*5.2. Semigroup*

In line with (37), let us split the unitary brace (34) into two or combine two brace into one, then delete the symbols of primitives Ψ and Φ from there. As was pointed out above, they are not necessary at this stage. By replacing the notation of upper cardinals (34) with pairs (∞1, <sup>∞</sup>2 ) and (∞1 , <sup>∞</sup>2 ), upon the union, one obtains

$$(\infty'\_{1}, \infty'\_{2}) \cup (\infty''\_{1}, \infty''\_{2}) = \left(\infty'\_{1} + \infty''\_{1}, \infty'\_{2} + \infty''\_{2}\right) \,. \tag{41}$$

Here, addition + obviously satisfies the properties (38). If the cardinal "∞-coordinates" are replaced with the "finite percentages" (<sup>κ</sup>, S) introduced above, i.e., if one puts

$$\begin{cases} \varkappa = \frac{\varpi\_1}{\varpi\_1 + \varpi\_2}, & \mathfrak{S} = \infty, +\infty \\ \end{cases}, \qquad \left\{ \varpi\_1 = \varkappa \mathfrak{S}, \quad \varpi\_2 = (1 - \varkappa)\mathfrak{S} \right\} \tag{42}$$

as in (33), then Rule (41) acquires the form of a number composition:

$$(\varkappa', \mathfrak{G}') \circ (\varkappa', \mathfrak{G}'') = \left(\frac{\varkappa' \mathfrak{G}' + \varkappa'' \mathfrak{G}''}{\mathfrak{G}' + \mathfrak{G}''}, \mathfrak{G}' + \mathfrak{G}''\right). \tag{43}$$

The commutativity/associativity properties of operation *◦* hold here due to the birationality of (42). Then, the formal application of <sup>Σ</sup>-postulate S + S → ∞ breaks, however, the symmetry () ↔ () and associativity of *◦* since

$$(\varkappa', \mathfrak{G}') \circ (\varkappa', \mathfrak{G}') \quad \mapsto \quad \varkappa' \circ \varkappa' = \mathfrak{s} \cdot \varkappa' + (1 - \mathfrak{s}) \cdot \varkappa'', \qquad \mathsf{s} := \frac{\mathfrak{G}'}{\mathfrak{G}' + \mathfrak{G}''} \tag{44}$$

and *s* is an undefined parameter. The consequence of the same kind holds true for the <sup>f</sup>-components of pairs (35), for which a convex *w*-combination of the statistical weights does arise:

$$(\mathfrak{t}'\_1, \mathfrak{t}'\_2, \dots) \circ (\mathfrak{t}''\_1, \mathfrak{t}''\_2, \dots) = : (\mathfrak{T}' \circ \overline{\mathfrak{T}}') = w \cdot \overline{\mathfrak{T}}' + (1 - w) \cdot \overline{\mathfrak{T}}'', \qquad w := \frac{\Sigma'}{\Sigma' + \Sigma''}.\tag{45}$$

At the same time, the splitting (41) is no more than an "intrinsic reshuffle" of one and the same ⎧⎩Ξ⎫⎭-brace, which "knows nothing" about the concept of a number (numbers *s*, *w*), much less about the concept of observation or its numerical form. Therefore, mathematics of the ensemble structures should be independent of any representation for (37) by such operations as (43). Composition ⎧⎩Ξ ⎫⎭ *◦* ⎧⎩Ξ ⎫⎭ = ⎧⎩Ξ⎫⎭ should be determined solely by its constituents (f,κ) and (f,κ), i.e., such numbers as (*s*, *w*) must not appear here.

**Remark 8.** *In classical statistics, the foregoing has an analog as indifference of data on events to the way of gathering and layout thereof. For example,* (2, 3)+(1, 4) ≡ (0, 6)+(3, 1) ≡··· =: data*. Then, the observation proper is being created by the scheme* data (3, 7) %→ - 3 3+7 , 7 3+7 = (0.3, 0.7)=(<sup>f</sup>1, <sup>f</sup>2) =: observ*. Parameters such as w can appear in* ⎧⎩Ξ⎫⎭ *only if, prior to any of the* ∪*-unions* (37)*, a construction similar to* (23) *has been fixed. That is, the invariantly number-free brace* (37) has been supplemented by an external number *w and ratio w* : (1 − *<sup>w</sup>*)*. The correction* ⎧⎩Ξ⎫⎭ ⎧⎩Ξ ⎫⎭(*w*), ⎧⎩Ξ ⎫⎭(1−*<sup>w</sup>*) *of the theory, related to this number and to arrays* (23)*, is very well known. This is a w-statistical mixture* {(*w*; *ψ*),(<sup>1</sup> − *w*; *ψ*)} *of wave functions, accompanied by a formalization in terms of the statistical operator w* · |*ψ ψ*| + (1 − *<sup>w</sup>*)· |*ψ ψ*|*.*

Now, to ensure that numerical (<sup>f</sup>,<sup>κ</sup>)-realization (35) of ensemble brace (32) inherits quantum empiricism (O, M) and structural properties (37) and (38) properly, we reassign the quantities (<sup>f</sup>,<sup>κ</sup>) with a "percentage meaning" and replace them with different numbers [*λ*, *μ*]:

$$\{\Xi\} = \left\{ \begin{bmatrix} \boldsymbol{\upmu}\_{1} \\ \boldsymbol{\upmu}\_{1} \end{bmatrix} \underline{\mathbf{a}}\_{1} , \begin{bmatrix} \boldsymbol{\upmu}\_{2} \\ \boldsymbol{\upmu}\_{2} \end{bmatrix} \underline{\mathbf{a}}\_{2} , \dots \right\} \tag{46}$$

(this important move will be touched upon once again in Section 7.1). In so doing, each pair [*μ λ*], [*μ λ*] behaves as a whole, and, under coinciding *αs*, the pairs are endowed with a composition [*μ λ*] ⊕ [*μ λ*] that is to be commutative. Along with this, if symbol 0 denotes a composition of objects (46), it should obviously copy properties (38):

$$(\Xi) \circledast (\Psi) = (\Psi) \circledast (\Xi), \qquad \left( (\Xi) \circledast (\Psi) \right) \circledast (\Phi) = (\Xi) \circledast \left( (\Psi) \circledast (\Phi) \right) \dots$$

The finite ensembles are vanishingly small in their contribution into infinite ones (Σ-postulate), i.e., elements of the ⎧ ⎩Ξ ⎫ ⎭-family, as infinite sets, are considered modulo finite ensembles. Once again, the "finitely many" is forbidden in theory. As soon as we put the numbers of *α*1, of *α*2, . . . to be finite, we immediately obtain the numerical distinguishability *n*1 = *n*2,..., i.e., the act of macro-observation. Let us designate the image of finite ensembles as ⎧ ⎩0 ⎫ ⎭, and, due to property ⎧ ⎩Ξ ⎫ ⎭ 0 ⎧ ⎩0 ⎫ ⎭ = ⎧ ⎩Ξ ⎫ ⎭, it is naturally referred to as zero. The collection (46) itself has also been formed by the ∪-combining the ingredients

$$\left\{ \left[ \begin{smallmatrix} \boldsymbol{\mu}\_{1} \\ \boldsymbol{\lambda}\_{1} \end{smallmatrix} \right] \underline{\mathfrak{a}}\_{1}, \left[ \begin{smallmatrix} \boldsymbol{\mu}\_{2} \\ \boldsymbol{\lambda}\_{2} \end{smallmatrix} \right] \underline{\mathfrak{a}}\_{2}, \dots \right\} \equiv \left\{ \left[ \begin{smallmatrix} \boldsymbol{\mu}\_{1} \\ \boldsymbol{\lambda}\_{1} \end{smallmatrix} \right] \underline{\mathfrak{a}}\_{1} \right\} \cup \left\{ \left[ \begin{smallmatrix} \boldsymbol{\mu}\_{2} \\ \boldsymbol{\lambda}\_{2} \end{smallmatrix} \right] \underline{\mathfrak{a}}\_{2} \right\} \cup \left\{ \begin{smallmatrix} \boldsymbol{\mu}\_{2} \\ \boldsymbol{\lambda}\_{2} \end{smallmatrix} \right\} \cup \left\{ \begin{smallmatrix} \boldsymbol{\mu}\_{3} \\ \boldsymbol{\lambda}\_{3} \end{smallmatrix} \right\}$$

which is why the same symbol 0 may be freely used between objects with different *αs*:

$$\cdot \cdot = \left\{ \left[ \begin{smallmatrix} \lambda\_1\\ \lambda\_1 \end{smallmatrix} \right] \underline{\mathfrak{a}}\_1 \right\} \uplus \left\{ \left[ \begin{smallmatrix} \mu\_2\\ \lambda\_2 \end{smallmatrix} \right] \underline{\mathfrak{a}}\_2 \right\} \uplus \cdot \cdot \; \cdot \; \; . $$

For the sake of brevity, we omit the redundant curly brackets further, redefining

$$\{\Xi\}^{\mu\_1} := \begin{bmatrix} \,^{\mu\_1} \\ \,^{\lambda\_1} \end{bmatrix} \mathfrak{K}^1 \noplus \begin{bmatrix} \,^{\mu\_2} \\ \,^{\lambda\_2} \end{bmatrix} \mathfrak{K}^2 \nrightarrow \begin{array}{c} \cdots \\ \end{array} . \tag{47}$$

As a result, we have had that the set-theoretic prototypes (26) and (27), (32) of states (11) do invariantly exist in the form of every possible ∪-decomposition. Thus, in dealing with the only instrument A , one reveals the following property.

• For each observation A , the set of ⎧ ⎩Ξ ⎫ ⎭ A -objects forms an infinite commutative semigroup G with respect to operation 0.

An internal (beyond the observation) nature of ⎧ ⎩Ξ ⎫ ⎭ A -objects (47) is characterized by commutative superpositions ⎧ ⎩Ξ ⎫ ⎭ A 0 ⎧ ⎩Ξ ⎫ ⎭ A thereof, which are independent of the classical composition of observational f-statistics.
