{e}_2, e_3\}\Rightarrow \bar{u}, \;\; \{\bar{e}'_0, \bar{e}'_1, \bar{e}'_2, \bar{e}'_3\}\Rightarrow \bar{d},\;\; \{\bar{e}''_0, \bar{e}''_1, \bar{e}''_2, \bar{e}''_3\}\Rightarrow \bar{s}. \end{aligned}$$
(6.2)

we realize mesons and baryons in Gell-Mann model by use of the Golois extension. Similarly, in the Kobayashi–Masukawa model—by use of the binary Galois extension structure of \({\mathfrak {su}}(3)\) [27].

Noticing the duality for the 3-generation structures of quarks responsible for flavour; cf. [29]:

the binary structure of u, c, b \(\iff \) the ternary structure of u, c, b

the binary structure of d, s, t \(\iff \) the ternary structure of d, s, t

we arrive at a cumulative scheme for elementary particles including quarks, leptons and bozons, and generation of Matter (Fig. 5) as well as the corresponding 3-generations, more precisely: 3 objects consisting of 2 collections of 2 particles, constructed from 2 proper generations, more precisely from:

  • 2 objects consisting of 3 collections of 2 particles (in each case),

  • 2 objects consisting of 2 collections of 3 particles (in either case),

  • 1 object consisting of 3 collections of 2 particles (in each case) and 1 object consisting of 2 collections of 3 particles (in either case).

Fig. 5
figure 5

A cumulative scheme for elementary particles including quarks, leptons and bozons, and generations of matter

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It is natural to give an example of a corresponding 3-generation (Fig. 6).

For introducing colours in terms of noncommutative Galois extensions we use the nonion extension of \({\mathfrak {su}}(3)\); cf. the next section.

Summing up, we replace the programme – by the following

\(\square _{13}\) :

binary and ternary elementary particles

\(\square _{14}\) :

successive extensions of the related binary and ternary extensions on \({\mathfrak {su}}(3)\)

\(\square _{15}\) :

relationships with the Gell-Mann model [13, 14] and Kobayashi–Masukawa model [28]

\(\square _{16}\) :

binary and ternary Pauli exclusion principles

\(\square _{17}\) :

quark realization of mezons by the use of the binary Galois extension on \({\mathfrak {su}}(3)\)

\(\square _{18}\) :

construction of quark models of mesons

\(\square _{19}\) :

construction of quark models for baryons

\(\square _{20}\) :

the 3-generations of quarks

\(\square _{21}\) :

adjoint representations

\(\square _{22}\) :

duality between the collections of corresponding binary and ternary Dirac operators

\(\square _{23}\) :

generation of colours and the identification problem

The steps \(\square _{13}\)–\(\square _{23}\) may be composed in the scheme shown on Fig. 7.

Fig. 6
figure 6

An example of a corresponding 3-generation constructed from 2 proper generations (proper in the sense described above)

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Fig. 7
figure 7

Ternary approach related to the nonion algebra for quarks and elementary particles

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7 Introduction of Colours

If we wish to introduce the concept of colours, we may consider the succesive ternary extension

Indeed, is the quaternionic algebra, so making its extension by \(T_2\) as in (8) we get . Next, making the extension by the nonion algebra we arrive at , as desired. This means that we apply the nonion extension of \({\mathfrak {su}}(3)\); cf. Sects. 1, 2, and 3:

$$\begin{aligned} \tilde{\sigma }_ie_k\tilde{\sigma }_i^{-1}=e_k'', \;\;i,k=1,2,3 \end{aligned}$$
(7.1)

with \(e_k\), \(e_k'',\) \(k=1,2,3\), as in (17) and, in analogy to (21) and (22):

$$\begin{aligned} \tilde{\sigma }_1=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}1&{}0\\ 0&{}0&{}1\\ 1&{}0&{}0 \end{array} \right) ,\;\; \tilde{\sigma }_2=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}{\mathbf {j}}^2&{}0\\ 0&{}0&{}{\mathbf {j}}\\ 1&{}0&{}0 \end{array} \right) ,\;\; \tilde{\sigma }_3=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}{\mathbf {j}}&{}0\\ 0&{}0&{}{\mathbf {j}}^2\\ 1&{}0&{}0 \end{array} \right) ,\;\;{\mathbf {j}}^3=1,\;\;{\mathbf {j}}\ne 1. \end{aligned}$$

For \(\tilde{\sigma }_1e_k\tilde{\sigma }_1^{-1}\) we have \(e_k\) and \(e_k''\), \(k=1,2,3,\) as in (15) and

$$\begin{aligned} \left\{ \begin{array}{l} \tilde{\sigma }_1e_1\tilde{\sigma }_1^{-1}=e_1',\;\; \tilde{\sigma }_1e_2\tilde{\sigma }_1^{-1}=e_1',\;\; \tilde{\sigma }_1e_3\tilde{\sigma }_1^{-1}=e_3', \\ \tilde{\sigma }_1e_1'\tilde{\sigma }_1^{-1}=e_1'',\;\; \tilde{\sigma }_1e_2'\tilde{\sigma }_1^{-1}=e_2'',\;\; \tilde{\sigma }_1e_3'\tilde{\sigma }_1^{-1}=e_3'', \\ \tilde{\sigma }_1e_1''\tilde{\sigma }_1^{-1}=e_1,\;\; \tilde{\sigma }_1e_2''\tilde{\sigma }_1^{-1}=-e_2,\;\; \tilde{\sigma }_1e_3''\tilde{\sigma }_1^{-1}=e_3. \\ \end{array} \right. \end{aligned}$$
(7.2)
Fig. 8
figure 8

Generation of transformations \(\tilde{\sigma }_1\), \(\tilde{\sigma }_2\), \(\tilde{\sigma }_3\) by the nonion extension of \({\mathfrak {su}}(3)\)

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For \(\tilde{\sigma }_2e_k\tilde{\sigma }_2^{-1}\), \(k=1,2,3\), we have

$$\begin{aligned}&\tilde{e}_1=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}i&{}0\\ i&{}0&{}0\\ 0&{}0&{}0 \end{array} \right) ,\;\; \tilde{e}_2=i\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}{\mathbf {j}}&{}0\\ {\mathbf {j}}^2&{}0&{}0\\ 0&{}0&{}0 \end{array} \right) ,\;\; \tilde{e}_3=\left( \begin{array}{c@{\quad }c@{\quad }c}-i&{}0&{}0\\ 0&{}0&{}0\\ 0&{}0&{}i \end{array} \right) =e'_3, \\&\tilde{\tilde{e}}_1=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}0&{}{\mathbf {j}}\\ 0&{}0&{}0\\ {\mathbf {j}}^2&{}0&{}0 \end{array} \right) ,\;\; \tilde{\tilde{e}}_2=i\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}0&{}{\mathbf {j}}\\ 0&{}0&{}0\\ {\mathbf {j}}^2&{}0&{}0 \end{array} \right) ,\;\; \tilde{\tilde{e}}_3=\left( \begin{array}{c@{\quad }c@{\quad }c}0&{}0&{}0\\ 0&{}1&{}0\\ 0&{}0&{}-1 \end{array} \right) =ie''_3, \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} \tilde{\sigma }_2e_1\tilde{\sigma }_2^{-1}=\tilde{e}_1',\;\; \tilde{\sigma }_2e_2\tilde{\sigma }_2^{-1}=\tilde{e}_2,\;\; \tilde{\sigma }_2e_3\tilde{\sigma }_2^{-1}=\tilde{e}_3, \\ \tilde{\sigma }_2\tilde{e}_1\tilde{\sigma }_2^{-1} =\tilde{\tilde{e}}_1,\;\; \tilde{\sigma }_2\tilde{e}_2\tilde{\sigma }_2^{-1}=\tilde{\tilde{e}}_2,\;\; \tilde{\sigma }_2\tilde{e}_3\tilde{\sigma }_2^{-1}=\tilde{\tilde{e}}_3=-ie_3'', \\ \tilde{\sigma }_2\tilde{\tilde{e}}_1\tilde{\sigma }_2^{-1}=e_1,\;\; \tilde{\sigma }_2\tilde{\tilde{e}}_2\tilde{\sigma }_2^{-1}=e_2,\;\; \tilde{\sigma }_2\tilde{\tilde{e}}_3\tilde{\sigma }_2^{-1}=ie_3. \\ \end{array} \right. \end{aligned}$$
(7.3)

For \(\tilde{\sigma }_3e_k\tilde{\sigma }_3^{-1}\), \(k=1,2,3\), we have

$$\begin{aligned}&e^*_{1}=i\left( \begin{array}{c@{\quad }c@{\quad }c}0 &{} 0 &{} {\mathbf {j}} \\ 0 &{} 0 &{} 0 \\ {\mathbf {j}}^2 &{} 0 &{} 0 \end{array} \right) ,\;\; e^*_2=i\left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} 0 &{} {\mathbf {j}} \\ 0 &{} 0 &{} 0 \\ -{\mathbf {j}}^2&{}0&{}0 \end{array} \right) ,\;\; e^*_3=\left( \begin{array}{c@{\quad }c@{\quad }c} -i &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} i \end{array} \right) =e'_3, \\&e^{**}_1=i\left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} {\mathbf {j}} \\ 0&{}{\mathbf {j}}^2&{}0 \end{array} \right) ,\;\; e^{**}_2=i\left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} 0 &{} {\mathbf {j}}\\ 0 &{} 0 &{} 0 \\ {\mathbf {j}}^2&{}0&{}0 \end{array} \right) ,\;\; e^{**}_3=\left( \begin{array}{c@{\quad }c@{\quad }c} i &{} 0 &{} 0 \\ 0 &{} - i &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right) =ie_3, \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} \tilde{\sigma }_3e_1\tilde{\sigma }_3^{-1}=e^{*}_1,\;\; \tilde{\sigma }_3e_2\tilde{\sigma }_3^{-1}=e^{*}_2,\;\; \tilde{\sigma }_3e_3\tilde{\sigma }_3^{-1}=e^{*}_3, \\ \tilde{\sigma }_3e^{*}_1\tilde{\sigma }_3^{-1}=e^{**}_1,\;\; \tilde{\sigma }_3e^{*}_2\tilde{\sigma }_3^{-1}=e^{**}_2,\;\; \tilde{\sigma }_3e^{*}_3\tilde{\sigma }_3^{-1}=e^{**}_3=e_3, \\ \tilde{\sigma }_3e^{**}_1\tilde{\sigma }_3^{-1}=e_1,\;\; \tilde{\sigma }_3e^{**}_2\tilde{\sigma }_3^{-1}=e_2,\;\; \tilde{\sigma }_3e^{**}_3\tilde{\sigma }_3^{-1}=e_3. \\ \end{array} \right. \end{aligned}$$
(7.4)

the rules (7.1) or (7.2)–(7.4) for the transformations \(\tilde{\sigma }_1\), \(\tilde{\sigma }_2\), \(\tilde{\sigma }_3\) are illustrated by the schemes in Fig. 8.

Fig. 9
figure 9

The concept of introducing colours for elementary particles via the transformations \(\tilde{\sigma }_1\), \(\tilde{\sigma }_2\), \(\tilde{\sigma }_3\) governing the nonion extension of \({\mathfrak {su}}(3)\)

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The Dirac-like operators for

$$\begin{aligned}&(e'_0,e'_1,e'_2,e'_3),\;\; (e''_0,e''_1,e''_2,e''_3),\;\; (\tilde{e}_0,\tilde{e}_1,\tilde{e}_2,\tilde{e}_3),\\&(\tilde{\tilde{e}}_0,\tilde{\tilde{e}}_1,\tilde{\tilde{e}}_2,\tilde{\tilde{e}}_3),\;\;(e^{*}_0,e^{*}_1,e^{*}_2,e^{*}_3),\;\; (e^{**}_0,e^{**}_1,e^{**}_2,e^{**}_3) \end{aligned}$$

can be introduced in an analogous way as shown in Fig. 9. The proposed approach can be naturally modified for more general quark models what is planned for a future paper.

It is worth-while to notice several other important applications of Galois extensions and noncommutative Galois theory in contemporary physics [30,31,32,33,34,35,36,37]. Also, because of an experimental confirmation of the existence of the Higgs boson [38,39,40,41], it is natural to extend our approach combining the Kobayashi–Masukawa model with the BEH-mechanism (here B stands for Robert Brout, E for Franois Englert, Nobel Prize 2013, and H for Peter W. Higgs, Nobel Prize 2013) [42,43,44,45,46,47]. In this context we can see an elegant interpretation of considering our approach parallelly in the theories of condensed matter and elementary particles—the appearance of the Nambu [48,49,50]–Goldstone [51] boson and the Meissner effect in superconductivity within the Ginzburg–Landau model: temperature-dependent vector potential and photons getting the mass. Around 1960 Yôichirô Nambu (Nobel Prize 2008) extended some ideas from superconductivity to particle physics. Jeffrey Goldstone introduced a complex massive scalar field and gave a theorem on appearance of a new massless scalar (Nambu–Goldstone) boson. Note that Murray Gell-Mann (Nobel Prize 1969) and Yuval Ne’eman in connection with the symmetry group SU(3) introduced the new quantum number called flavour [13].

8 Conclusions

For conclusions, we combine the step \(\bullet _6\) of Sect. 2 in our previous paper [1] and the steps \(\bullet _{13}\)–\(\bullet _{23}\) formulated there in Sect. 6 with the steps –:

figure b

The steps \(\bullet _6\), \(\bullet _{13}\)–\(\bullet _{23}\), – may be composed in the scheme shown on Fig. 10.

Fig. 10
figure 10

Quaternary approach related to the quindenion algebra, for crystallographic lattices

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Other new ideas are provided by Perk [55] and Au-Yang and Perk [56], kindly communicated to us in a private letter of Prof. Perk (21.08.,2014).