On functional equations and distributive second order formulae with specialized quantifiers
The structure of invertible algebras with distributive second order formulae with specialized quantifiers is given. As a consequence, the applications for solutions of the some functional equations of distributivity on quasigroups are provided.
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irk-123456789-1883622023-02-26T01:26:57Z On functional equations and distributive second order formulae with specialized quantifiers Movsisyan, Yu. The structure of invertible algebras with distributive second order formulae with specialized quantifiers is given. As a consequence, the applications for solutions of the some functional equations of distributivity on quasigroups are provided. 2018 Article On functional equations and distributive second order formulae with specialized quantifiers / Yu. Movsisyan // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 269–285. — Бібліогр.: 76 назв. — англ. 1726-3255 2010 MSC: 20N05, 20N02, 03C85, 03C05. http://dspace.nbuv.gov.ua/handle/123456789/188362 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The structure of invertible algebras with distributive second order formulae with specialized quantifiers is given. As a consequence, the applications for solutions of the some functional equations of distributivity on quasigroups are provided. |
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On functional equations and distributive second order formulae with specialized quantifiers |
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On functional equations and distributive second order formulae with specialized quantifiers |
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on functional equations and distributive second order formulae with specialized quantifiers |
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On functional equations and distributive second order formulae with specialized quantifiers / Yu. Movsisyan // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 269–285. — Бібліогр.: 76 назв. — англ. |
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Algebra and Discrete Mathematics |
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2025-07-16T10:23:19Z |
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“adm-n2” — 2018/7/24 — 22:32 — page 269 — #107
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 2, pp. 269–285
c© Journal “Algebra and Discrete Mathematics”
On functional equations and distributive second
order formulae with specialized quantifiers∗
Yuri Movsisyan
Communicated by A. Zhuchok
Abstract. The structure of invertible algebras with distribu-
tive second order formulae with specialized quantifiers is given. As
a consequence, the applications for solutions of the some functional
equations of distributivity on quasigroups are provided.
Introduction
Let A be a binary operation on the set Q and A′ be a binary operation
on the set Q′. Operations A and A′ are called isotopic if there exist
bijective mappings α, β, γ : Q→ Q′ such, that:
γA(x, y) = A′(αx, βy)
or
A(x, y) = γ−1A′(αx, βy)
for every x, y ∈ Q. The groupoids Q(A) and Q′(A′) are called isotopic if
the operation A and A′ are isotopic [5, 6].
A groupoid Q(·) is called a quasigroup if the equations a · x = b and
y · a = b have unique solutions x, y ∈ Q for every a, b ∈ Q [5, 6, 11, 16,
∗This research is supported by the State Committee of Science of the Republic of
Armenia, grant: 10-3/1-41.
2010 MSC: 20N05, 20N02, 03C85, 03C05.
Key words and phrases: functional equation, hyperidentity, second order formula,
quasigroup, moufang loop, invertible algebra, isotopy.
“adm-n2” — 2018/7/24 — 22:32 — page 270 — #108
270 Functional equations and second order formulae
17,31,35, 57, 58, 66, 68]. The algebra (Q; Σ) with quasigroup operations is
called an invertible algebra [10,43,45,46].
A quasigroup Q(·) with a unit (identity element) is called a loop. A
loop Q(·) is called a Moufang loop [11,41,57] if it satisfies the identity:
(x · y) · (z · x) = (x · (y · z)) · x.
A commutative Moufang loop is defined by the following identity
[11,57]:
(x · x) · (y · z) = (x · y) · (x · z).
A quasigroup Q(·) is called distributive [11, 57] if it satisfies the fol-
lowing identities of distributivity:
x · (y · z) = (x · y) · (x · z),
(x · y) · z = (x · z) · (y · z).
For functional equations in algebra, logics, real analysis and topology
see [1–3,10,26–30,37–39].
The problems of solution of the following two general functional equa-
tions of distributivity
A(x,B(y, z)) = H(G(x, y),K(x, y)),
A(B(y, z), x) = H(G(y, x),K(z, x))
on quasigroups are the unsolved problems of quasigroup theory [1–3,10,26].
Moreover, the following functional equations still remain unsolved on
quasigroups, as well:
A(x,A(y, z)) = H(G(x, y),K(x, y)),
A(A(y, z), x) = H(G(y, x),K(z, x)),
A(x,A(y, z)) = H(K(x, y),K(x, y)),
A(A(y, z), x) = H(K(y, x),K(z, x)),
A(x,B(y, z)) = B(A(x, y), A(x, z)),
A(B(y, z), x) = B(A(y, x), A(z, x)).
Here, the solution is unknown even in the case A = B.
Note that the solution of the following systems of functional equations
are also unknown on quasigroups:
{
A(x,B(y, z)) = B(A(x, y), A(x, z)),
A(B(y, z), x) = B(A(y, x), A(z, x)),
“adm-n2” — 2018/7/24 — 22:32 — page 271 — #109
Yu. Movsisyan 271
and
{
A(x,B(y, z)) = B(A(x, y), A(x, z)),
B(A(y, z), x) = A(B(y, x), B(z, x)).
Here, in the case of A = B, the solution follows from [9]. Namely, it is
proved in [9] that every distributive quasigroup is isotopic to a certain
commutative Moufang loop.
The following propositions concerning to the solution of the related
functional equations are consequences from the main result of current
paper.
1) If quasigroupsQ(A),Q(K) and groupoidQ(H) satisfies the following
identities:
A(x,A(y, z)) = A(A(x, y), A(x, y)),
A(A(y, z), x) = H(K(y, x),K(z, x)),
then Q(A) and Q(K) are isotopic to a commutative Moufang loop.
2) If quasigroupsQ(A),Q(K) and groupoidQ(H) satisfies the following
identities:
A(A(y, z), x) = A(A(y, x), A(z, x)),
A(x,A(y, z)) = H(K(x, y),K(x, z)),
then Q(A) and Q(K) are isotopic to a commutative Moufang loop.
3) If quasigroups Q(A), Q(K), Q(K ′) and groupoids Q(H) and Q(H ′)
satisfies the following identities:
A(x, x) = x,
A(x,A(y, z)) = H(K(x, y),K(x, z)),
A(A(y, z), x) = H ′(K ′(y, x),K ′(z, x)),
then Q(A), Q(K) and Q(K ′) are isotopic to a commutative Moufang loop.
1. Auxiliary results and concepts
Lemma 1. If a binary algebra Q(A,B,H,K) satisfies the following iden-
tity:
A(x,B(y, z)) = H(K(x, y),K(x, z)), (1)
where Q(A) and Q(K) are quasigroups, Q(B) and Q(H) are groupoids,
then A and K are isotopic to a quasigroup operation A0, and the operations
B and H are isotopic to an idempotent operation B0 such that:
A0(x,B0(y, z)) = B0(A0(x, y), A0(x, z)). (2)
“adm-n2” — 2018/7/24 — 22:32 — page 272 — #110
272 Functional equations and second order formulae
Besides:
A(x,B0(y, z)) = B
L−1
A
0 (A(x, y), A(x, z)), (3)
where LA(x) = A(0, x), while the element 0 is an arbitrary fixed element
of Q, and
B
ϕ
0 (x, y) = ϕ−1B0(ϕx, ϕy).
In particular, if the operation B is idempotent, then
A(x,B(y, z)) = BL−1
A (A(x, y), A(x, z)). (4)
In addition, if Q(B) is a quasigroup, then Q(H) also is a quasigroup.
Proof. If making the substitution y = z in the equality (1), we obtain:
A(x,ΘB(y)) = ΘHK(x, y), (5)
where ΘB(y) = B(y, y). Let RA(x) = A(x, 0).
If in equality (5) y = 0, then
A(x,ΘB(0)) = ΘHK(x, 0),
i.e.
S(x) = ΘHRK(x),
where S(x) = A(x,ΘB(0)) is a bijection. Hence, ΘH = SR−1
K also is a
bijection. If in equality (5) x = 0, then
LAΘB(y) = ΘHLK(y),
ΘB = L−1
A ΘHLK ,
i.e. ΘB is a bijection. According to (5), the operations A and K are
isotopic and
K(x, y) = Θ−1
H A(x,ΘBy).
If in the equality (1) x = 0, then
LAB(y, z) = H(Lky, LKz),
H(y, z) = LAB
(
L−1
K y, L−1
K z
)
.
Let us consider the following new operations:
B0(x, y) = B
(
Θ−1
B x,Θ−1
B y
)
,
“adm-n2” — 2018/7/24 — 22:32 — page 273 — #111
Yu. Movsisyan 273
and
A0(x, y) = L−1
A A(x, y).
Since x = B
(
Θ−1
B x,Θ−1
B x
)
, the operation B0 is idempotent (and isotopic
to B).
Substituting the values of operationsK andH in identity (1), we obtain
the equalities (2) and (3). If, in addition, the operation B is idempotent,
then ΘB is the identical mapping and B0 = B. Hence, from (3), (4) follows
.
Lemma 2. If a binary algebra Q(A,B,H,K) satisfies the following iden-
tity:
A(B(y, z), x) = H(K(y, x),K(z, x)), (6)
where Q(A) and Q(K) are quasigroups, Q(B) and Q(H) are groupoids,
then A and K are isotopic to a quasigroup operation A0, and operations
B and H are isotopic to an idempotent operation B0 such that:
A0(B0(y, z), x) = B0(A0(y, x), A0(z, x)). (7)
Besides:
A(B0(y, z), x) = B
R−1
A
0 (A(y, x), A(z, x)), (8)
where RA(x) = A(x, 0), while the element 0 is an arbitrary fixed element
of Q. In particular, if the operation B is idempotent, then
A(B(y, z), x) = BR−1
A (A(y, x), A(z, x)). (9)
In addition, if Q(B) is a quasigroup, then Q(H) also is a quasigroup.
According to [43, 45–47, 49], a hyperidentity (or ∀(∀)-identity) is a
universal second-order formula of the following type:
∀X1, . . . , Xm∀x1, . . . , xn (w1 = w2),
where X1, . . . , Xm are functional variables, and x1, . . . , xn are object vari-
ables in the words (terms): w1, w2. Hyperidentities are usually written
without quantifiers: w1 = w2. We say that the hyperidentity w1 = w2
is satisfied in the algebra (Q; Σ) (or the algebra (Q; Σ) satisfies the
hyperidentity w1 = w2) if this equality is valid, whenever every ob-
ject variable xi and every functional variable Xj in it is replaced by
any element from Q and by any operation of the corresponding arity
from Σ respectively (supposing the possibility of such replacement) (also
“adm-n2” — 2018/7/24 — 22:32 — page 274 — #112
274 Functional equations and second order formulae
see [10, 12–14, 19, 22, 23, 25, 33, 42, 65, 67, 70]). If m > 1, the hyperiden-
tity is called non-trivial. The number m is called functional rank of the
hyperidentity. (For the second order formulae see [18,33–35].)
For example, a binary idempotent algebra satisfies the hyperidentity
of idempotency:
X(x, x) = x.
The mode [61] is an idempotent algebra with the hyperidentity of mediality:
X(Y (x, y), Y (u, v)) = Y (X(x, u), X(y, v)).
A distributive bisemilattice (multisemilattice) [24] is an algebra with
semilattice operations satisfying the hyperidentity of distributivity:
X(x, Y (y, z)) = Y (X(x, y), X(x, z)).
Binary algebras with the hyperidentity of associativity:
X(x, Y (y, z)) = Y (X(x, y), z)
under the name of Γ-semigroups (or gamma-semigroups), doppelsemi-
groups and doppelalgebras also were considered by various authors [4, 8,
32,53,56,60,62–64,73,74]. In addition, the hyperidentity of associativity
is satisfied in commutative dimonoids (see, e.g., [75], [76]).
For classification of hyperidentities in invertible and related algebras
see [43, 45–47,50].
For categorical definition of a hyperidentity, in [43] (bi)homomor-
phisms between two algebras (Q; Σ) and (Q′; Σ′) are defined as the pairs
(ϕ; ψ̃) of mappings:
ϕ : Q→ Q′, ψ̃ : Σ → Σ′, |A| = |ψ̃A|,
with the following condition:
ϕA(a1, . . . , an) = (ψ̃A)(ϕa1, . . . , ϕan)
for any A ∈ Σ, a1, . . . , an ∈ Q, |A| = n. Hyperidentities are "identities" of
algebras in the category of (bi)homomorphisms (ϕ; ψ̃). (More about the
application of such morphisms in the cryptography can be found in [7].)
The set of all binary operations defined on the set Q is denoted by
F2
Q, and we consider the following two operations on this set:
A ·B(x, y) = A(x,B(x, y)),
A ◦B(x, y) = A(B(x, y), y),
“adm-n2” — 2018/7/24 — 22:32 — page 275 — #113
Yu. Movsisyan 275
where A,B ∈ F2
Q, x, y ∈ Q. These operations (·) and (◦) are called the
right and left multiplications of binary operations (functions), and they
were studied in the works of various authors [15,21,36,44,46,48,51,52,54,
59,69,71,72].
Lemma 3. The set F2
Q forms a monoid under the right (and left) mul-
tiplication of binary operations. These two semigroups are isomorphic.
The identity element of the semigroup F2
Q(·) is E ∈ F2
Q and it is defined
by the rule: E(x, y) = y for all x, y ∈ Q, and the identity element of the
semigroup F2
Q(◦) is F ∈ F2
Q, and it is defined by the rule: F (x, y) = x
for all x, y ∈ Q. The mapping A → A∗ is the isomorphism of these two
semigroups, where A∗(x, y) = A(y, x) for all x, y ∈ Q. �
Corollary 1. The set of idempotent binary operations on Q is a subsemi-
group in the semigroups F2
Q(·) and F2
Q(◦). �
The binary operation A ∈ F2
Q is the right (left) invertible one if the
equation A(a, x) = b (A(y, a) = b) has the unique solution x ∈ Q (y ∈ Q)
for every a, b ∈ Q. Unique solutions x, y ∈ Q are usually denoted by
x = A−1(a, b) and y =−1 A(b, a). Hence,
A ·A−1 = A−1 ·A = E ,
for the right invertible operation A, and we have:
−1A ◦A = A ◦−1 A = F
for the left invertible operation A. The operation A−1 (or −1A) is a right
(or left) invertible for a right (or left) invertible operation A ∈ F2
Q and:
(
A−1
)
−1
= A =−1
(
−1A
)
;
It is evident that if A is right (or left) invertible then A∗ is left (right)
invertible and:
(
A−1
)
∗
=−1 (A∗),
(
−1A
)
∗
= (A∗)−1 .
The binary operation A ∈ F2
Q is invertible if it is right and left invertible,
i.e. Q(A) is a quasigroup. In this case:
(
−1
(
A−1
))
−1
=−1
(
(
−1A
)
−1
)
= A∗ .
The set of all right (left) binary invertible operations on the set Q is
denoted by Fr
Q (and F ℓ
Q).
“adm-n2” — 2018/7/24 — 22:32 — page 276 — #114
276 Functional equations and second order formulae
Lemma 4. The set Fr
Q is a group under the right multiplication of binary
operations. The set F ℓ
Q is a group under the left multiplication of binary
operations. These two groups are isomorphic too. �
The concept of the right (left) invertibility can be defined via or-
thogonality of operations, as well [17,20]. For applications of right (left)
invertible operations in geometry and topology (knot theory) see [40, 55].
For right (left) loops see [68].
2. Main results
An invertible algebra (Q; Σ) is called Dl-algebra (Dr-algebra) if it
satisfies the following two conditions:
a) In the algebra (Q; Σ) the following hyperidentity of the left (right)
distributivity is satisfied:
X(x,X(y, z)) = X(X(x, y), X(x, z)) (10)
(X(X(y, z), x) = X(X(y, x), X(z, x))); (11)
b) In the algebra (Q; Σ) the following ∀∃∗∃∗∗(∀)-identity of the left
(right) distributivity is satisfied:
∀X,Y ∃∗X ′∃∗∗Y ′∀x, y, z(X(Y (y, z), x) = X ′(Y ′(y, x), Y ′(z, x))) (12)
(∀X,Y ∃∗X ′∃∗∗Y ′∀x, y, z(X(x, Y (y, z)) = X ′(Y ′(x, y), Y ′(x, z)))), (13)
where ∀X,Y means "for every values of X,Y ∈ Σ", ∃∗X ′ means "there
exists an operation on Q" and ∃∗∗Y ′ means "there exists a quasigroup
operation on Q". Thus, (12) and (13) are the second order formulae with
specialized quantifiers (see [33]).
Examples. 1) Let Q(+, ·) be a field and for every a ∈ Q:
Aa(x, y) = (1− a)x+ ay.
If Σ = {Aa|a ∈ Q} then the algebra (Q; Σ) is an Dl- and Dr-algebra.
2) Let Q(A) be a distributive quasigroup.
If Σ = {A,A−1,−1A,−1(A−1), (−1A)−1, A∗}, where A∗(x, y) = A(y, x) for
all x, y ∈ Q, then the algebra (Q; Σ) is an Dl- and Dr-algebra (also see
[52]).
3) Let Q(+, ·) be a field and:
Ai(x, y) = aix+ biy + ci,
“adm-n2” — 2018/7/24 — 22:32 — page 277 — #115
Yu. Movsisyan 277
where ai, bi, ci ∈ Q and ai 6= 0, bi 6= 0, ai + bi 6= 0. If ΣI = {Ai| i ∈ I},
then the algebra (Q; ΣI) satisfies the formulae (12) and (13).
Special cases of Dl-algebras and Dr-algebras were considered in [46]
(Theorems 4.3, 4.3’). Namely, in [46], the invertible algebras with the
following hyperidentities are characterized:
X(x,X(y, z)) = X(X(x, y), X(x, z)),
X(Y (y, z), x) = Y (X(y, x), X(z, x)),
and the invertible algebras with the following hyperidentities are charac-
terized:
X(X(y, z), x) = X(X(y, x), X(z, x)),
X(x, Y (y, z)) = Y (X(x, y), X(x, z)).
In [46] is considered the dual case too.
Now we prove the following more general results.
Theorem 1. 1) If (Q; Σ) is a Dl-algebra, then the quasigroups Q(A),
A ∈ Σ, are distributive and hence are isotopic to commutative Moufang
loops. Every Dl-algebra satisfies the following non-trivial hyperidentity:
X(Y (X(y, z), z), x) = X(Y (X(y, x), X(z, x)), X(z, x)). (14)
2) If (Q; Σ) is a Dr-algebra, then the quasigroups Q(A), A ∈ Σ are
distributive and hence are isotopic to commutative Moufang loops. Every
Dr-algebra satisfies the following non-trivial hyperidentity:
X(x, Y (y,X(y, z))) = X(X(x, y), Y (X(x, y)X(x, z))). (15)
Proof. 1) From the hyperidentity (10) of the left distributivity it follows
that every quasigroup Q(A), A ∈ Σ, is idempotent. If in the formula (12)
X = A ∈ Σ, Y = B ∈ Σ, X ′ = H, Y ′ = K, we obtain:
A(B(y, z), x)) = H(K(y, x),K(z, x)).
According to Lemma 2 (equation (9)) we have:
A(B(y, z), x) = BR−1
A (A(y, x), A(z, x)). (16)
If we substitute z = x in (16), we obtain:
A(B(y, x), x) = BR−1
A (A(y, x), x). (17)
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278 Functional equations and second order formulae
In particular, by putting B = A, we have:
A(A(y, x), x) = AR−1
A (A(y, x), x),
or
A(u, x) = AR−1
A (u, x),
i.e. AR−1
A = A. Then substituting B = A in (16), we obtain a right
distributive identity for any quasigroup operation A ∈ Σ:
A(A(y, z), x) = A(A(y, x), A(z, x)).
Hence, the quasigroup Q(A) is distributive for every operation A ∈ Σ.
According to [9] every distributive quasigroup Q(A) is isotopic to
certain commutative Moufang loop Q(+
A
):
x+
A
y = A
(
R−1
A x, L−1
A y
)
,
where RA and LA are automorphisms of Q(A) and automorphisms of the
loop Q(+
A
), RALA = LARA and RA(x)+
A
LA(x) = x for any x ∈ Q.
Now we prove that hyperidentity (14) is satisfied in the Dℓ-algebra
(Q; Σ).
From equality (17) we have:
A ◦B = BR−1
A ◦A,
and
BR−1
A = A ◦B ◦ −1A.
Hence, according to equality (16), we obtain:
A(B(y, z), x) = (A ◦B ◦ −1A)(A(y, x), A(z, x)),
A(B(y, z), x) = A(B ◦ −1A(A(y, x), A(z, x)), A(z, x)),
A(B(y, z), x) = A(B(−1A(A(y, x), A(z, x)), A(z, x)), A(z, x)),
A(B(y, z), x) = A(B(A(−1A(A(y, z), x), A(z, x)), A(z, x)).
A(B(A(y, z), z), x) = A(B(A(−1A(A(y, z), z), x)A(z, x)), A(z, x)),
A(B(A(y, z), z), x) = A(B(A(y, x), A(z, x)), A(z, x));
Thus, in the algebra (Q; Σ), the hyperidentity (14) is satisfied.
2) Analogically, by using the equality BL−1
A = A · B · A−1, we prove
property 2).
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Yu. Movsisyan 279
Corollary 2. If quasigroups Q(A), Q(K) and groupoid Q(H) satisfies
the following identities:
A(x,A(y, z)) = A(A(x, y), A(x, z)),
A(A(y, z), x) = H(K(y, x),K(z, x)),
then Q(A) and Q(K) are isotopic to a commutative Moufang loop.
Proof. We applied Theorem 1 for Σ = {A}, and used Lemma 2.
Corollary 3. If quasigroups Q(A), Q(K) and groupoid Q(H) satisfies
the following identities:
A(A(y, z), x) = A(A(y, x), A(z, x)),
A(x,A(y, z)) = H(K(x, y),K(x, z)),
then Q(A) and Q(K) are isotopic to a commutative Moufang loop.
Proof. We applied Theorem 1 when Σ = {A}, and used Lemma 1.
Corollary 4. If quasigroups Q(A), Q(K), Q(K ′) and groupoids Q(H)
and Q(H ′) satisfies the following identities:
A(x, x) = x,
A(x,A(y, z)) = H(K(x, y),K(x, z)),
A(A(y, z), x) = H ′(K ′(y, x),K ′(z, x)),
then Q(A), Q(K) and Q(K ′) are isotopic to a commutative Moufang loop.
Proof. We applied the proof of Theorem 1 when Σ = {A}, and use
Lemma 1 and Lemma 2. Note that in the proof of Theorem 1 we used the
idempotency of the operations of Σ.
Let Σl be the set of commutative Moufang loop operations correspond-
ing to the quasigroup operations from Dl-algebra (Q; Σ) according to the
previous Theorem. We obtain a new algebra (Q; Σl).
Let Σr be the set of commutative Moufang loop operations correspond-
ing to the quasigroup operations from Dr-algebra (Q; Σ) according to the
previous Theorem, too. We obtain an algebra (Q; Σr).
Our final result shows the connection (through the hyperidentity)
between the operations from Σl and Σr.
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280 Functional equations and second order formulae
Theorem 2. If (Q; Σ) is a Dl-algebra (Dr-algebra), then the algebra
(Q; Σl) (algebra (Q; Σr)) satisfies the following non-trivial hyperidentity:
X(x, Y (x,X(y, z))) = X(Y (x, y), Y (x, z)). (18)
Proof. 1) Let (Q; Σ) be a Dl-algebra. First, we prove the following equality:
BR−1
A
R
A
(a) = B
for every A,B ∈ Σ, where RA(a) and RA are defined for any element
a ∈ Q and the arbitrary fixed element 0 ∈ Q by the rule:
RA(a)(x) = A(x, a),
RA(x) = A(x, 0).
By putting x = a in identity (16), we have:
A(B(y, z), a) = BR−1
A (A(y, a), A(z, a)),
RA(a)B(y, z) = BR−1
A (RA(a)y,RA(a)z),
RA(a)B
(
R−1
A (a)y,R−1
A (a)z
)
= BR−1
A (y, z),
BR−1
A
(a) = BR−1
A ,
BR−1
A
R
A
(a) = B;
Thus, the mapping ϕ = R−1
A RA(a) is an automorphism of quasigroup
Q(B) for any operation B ∈ Σ. We have:
RAϕx = RA(a)(x) = A(x, a) = RA(x)+
A
LA(a),
ϕ(x) = x+
A
R−1
A LA(a) = x+
A
b,
where b = R−1
A LA(a). Then:
ϕB(x, y) = B(ϕx, ϕy),
B(x, y)+
A
b = B(x+
A
b, y+
A
b),
B(x, y)+
A
z = B(x+
A
z, y+
A
z),
where z = b is an arbitrary element of Q. Thus,
(RBx+
B
LBy)+
A
z = RB(x+
A
z)+
B
LB(y+
A
z). (19)
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Yu. Movsisyan 281
The identity element of the loop Q(+
A
) is the element A(0, 0) = 0, and the
identity element of the loop Q(+
B
) is the element B(0, 0) = 0. Substituting
x = 0 or y = 0 in equality (19), we obtain:
LBy+
A
z = RBz+
B
LB(y+
A
z),
RBx+
A
z = RB(x+
A
z)+
B
LBz,
LB(y+
A
z) = (−RBz)+
B
(LBy+
A
z),
RB(x+
A
z) = (−LBz)+
B
(RBx+
A
z),
where (−x)+
B
(x+
B
y) = y. From equality (19) we obtain:
(RBx+
B
LBy)+
A
z = ((−LBz)+
B
(RBx+
A
z))+
B
((−RBz)+
B
(LBy+
A
z)),
(x+
B
y)+
A
z = (LB(−z)+
B
(x+
A
z))+
B
(RB(−z)+
B
(y+
A
z)).
(20)
In the commutative Moufang loop Q(+
B
) the following equality [9] is
satisfied:
(LBu+
B
v)+
B
(RBu+
B
w) = u+
B
(v+
B
w)
for any u, v, w ∈ Q.
Applying this identity in equality (20) we obtain:
(x+
B
y)+
A
z = (−z)+
B
((x+
A
z)+
B
(y+
A
z)),
z+
B
((x+
B
y)+
A
z) = (x+
A
z)+
B
(y+
A
z),
z+
B
(z+
A
(x+
B
y)) = (z+
A
x)+
B
(z+
A
y);
Hence, in the algebra (Q; Σl) the hyperidentity (18) is satisfied.
2) Analogously, by using the equality:
BL−1
A
L
A
(a) = B,
it is proved that if (Q; Σ) is a Dr-algebra, then the algebra (Q; Σr) satisfies
the hyperidentity (18).
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Contact information
Yu. Movsisyan Yerevan State University, Alex Manoogian 1,
Yerevan 0025, Armenia
E-Mail(s): movsisyan@ysu.am
Web-page(s): www.ysu.am
Received by the editors: 14.05.2017
and in final form 02.06.2018.
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