Balanced identities in algebras of quasigroups

Belousov, V. D.

doi:10.1007/bf01832735

Cited by 21 publications

(62 citation statements)

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“…Proof. Substituting the values of the functional variables F 1 , … , F 6 given by (8) into both sides of the functional equation of pseudomediality (2), we obtain the same expression.…”

Section: Theorem 2 a General Parastrophically Uncancelable Quadraticmentioning

confidence: 79%

“…In the present paper, we continue the investigation of quadratic equations analyzed in the aforementioned and some other works (see, e.g., [8,9]). First, we develop a new conceptual apparatus allowing one to classify quadratic parastrophically uncancelable functional equations for three and four object variables up to parastrophic equivalence.…”

Section: 548mentioning

confidence: 94%

“…A functional equation in which each object variable has exactly two appearances is called quadratic. Examples of quadratic functional equations are such well-known functional equations as the equations of associativity, mediality, and transitivity, balanced equations, etc.In the present paper, we continue the investigation of quadratic equations analyzed in the aforementioned and some other works (see, e.g., [8,9]). First, we develop a new conceptual apparatus allowing one to classify quadratic parastrophically uncancelable functional equations for three and four object variables up to parastrophic equivalence.…”

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confidence: 94%

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On the classification of functional equations on quasigroups

Sokhats'kyi

2004

Ukr Math J

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512.548We consider functional equations over quasigroup operations. We prove that every quadratic parastrophically uncancelable functional equation for four object variables is parastrophically equivalent to the functional equation of mediality or the functional equation of pseudomediality. The set of all solutions of the general functional equation of pseudomediality is found and a criterion for the uncancelability of a quadratic functional equation for four object variables is established.One of methods for the investigation of algebras and algebraic operations is the method of functional equations. This method was especially extensively developed in the theory of quasigroups after the determination of all solutions of the functional equation of associativity. This result was further developed in [1 -7].If at least one of the object variables of a functional equation has only one appearance and this equation has a solution in the set of quasigroup operations of some set Q, then this set is one-element. For this reason, we assume that each object variable in a functional equation has at least two appearances. A functional equation in which each object variable has exactly two appearances is called quadratic. Examples of quadratic functional equations are such well-known functional equations as the equations of associativity, mediality, and transitivity, balanced equations, etc.In the present paper, we continue the investigation of quadratic equations analyzed in the aforementioned and some other works (see, e.g., [8,9]). First, we develop a new conceptual apparatus allowing one to classify quadratic parastrophically uncancelable functional equations for three and four object variables up to parastrophic equivalence. For example, we prove that, in the case of three object variables, only one class of equivalence containing the functional equation of general associativity exists. In the case of four object variables, we have two such classes represented by the general functional equations of mediality and pseudomediality, respectively. We find the set of solutions of the functional equation of pseudomediality (the equation of mediality was solved in [1, 10]) and establish a convenient criterion for the parastrophic uncancelability of quadratic functional equations for four different object variables.The results of this work were reported at the Vinnitsa seminar on algebra and discrete mathematics and at the 3rd and 4th International Conferences held in Ukraine [11,12].Recall that a groupoid ( Q; · ) in which each of the equations x · a = b and a · y = b has a unique solution for all values of a, b ∈ Q is called a quasigroup. This means that, for each quasigroup operation ( · ) on the set Q, the equalities b / a : = x and a \ b : = y define the operations of left ( / ) and right ( \ ) divisions of the operation ( · ). Therefore, a quasigroup can also be defined as an algebra ( Q; ·, /, \ ) in which the following identities are satisfied: ( x / y ) · y = x, ( x·y ) / y = x, x · ( x \ y ) = y, and x \ ( x·y ) = y.The...

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“…Proof. Substituting the values of the functional variables F 1 , … , F 6 given by (8) into both sides of the functional equation of pseudomediality (2), we obtain the same expression.…”

Section: Theorem 2 a General Parastrophically Uncancelable Quadraticmentioning

confidence: 79%

Section: 548mentioning

confidence: 94%

mentioning

confidence: 94%

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On the classification of functional equations on quasigroups

Sokhats'kyi

2004

Ukr Math J

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“…Later, such algebras were invertigated by Aczel [1], Belousov [2,4], Movsisyan [8][9][10][11], Sade [15], and others.…”

Section: Introductionmentioning

confidence: 99%

“…A quasigroup (Q; ·) of the form xy = ϕx + a + ψy, where (Q; +) is a group, ϕ and ψ are automorphisms (antiautomorphisms) of (Q; +), and a is a fixed element of Q, is called a linear (alinear) quasigroup over the group (Q; +) (see [2,6]). …”

Section: Introductionmentioning

confidence: 99%