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Concordance of sense

Reading time17 min
Original author: Сергей Пшеничников

In [1,2,3] texts (sign sequences with repetitions) were transformed (coordinated) into algebraic systems using matrix units as word images. Coordinatization is a necessary condition of algebraization of any subject area. Function (arrow) (7) in [1]) is a matrix coordinatization of text. One can perform algebraic operations with words and fragments of matrix texts as with integers, but taking into account the noncommutativity of multiplication of words as matrices. Structurization of texts is reduced to the calculation of ideals and categories of texts in matrix form.

This article defines the concept of a matrix word in context. Words-signs in repetition may have different fragments of text between them (contexts), and words that are the same in spelling and sound - have different senses (as homonyms). In a text, all repeated words can be homonyms if their contexts differ by an appropriate measure (modulo). Conversely, words different in spelling and sound can have similar contexts and different measures of synonymy. The frequency of keywords in semantic analysis is more appropriately defined as the frequency of contexts comparable by an appropriate measure than as the frequency of word-signs, like letters of the alphabet. When calculating the semantic frequency of words taking into account the context, different word-signs with the same contexts should be summed in the frequency calculation and, conversely, the same word-signs with different contexts should be excluded.

Matrix words are complemented by context multipliers. These multipliers due to the properties of matrix units do not lead to the change of words as signs but contain signs that affect the sense of the defined words. Context multipliers are present in matrix words, but do not affect the signs. Multipliers contain relations (according to Frege) with other signs (part of the properties of these signs is their sense in a given context). The semantic similarity and difference of words can then be calculated by comparing (matching) these multipliers-contexts.

To perform algebraic operations with matrix words in context, concordance (concordance) - a semantic concordance of signs and text fragments, which depends on the measure (module) of concordance - is required. Matrix words can add up to a text if their contexts have a common measure (module). The invariants of matrix texts that retain their sense when words and text fragments are replaced by consonant ones are increasing and decreasing Noether chains. Noether chains allow to make systems of algebraic equations for transformations of texts preserving their sense.

The word in context

Suppose there are two repeated words E_{i_1,j} and E_{i_2,j} (the second j coordinate is the number from the dictionary, the first coordinates i1 and i2 – are the word numbers in the text) and a fragment of the matrix text F_{i_1,i_2}^j between these words (context):

F_{i_1,i_2}^j=E_{i_1+1,k_1} + \ldots + E_{i_2-1,k_n}, \ \ \ \ \ \ \ \ \ \ (1)

where each km – is the number of the word in the dictionary (9) in [1], . Because of the coordinate rule (7) in [1] any km in (1):

k_m < i_2, \ \ \ \ \ \ \ \ \ \ \ (2)


k_1 \leq i_1 + 1, \ldots , k_n \leq i_2 - 1, \ \ \ \ \ \ \ \ \ \ \ (3) i_2 > i_2 - 1, \ldots, i_2 > i_1 + 1. \ \ \ \ \ \ \ \ (4)

In the case of i2 = i1 + 1 fragment zero. For example, in the polychrome "..." in (1) the context of each dot is missing, and then the sense (context) has not each dot, but three dots as a whole, like a word (sign) in the dictionary. In this case the dot is also a sign from the dictionary of the text. Between two dots that are not adjacent, there is a non-zero text fragment (a sentence, as the corresponding context of each dot). Thus, even dots in the text, although they look the same, have different sense-context (as homonyms). Similarly, signs of paragraphs, paragraphs, and, generally speaking, all words have different senses in the text if they are repeated. Conversely, if words have the same appropriate measure (modulo) of context, but these words are different as signs, then they can be considered close in sense (synonyms). For example, «...», «so on», «etc».

Highly likely, in order to achieve universal incomprehension among the builders of the Tower of Babel, it was superfluous to force them to speak different languages. There is no universal understanding in a single contextual language, either; we need semantic (contextual) interpreters.

In short E_{i_2,j} in the context of F_{i_1,i_2}^j is called an expression:

E_{i_2,j} = (F_{i_1,i_2}^j+ E) E_{i_2,j}, \ \ \ \ \ \ \ \ \ (5)

where E is a unit matrix. Because of (2):

F_{i_1,i_2}^j E_{i_2,j}= 0. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)

The product to the right of any summand F_{i_1,i_2}^j from (1) by E_{i_2,j} is zero.

Multiplier (F_{i_1,i_2}^j+E) does not cause (6) to change the sign of E_{i_2,j}, but can be used to compare two (not necessarily repetitive) words E_{i_1,i_2} and E_{i_3,i_4} by comparing their contexts F_{i_5,i_1}^{i_2} and F_{i_6,i_3}^{i_4}. This semantic comparison of words in the text by context (sense) will hereafter be referred to as concordance (concordance) by sense of words.

Word concordance

Let there be two words E_{i_2,j_1} and E_{i_4,j_2} numbered j_1 and j_2 from the right text dictionary D_R in contexts F_{i_2,i_1}^{j_1} and F_{i_3,i_4}^{j_2} between pairs of repeating words:

E_{i_2,j_1}= \left( F_{i_2,i_1}^{j_1} D_{1R} + E \right)E_{i_2,j_1}, \ \ \ \ \ \ \ \ (7)E_{i_4,j_2}= \left( F_{i_3,i_4}^{j_2} D_{2R} + E \right)E_{i_4,j_2}, \ \ \ \ \ \ \ \ (8)

Where D_{1R} and D_{2R} -- right context dictionaries F_{i_2,i_1}^{j_1} and F_{i_3,i_4}^{j_2}, i_1, i_2 and i_3, i_4 -- numbers by pairs of repeating words. Hereafter, all dictionaries are taken as right-handed and the R index is not specified.

Two words can be concordant (agreed) both by the intersection of the word contexts (2) in [3] and by the association (3) in [3]. In what follows, only the intersection of contexts will be considered. Algebraically descriptions for union and intersection coincide. For application, their purpose is different. A human, due to natural physical limitations, can hold only a few entities (about seven) at a time in the process of understanding. Such an operation of thinking as abstraction is used to reduce the variety of the world to this number. Concordance by intersection is a mathematical explication of the process of abstraction in the form of reduction (4) in [3]. The limiting case of abstract concepts of natural language is logical categories (Aristotle, Kant, Hegel). Hierarchical continuity of concepts (words) is necessary for construction of part-whole relations (relations of understanding).

Concordance by association (3) in [3] increases essences. But their number matters only for humans. For machine languages this restriction is not essential. Therefore, concordance on unification can be applied to interaction of machines as well as to future collective mind of human population (according to P.G. Kuznetsov), for which it is necessary to create technologies of collective understanding. At present, acceptable understanding is achieved in teams of programmers. For collectives of five or more, such as physicians (according to T. and B. Buzan), there is no single term that they understand in the same way. In mathematics, seemingly the universal language of mankind, with ideal objects not changing in time (P.G. Kuznetsov), specialization has reached such a level that territorially distributed teams of three or four people understand each other completely.

Concordance by intersection will be called simply concordance. Two words (7) and (8) are concordant (consistent) \dot{\sim}(«dot over tilde») on the intersection of the right-hand dictionaries D_{1R} and D_{2R} contexts F_{i_1,i_2}^{j_1} and F_{i_3,i_4}^{j_2}:

E_{i_2,j_1}\dot{\sim} E_{i_4,j_2}(\mathrm{mod}{D_1D_2}), \ \ \ \ \ \ (9)

if the intersection of two dictionaries:

 D_1D_2 \ne 0 \ \ \ \ \ \ \ \ \ (10)

Expression (9) means that the words E_{i_2,j_1} and E_{i_4,j_2} are similar in the sense that their contexts F_{i_1,i_2}^{j_1} and F_{i_3,i_4}^{j_2} have a common vocabulary D_1D_2. The consistent contexts are the contexts after the reduction (4) in [3]:

F_{i_1,i_2}^{j_1} D_1D_2 \ \ \ \ \ \ \ \ (11)F_{i_3,i_4}^{j_2} D_1D_2  \ \ \ \ \ \ \ \ (12)

Each reduced context contains all the words from the dictionary D_1D_2. Indeed, for any word E_{i_5,j_3}, available in F_{i_1,i_2}^{j_1}, but absent in the F_{i_3,i_4}^{j_2}:

\left(E_{i_5,j_3} + F_{i_1,i_2}^{\star j_1}\right) D_1D_2= F_{i_1,i_2}^{\star j_1} D_1D_2, \ \ \ \ \ \ \ (13)

where F_{i_1,i_2}^{\star{}j_1} -- part of the context F_{i_1,i_2}^{j_1} after deleting the word E_{i_5,j_5}. N words are concordant if each pair is concordant:

E_{i_1,j_1}\dot{\sim} E_{i_2,j_2}(\mathrm{mod}D_1D_2 \ldots D_N), \ \ \ \ \ \ \ \ (14)

and the work of the dictionaries

D_1D_2\ldots D_N \ne 0 \ \ \ \ \ \ \ \ \ \ (15)

Concordance ratio \dot{\sim} is an equivalence relation since the reflexivity and symmetry conditions for the matrices are satisfied, and the transitivity of the relation follows from (14) and (15).

The measure (module) of concordance is (15). It is this modulus that explains the appearance of the term "modulo concordance" by analogy with the term "modulo comparison" for integers. Just as different integers can be equal modulo, so different (as characters) words in a text can be equivalent (interchangeable) modulo concordance. This means that if words have concordant contexts, then the words have concordant sense and can be considered equivalent (interchangeable in sense in the text).

Words E_{i_1,j_1},\ldots,E_{i_n,j_n} and their sums can be concordant modulo. On concordance relations, like equality and comparison modulo, it is possible to compose systems of concordance equations. The unknowns can be definable and determinable words, concordance moduli, contexts, and text fragments. Concordance equations allow to calculate answers to such questions: in what sense (here the unknown is the module of concordance) are words and texts concordant? If the sense (modulus) is given, what set of words do we replace with other words? In this way, it is possible to compute word definitions and sense versions of texts. Find interchangeable words, compute semantic markup and text structuring, annotation text drafts, and semantic text translation (even of the same language). New functions of text editors and readers, messengers and social networks can be based on these computational capabilities. In the latter case, it is possible, by compiling a personal contextual dictionary of a user-participant according to his messages, to accompany communication with semantic translation of text and sound through the personal contextual languages of other participants.

Concordant addition

The concordant addition of the pair of words (7) and (8) is the expression:

E_{i_2,j_1}+ E_{i_4,j_2}= \\ = \left[ \left(F_{i_1,i_2}^{j_1} +       F_{i_3,i_4}^{j_2}\right)D_1D_2 + E\right]       \left(E_{i_2,j_1}+ E_{i_4,j_2}\right) \ \ \ \ \ \ \ \ (16)

At the same time, according to (6):

 F_{i_1,i_2}^{j_1} E_{i_2,j_1}= 0 \ \ \ \ \ \ \ (17)F_{i_3,i_4}^{j_2} E_{i_4,j_2}= 0 \ \ \ \ \ \ \ (18)

Because F_{i1,i2}^{j_1}D_1D_2 and F_{i_3,i_4}^{j_2}D_1D_2 - are parts of fragments F_{i_1,i_2}^{j_1} and F_{i_3,i_4}^{j_2}, then:

F_{i_1,i_2}^{j_1} D_1D_2E{i_2,j_1}= 0 \ \ \ \ \ \ \ (19)F_{i_3,i_4}^{j_2} D_1D_2E_{i_4,j_2}= 0 \ \ \ \ \ \ \ (20)

Thus, \left[\left(F_{i_1,i_2}^{j_1}+F_{i_3,i_4}^{j_2}\right)D_1D_2+E\right] is a concordant context for the sum of words. The matching module is a common dictionary of two contexts D_1D_2. Concordant addition of n words:

E_{i_1,j_1}+ \ldots + E_{i_n,j_n}= \\ = \left[\left(F_{\ldots,i_1}^{j_1} +\ldots +F_{\ldots,i_n}^{j_n}\right)D_1\cdot \ldots \cdot D_n + E\right] \times \\ \times  \left(E_{i_1,j_1}+ \ldots + E_{i_n,j_n}\right), \ \ \ \ \ \ \ \ \ \ \ (21)

where the ellipses in the indices F_{\ldots,i}^j means the number of the repeating word j to the left of the number i, D_1\ldots{}D_n -- the product of the right context dictionaries F_{\ldots{},i_1}^{j_1},\ldots{},F_{\ldots{},i_n}^{j_n}.

The word in a refined context

Two words are concordant (9) if the right dictionaries of their contexts have a non-zero overlap (10). But each word of these contexts is also a word in the context (5). Therefore, mutual concordance of the defined word with the defining words is necessary. According to V.A. Lefebvre, this reflexivity is the cause of the ambiguity of natural language and texts' interpretations («I think that they think that I think that ...»).

A mathematical explication of reflexion is the latent semantic nonlinearity of linearly ordered word-signs. Perhaps, in the future, linguistic texts will cease to be linear and oneedimensional. Note texts, for example, are 5-dimensional, although they can also be transposed into one-dimensional stan-"thread", but this will turn note texts into monstrously incomprehensible codes with dictionaries comparable to the dictionaries of language texts. Such one-dimensional music texts, like language texts, would require a semantic gestalt translation, not just a personal intonation translation, as for 5-dimensional music texts. In a future multidimensional language text, it will be possible to point to the sense chains of revealing the sense of words and text fragments, rather than to recognize them intuitively or with the help of fast reading know-how.

Context F_{i_1,i_2}^j (1) in the definition of the word (5) can be regarded as the concordant sum of matrix words (21), since each summand word in (1) also has its own context. Then the word in such a refined context for (5) has the form:

E_{i_2,j} = \left(F_{i_1,i_2}^j + E\right) E_{i_2,j} =\\  =\left[\left[\left(F_{\ldots,i_1}^{j_1} + \ldots + F_{\ldots,i_n}^{j_n}\right) \times \\ \times D\left(F_{\ldots,i_1}^{j_1}\right)D  \left(F_{\ldots,i_2}^{j_2}\right) \ldots D\left(F_{\ldots,i_n}^{j_n}\right) + E\right]\times \\ \times F_{i_1,i_2}^j D\left(F_{i_1,i_2}^j\right) + E\right] E_{i_2,j}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (22)


 D\left(F_{\ldots,i_1}^{j_1}\right), \ldots , D\left(F_{\ldots,i_n}^{j_n}\right) \ \ \ \ \ \ \ \ \ \ \ \ \ (23)

-context dictionaries F_{\ldots,i_1}^{j_1},\ldots,F_{\ldots,i_n}^{j_n},

 D\left(F_{i_1,i_2}^j\right) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (24)

-fragment-context dictionary F_{i_1,i_2}^j.

The word in the refined context (22) is a matrix bilinear form in F.

Two words of the form (22) are concordant in the refined contexts if the intersection (product) of all vocabularies of all contexts of both words

D_1D_2 \ne 0, \ \ \ \ \ \ \ (25)

where D_1 and D_2 -- the products of all dictionaries (23) and (24) of the first and second word.

There can be n words concordant on refined contexts if each pair is concordant. The module of concordance is the product of all vocabularies of all contexts of all forms.

There can be concordant sums of words (text fragments) (21) over refined contexts if each pair of sums is concordant.

A pair of word sums is concordant if the product of the vocabularies of all contexts of all words of the pair of sums is different from zero.

If the modulus of concordance, as the product of the dictionaries of all refined contexts of all words as bilinear forms (22), is nonzero, then the text of these words is concordant.

Concordance classes

All words and fragments of the matrix text can be decomposed into concordance classes. Each word E_{i_2,j} numbered i_2 to the text in the form (22) corresponds to the multiplier on the left:

\left[     \left[         \left(F_{\ldots,i_1}^{j_1} +\ldots +F_{\ldots,i_n}^{j_n}         \right) + E     \right]F_{i_1,i_2}^j+ E \right].   \ \ \ \ \ \ \ \ \ \ \ (26)

To each text fragment F_i, as any F in (25), corresponds to its vocabulary D_i

F_iD_i= F_i \ \ \ \ \ \ \ \ \ \ \ \ (27)

The multipliers (26) on the left for E_{i_2,j} in (22), as well as D_i. To the right for F in (27), exist, but do not change E_{i_2,j} or F_i. In this case, the multipliers are uniquely determined from the text by its fragments. The absence of multiplier influence on signs is a necessary condition, but not sufficient for concordance relations. A sufficient condition is that those not affecting the signs E_{i,j} and F_i the multipliers (25) on the left and D_i to the right (26) are a single-valued function (property) of the text.

To each pair of words E_{i_1,j_1} and E_{i_2,j_2} in the form (22) with the numbers i_1 and i_2 in the text corresponds to the module \kappa_{i_1,i_2} (kappa) concordance is the product of all vocabularies of all refined contexts of both words (25).

Each pair of text fragments F_i and F_j corresponds to the module \kappa_{i,j} concordance is the product of all dictionaries of all refined contexts of all words.

To each pair E_{i_1,j_1} and F_j of the word form (22) and the text fragment corresponds to the module \kappa_{i_1,j} concordance –- the product of all dictionaries of all refined contexts E_{i_1,j_1} and F_j.

Conversely each module, \kappa_K (class name) corresponds to a set of refined contexts, a set of words corresponding to these contexts according to (22) and a set of text fragments having a vocabulary equal to \kappa_K. All these three sets are mutually concordant and all their elements are elements of one concordance class \kappa_K.

The set of all concordance classes modulo \kappa_K is the Boolean set of all n words of the text dictionary or all its partial sums (fragment dictionaries). The number of all partial sums 2^n.

The belonging of such elements to one class means that there exist matrices of transformation of elements into each other. Indeed, if the set of refined contexts, the set of words corresponding to these contexts according to (22) and the set of text fragments have one vocabulary equal to , then all these elements are similar to each other (20) in [1]. In this case, the common object of transformations in refined contexts and text fragments are matrix polynomials (31) in [1].

Reciprocal transformations of refined contexts, words corresponding to these contexts and text fragments having the vocabulary equal to \kappa_K, the following:

1.Conversion of a pair of refined contexts of the form (26)

\begin{split}         F_1 = \left[          \left(F_1^1+\ldots +F_n^1 \right)D_{1R} \ldots D_{nR} + E\right]F_1 \\ F_2 = \left[          \left(F_1^2+\ldots +F_n^2 \right)D_{1R} \ldots D_{nR} + E\right]F_2 \end{split} \ \ \ \ \ \ \ \ \ \ (28)

Let there be two matrix texts (28). Because they belong to the same class, they have the same modulus \kappa_i or, what is the same, have the same right dictionaries. But matrix texts having the same dictionaries form ideals (multiples of the dictionary) by (37) in [1]. There always exists a matrix polynomial whose multiplication on the left-hand side by one refined fragment (28) results in a refined fragment of the form (28):

 F_1 = F_{1,2}F_2 =  \\ =          \left[\left(F_{1,2}F_1^2+\ldots +F_{1,2}F_n^2\right)D_{1R} \ldots D_{nR} + E\right]F_{1,2}F_2  \ \ \ \ \ \ \ \ (29)

To the precision of this matrix multiplier F_{1,2} the two refined fragments are indistinguishable (interchangeable).

2.Conversion of words in a refined context of the form (22). Let there be two words:

\begin{split}     E_{i_1,j_1} = (F_1 + E) E_{i_1,j_1} \\  E_{i_2,j_2} = (F_2 + E) E_{i_2,j_2} \end{split} \ \ \ \ \ \ \ \ \ (30)

Since the words are concordant (have a common vocabulary \kappa_{1,2} , as the product of all dictionaries of all refined contexts (14)), then:

E_{i_1,j_1} \dot{\sim} E_{i_2,j_2}(\kappa_{1,2}) \ \ \ \ \ \ \ \ \ \ \ (31)

Like comparisons of integers, the concordance of matrix units (31) can be written through the equality:

 E_{i_1,j_1}= \left(F_{1,2}F_2 + E\right) F_{1,2}E_{i_2,j_2} E_{j_2,j_1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ (32)

3.Conversion of words and contexts.

Let there be a word and a context:

\begin{split}  E_{i_1,j_1}= (F_1 + E) E_{i_1,j_1}\\  F_2 = \left[\left(F_1^2+\ldots +F_n^2\right)D_1 \ldots D_n + E\right]F_2      \end{split}  \ \ \ \ \ \ (33)

Word and context (33) are concordant if they have a common modulus \kappa_{1,2} :

 E_{i_1,j_1} \dot{\sim} F_2(\kappa_{1,2}) \ \ \ \ \ \ (34)

Or in the record with equality:

 E_{i_1,j_1} = \left(F_{1,2}F_2 + E\right) F_{1,2}E_{j_2,j_1}, \ \ \ \ \ \ \ (35)

where F_{1,2}E_{j_2,j_1} is understood as a concordant transformation of words (32).

The transformation of words and text fragments reduces to (35), because text fragments are matrix polynomials (31) in [1], as well as contexts. This means that (34) is the formula for calculating the naming of a text fragment by a word belonging to the concordance class \kappa_{1,2}. And vice versa, word definition by text.

Noether sense chains

The concordance classes \kappa are distinguished by the words included in the dictionary \kappa . Let a sequence of dictionaries be given:

\kappa_1, \kappa_2, \ldots , \kappa_n, \ \ \ \ \ \ \ \ \ (36)

such that neighboring dictionaries are distinguished by one word E_{i,i} :

\kappa_i=\kappa_{i-1}+E_{i,i} \ \ \ \ \ \ \ \ \ (37)

Concordance class K_i (capital kappa) for each \kappa_i -- is the set of all words in the refined context, all refined contexts, and all text fragments with a common vocabulary \kappa_i. Elements K_i are mutually substitutable by formulas (29), (32) and (35).

Let there be classes of concordance K_i, corresponding to (36). Then:

K_1 \subset K_2 \subset \ldots K_{n-1} \subset K_n \ \ \ \ \ (38)

And vice versa,

K_1 \supset K_2 \supset \ldots K_{n-1} \supset K_n \ \ \ \ \ \ \ \ (39)

for these \kappa_i , that

 \kappa_{i-1}+ E_{i,i}=\kappa_i \ \ \  \ \ \ \ \ (40)

In dictionaries (36) and (37) there is an increase of words in the dictionary \kappa_i from left to right in (36). In dictionaries (36) and (40) -- there is a reduction.

The sequence of nonempty subsets K_1,K_2,\ldots,K_n (38) the corpus of texts compiled from D (the dictionary of the corpus of all texts) is ascending, because each of them is a subset of the next one.

Conversely, the sequence of subsets K_1,K_2,\ldots,K_n(39) is decreasing, since each of them contains the next subset.

A sequence is said to stabilize after a finite number of steps if there exists n such that for all m\geq{}n , K_n=K_m . This is the case for matrix texts -- there is no larger dictionary than the dictionary of all texts D. The set of subsets of a given set D (or K) satisfies the condition of breaking of increasing chains, since any increasing sequence becomes constant after a finite number of steps.

Any decreasing sequence (39) becomes constant after a finite number of steps, since the dictionary D has a minimal set - one word, hence the set of subsets (39) satisfies the condition of breaking of decreasing chains.

In general algebra, objects are called nether objects if they satisfy chain breaking conditions. Amalie Emmy Noether has made masterly use of the cliff-chain technique in her many cases. Objects such as concordance classes are also neoteric.

Noether chains can also be defined for word order in a text. Relative word order is essential for texts. For example, «incidental in the necessary» differs in sense from «necessary in the incidental» or «mom's dad» and «dad's mom». For musical texts and codes, the order of characters is as significant as the characters themselves.

The concordance module is a fragment of the vocabulary of the text. For a dictionary, the word order is insignificant. Therefore, the concordance class contains elements without taking into account the order of words in text fragments. The word order can be taken into account through the available subclasses of the concordance class as follows.

Let there be two words:

E_{i_1,j_1}=E_{i_1,i_1-1}E_{i_1-1,i_1-2}\ldots E_{2,1} E_{1,j_1}, \ \ \ \ \ (41) E_{i_2,j_2}= E_{i_2,i_1-1}E_{i_2-1,i_2-2}\ldots E_{2,1} E_{1,j_2}. \ \ \ \ \ \ \ \ (42)

Word E_{i_1,j_1} is in the text to the left of E_{i_2,j_2} , if there exists such a matrix unit:

 E_{i_2,i_1-1}E_{i_2-1,i_2-2} \ldots E_{i_1-1,i_1}, \ \ \ (43)

such that:

 E_{i_2,i_1-1}E_{i_2-1,i_2-2}\ldots E_{2,1} = \\ =   \left(E_{i_2,i_1-1}E_{i_2-1,i_2-2} \ldots E_{i_1-1,i_1} \right)  \left(E_{i_1,i_1-1} E_{i_1-1,i_1-2}\ldots E_{2,1} \right). \ \ \ \ (44)

In this case the set of matrix units:

\{E_{i_1,1}\} = \{E_{i_1,i_1-1}, \ldots, E_{2,1}\} \ \ \ \ (45)

is a subset of:

\{E_{i_2,1} \} = \{E_{i_2,i_2-1}, \ldots, E_{2,1}\} \ \ \ \ \ \ \ (46)\{E_{i_1,1}\} \subset \{E_{i_2,1}\}. \ \ \ \ \ \ \ \ \ \ (47)

If the word E_{i_1,j_1} is in the text to the left of E_{i_2,j_2} , then in terms of (47):

\{E_{i_1,j_1}\} \subset \{E_{i_2,j_2} \} \ \ \ \ \ \ \ (48)

Let there be a matrix polynomial:

 E_{i_1,j_1}+ E_{i_2,j_2}. \ \ \ \ \ \ \ \ \ \ (49)

Expression (50) defines a concordance class with the following description:

1.The elements of the class are polynomials having a dictionary E_{j_1,j_1}+E_{j_2,j_2} , with any first monomial coordinates.

2.A subclass of elements with such first coordinates that:

\{E_{i_1,j_1} \} \subset E_{i_2,j_2}

3.A subclass of elements with such first coordinates that:

\{E_{i_2,j_2}\} \subset \{E_{i_1,j_1}\}

For the matrix polynomial:

E_{i_1,j_1} + E_{i_2,j_2}+ \ldots + E_{i_n,j_n}

concordance class is defined by the dictionary (module) and consists of subclasses considering the order of words. The order subclasses are defined by ascending or descending Noether chains for the first coordinates of matrix monomials in the left dictionary texts (12) in [1]. Expression (50) corresponds to this definition of the left-hand dictionary. For the left-hand dictionaries there are also the Noether chains, as for the right-hand dictionaries (36).

The Noether chains for words and their order are semantic invariants of the text, preserved by appropriate concordant word substitutions in the text (retelling the text in one's own words), substitutions of fragments with words (abstracting and annotating), substitutions of words with fragments (bot-writing). The invariance comes from the fact that netter chains are constructed by left or right dictionaries of matrix polynomials. The invariance on the Noether chains of the right dictionaries means that the places of words in the text are not important for the sense of the text, what matters is the system of their context correspondence as a function of embedding (taking into account the order of words within n--grams). Invariant on the Noether chains of the left dictionaries means that for the structure of the text the words from the right dictionary are not important, the system of their structural correspondence as a function of embedding the left dictionaries of the text-forming fragments (structural pattern of the text) is important.

The text Noether chains are more preferable for semantic analysis than frequent keywords, because they take into account the contexts of words, and also reveal patterns of disclosure of the system of concepts in the text through the sequence of nesting of their content (context) - this is the above-mentioned hierarchical continuity of concepts (words). Logical, ethical and aesthetic categories of natural languages can be calculated as Noether chains of sense.

If the Noether semantic chains are defined as target functions (sequences of embeddings), it is possible to compose systems of equations on variables of bilinear forms (22). Because the variables in (22) are pairwise meshed with each other (pairwise nested in Noether chains), a system of quadratic equations on words in a refined context, their contexts and text-forming fragments as unknowns of such equations can be composed.

The equalizers of sense

In category theory, the following model is called an equalizer (a generalization of the equation) with respect to matrix text fragments. Let four object-fragments be given F_1D_1, F_2D_2, F_3D_3, F_4D_4, where D_1, D_2, D_3, D_4 -- fragment dictionaries. Objects F_1 and F_2 are connected by a pair of morphisms F_{1,2}^1 and F_{1,2}^2:

 F_2D_2 = F_{2,1}^1 F_1D_1, F_2D_2 = F_{2,1}^2 F_1D_1  \ \ \ \ \ \ (50)

This means that the dictionary D_2 -- this is part or all of the vocabulary D_1 . F_{2,1}^1 and F_{2,1}^2 may differ from each other because of the fact that in the F_1 there may be repetitions of words. Then there is no unambiguity in (10) -- the transformation of fragments (the result depends on which of the repeated words F_1 is used to convert a fragment into a word F_2 ). The third object-fragment F3 and morphism F3,1 (function) is called an equalizerF_{1,2}^1 and F_{1,2}^2 , if atF_1=F_{1,3}F_3 concordant F_{2,1}^1F_{1,3} and F_{2,1}^2F_{1,3}:

 F_{2,1}^1 F_{1,3} \dot{\sim} F_{2,1}^2 F_{1,3} \ \ \ \ \ (51)

At the same time, for any other object F_4 , satisfying the same requirements:

 F_{2,1}^1 F_{1,4} \dot{\sim} F_{2,1}^2 F_{1,4}, \ \ \ \ \ \ \ (52)

which and F_3 , there is a single morphism F_{3,4}:

 F_3 = F_{3,4}F_4, \ \ \ \ \ \ \ \ (53)

Such that:

 F_{1,3}F_{3,4}\dot{\sim} F_{1,4} \ \ \ \ \ \ \ \ (54)

An essential difference between the above definition for the equalizer of matrix fragments and the canonical definition of the equalizer for the Set category, for instance, is the replacement of the equality relation by the concordance relation. But since equality and concordance relations are equivalence relations (have properties of reflexivity, symmetry and transitivity), such replacement is admissible and satisfies the axioms of the category [3].

The reason for using concordance is as follows. For (51) it is required to find the third text fragment and its corresponding matrix polynomial-transformation F_{1,3} such that, when multiplied by it on the right, the ambiguity in (51) (F_{1,2}^1 or F_{1,2}^2) is eliminated. Since in the monomials of matrix polynomials F_{1,2}^1 or F_{1,2}^2 both coordinates refer to the position of words in the text, then F_{2,1}^1F_{1,3} and F_{2,1}^2F_{1,3} -- this is the concordant selection rule for repeated words, which eliminates the ambiguity in (51).

If words are considered in a refined context, the semantic distinction of repeated words in the text and their concordance on refined contexts are used to achieve this unambiguity.

A system of equations for fragments in refined contexts (a word is a special case of a fragment) can be composed in three ways:

1.According to the correlation of the concordance of text fragments in refined contexts (28) -- (35):

F_{i_1}^j\dot{\sim} F_{i_2}^j(\kappa_{i_1,i_2}),  \ \ \ \ \ \ \ \ (55)

where F_{i_1}^j and F_{i_2}^j different words and text fragments. For example, it is the concordance of the title of the text and the whole text or parts of the text (paragraphs, chapters, etc.), parts of the text (e.g., the abstract and the whole text, the first paragraphs of paragraphs, etc.). The listed combinations of fragments are denoted by the numbers j from (56) and are the corresponding numbers of equations in the equation systems of the text.

2.By the Noether chains of text fragments and their ordering. The equations in this case are recurrent and are defined by formulas (37) or (40). Recurrence on the first coordinates determines the sequence of text fragments (structural pattern of the text). Recurrence on the second coordinates determines the sequence of fragments by continuity of sense (contextual table of contents of the whole text and its sections). Each Netter chain defines an equation in a system of equations.

3.The combination of the two clauses above.

According to (22), systems of equations have the general form:

\sum_{i_1,i_2} F_{i_1}^j F_{i_2}^j =\sum_{i}F_{i}^j. \ \ \ \ \ \ \ (56)

Systems of equations (57) are either systems of linear or quadratic equations on F, depending on which fragments of F in (57) are taken as unknowns. The set and unknown quantities in (57) are matrices. For the linear case, there are matrix versions of the Gaussian method for solving systems of linear matrix equations. For systems of quadratic matrix equations, there is also a generalization of the Gaussian method of eliminating the unknowns and reductions in systems of equations with many unknowns to an equation with one unknown and formulas for the relation between the unknowns.

Exact linearization of equations

In [4,5] a method of exact linearization and solution of systems of nonlinear algebraic equations over the field of real numbers was developed. The system of quadratic equations is a particular judgement. You can reduce a system of quadratic equations to a system of linear equations without loss of generality or accuracy.

For example, let a quadratic equation be given ( a , b , c are real numbers):

ax^2 + bx + c = 0  \ \ \ \ (57)

and four matrix units (1) in [1]:

	E_{1,2} = \left\| {\begin{array}{*{20}{c}} 			0&1 \\  			0&0  	\end{array}} \right\|,\;\; 	E_{2,1} = \left\| {\begin{array}{*{20}{c}} 			0&0 \\  			1&0  	\end{array}} \right\|,\\\;\;E_{1,1} = {E_{1,2}}{E_{2,1}} = \left\| {\begin{array}{*{20}{c}} 			1&0 \\  			0&0  	\end{array}} \right\|,\\ \;\;{E_{2,2}} = {E_{2,1}}{E_{1,2}} = \left\| {\begin{array}{*{20}{c}} 			0&0 \\  			0&1  	\end{array}} \right\|, \ \ \ \ (58)

The matrix units (58) have the following properties:

\left(E_{1,2}\right)^2 = E_{1,2}E_{1,2} = \left\| {\begin{array}{*{20}{c}} 			0&0 \\  			0&0  	\end{array}} \right\|,\\  	\left(E_{2,1}\right)^2 = E_{2,1}E_{2,1} = \left\| {\begin{array}{*{20}{c}} 			0&0 \\  			0&0  	\end{array}} \right\|, \ \ \ \ \ (59) E_{1,2}E_{2,1} + E_{2,1} E_{1,2} = E , \ \ \ \ \ (60)\left(E_{1,2} + E_{2,1}\right)^2= E, \left(E_{1,1} - E_{2,2}\right)^2= E, \ \ \ \ (61) \left(E_{1,2} + E_{2,1}\right)\left(E_{1,1} - E_{2,2}\right)+ \left(E_{1,1} - E_{2,2}\right) \left(E_{1,2} + E_{2,1}\right)=0, \ \ \ \ \ (62)

where E -- unit matrix, E=\left\|{\begin{array}{*{20}{c}} 			1&0 \\  			0&1  	\end{array}} \right\|

From formulas (60) -- (63) it follows that the permutation properties of matrix pairsE_{1,2}, E_{2,1} and (E_{1,2}+E_{2,1}), (E_{1,1}-E_{2,2}) are opposite. Squares E_{1,2}, E_{2,1} are equal to the null matrix, and their sum of products in different orders (the anticommutator (61)) is equal to the unit matrix. Conversely, for the elements (E_{1,2}+E_{2,1}), (E_{1,1}-E_{2,2}) their squares are equal to the unit matrix, and the anticommutator is equal to the zero matrix.

If we use the properties of the Kronecker (direct) product of matrices:

\omega_1= (E_{1,2} + E_{2,1}) \otimes E_{1,2}, \omega_2= (E_{1,2} + E_{2,1}) \otimes E_{2,1}, \ \ \ \ \ \ (63) \alpha_1 = (E_{1,1} - E_{2,2}) \otimes (E_{1,2} + E_{2,1}) , \alpha_2 = (E_{1,1} - E_{2,2}) \otimes (E_{1,1} - E_{2,2}), \ \ \ \ \ (64)

then the linearized equation (58) is the expression:

B\Phi \equiv \left(\alpha_1 \sqrt{a_i}x + \omega_1b +\omega_2x + \alpha_2\sqrt{c}\right) \Phi = 0, \ \ \ \ (65)

where \Phi is a Cartan spinor (simplified, a nonzero column of, in general, complex numbers). The square of the matrix factor B in (65):

(\alpha_1\sqrt{a_i}x + \omega_1 b + \omega_2x + \alpha_2\sqrt{c}) (\alpha_1\sqrt{a_i} x + \omega_1 b +\omega_2 x + \alpha_2\sqrt{c}) = \\ =(ax^2+bx+c) E, \ \ \ (66)

where E is a 4x4 unit matrix. Properties of matrices \omega (64) in the product BB leave the product bx and remove ax^2 and c. Rearrangement properties (65) leave ax^2 and c , and remove bx in BB

In the theory of comparisons of integers, an analogy with logarithms is made for the index of a class of deductions. The sense of the transformation (57) in (66) can be conventionally represented as:

\sqrt{\sum{\ldots}}= \sum{\sqrt{\ldots}} \ \ \ \ \ \ \ \ (67)

The permutations of operations (67) over the field of real numbers are impossible, but over the algebra of unions (hypercomplex numbers) are natural. The elements \alpha and \omega (unions) are matrix generalizations of complex numbers, and exact linearizations (67) are possible, but the price to pay is that the coefficients \alpha and \omega in the linear in x equation (65) become noncommutative.

Algebra unions, exact linearization of systems of algebraic nonlinear equations over the field of real numbers, and the unionic generalization of the Gauss method of eliminating unknowns are described in detail in [4,5].

For the exact linearization and solution of systems of concordant equations (56) it is necessary that the symbols in (56) commute with unions \alpha and \omega , and the unknowns were squared in expression (56). The second requirement is necessary to exclude the unknowns, since \alpha^{-1}=\alpha, while \omega -- have no inverse. This requirement is easy to fulfill, because for matrix text fragments the F=F^2 (10) in [1]. The first requirement can be satisfied by using the property of the Kronecker (direct) product of matrices:

F_i^j\longrightarrow E \otimes F_i^j, F_k^j\longrightarrow F_k^j \otimes E

Fragments and unions E\otimes{}F_i^j , F_k^j\otimes{}E , \alpha and \omega and are permutable with each other due to a corresponding increase in the dimensionality of the matrix units used.


  1. S. B. Pshenichnikov. Algebra of text. Researchgate Preprint, 2021.

  2. S.B. Pshenichnikov. Algebra of text. examples. Researchgate Preprint, 2021.

  3. S. B. Pshenichnikov. Context category. Researchgate Preprint, 2021.

  4. P. G. Kuznetsov and S. B. Pshenichnikov. The spinor method of solving systems of nonlinear algebraic equations. Doklady Akademii nauk SSSR, 283(5):1073–1076, 1985.

  5. S. B. Pshenichnikov. The spinor method of expulsion and formulas for the tie between unknown variables in the systems of the nonlinear algebraic equations. Latvian Mathematical Yearbook, 30:150–161, 1986.

Initially published on researchgate

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