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8PHYSICS AND MATHEMATICS
TERNARY UNIVERSAL LOOPS SATISFYING QUADRATIC IDENTITIES OF THE LENGTH
THREE
Tarasevych A.
PhD student,
Khmelnitsky National University, Ukraine
Sokhatsky F.
Professor of the Department of Applied Mathematics Vasyl' Stus Donetsk National University, Ukraine
Abstract
In this article, it is proved that each quadratic identity of the length three in the variety of universal loops (i.e., each of its element is neutral) is equivalent to exactly one of the five given identities. The first identity defines the class of all universal loops, the second one defines the variety of the boolean skeins and the other three identities define three parastrophic varieties whose operations are repetition-free compositions of binary commutative middle loops. The obtained varieties are solutions of A. Krapez's functional equation F(x, x, y) = y and that is why it is a partial solution of his problem.
Keywords: ternary quasigroup, quadratic equation, universal loop, Steiner quasigroup, identity, middle loop.
Introduction
This article is the continuation of the articles [3, 4] and all necessary concepts and results can be found there.
In [3], the quadratic generalized functional equations of the length three on invertible functions (i.e., qua-sigroup operations) are studied: general solutions of each element from the family of pairwise parastrophically primarily non-equivalent equations have been found; a full proof of the classification theorem is given.
A quasigroup functional equation, whose functional variables are parastrophic, is an identity in the variety of quasigroups. A ternary quasigroup in which each element is neutral, is called universal loop. For example, a ternary Steiner quasigroup is exactly a totally symmetric universal loop.
In this article, the solutions of all representatives of the classification in the variety of all universal loops are given (Theorem 7). Using the obtained results, it is proved that each ternary quasigroup quadratic identity of the length three is equivalent to exactly one of the given identities (Theorem 3). The first identity defines the class of all universal loops, the second one defines the variety of the boolean skeins [2] and the other three identities define three parastrophic varieties whose operations are repetition-free compositions of binary commutative middle loops (Theorem 4).
Preliminaries
f. r
An algebra (Q; is called a quasigroup if the following identities are true:
(Xoy)oy = X, (Xoy)oy = X, x°(xcy) = y, Xo(xoy) = y.
An element 0 is called left, right, middle neutral for an invertible operation (o). if the respective identity is
true:
Oox = x, xoo = x, jcojc = 0.
in this case, the quasigroup (Q; 0) is a left, right, middle loop respectively. One can find a more detailed information in [6]. Note, the left parastrophe of a middle loop is a left loop and its right parastrophe is a right loop.
A quasigroup is called a group isotope if it is isotopic to a group, ff
xoy = a(x) + a +fi(y), a0 = /?0 = 0, (1)
where (Q; +, 0) is a group, then the tuple (+, x, (, a) is called a canonical decomposition defined by 0 of the group isotope (Q; o).
Theorem 1[7]. Each element of a group isotope uniquely defines its canonical decomposition.
Proposition 1. If a commutative middle loop (0\ 0) is a group isotope, then its canonical decomposition defined by 0 is
xoy = a(x) + a(y)
where (Q; +, 0) is a group of exponent two, i.e. it is a totally symmetric group.
Proof Let (Q,°, 0) be a commutative middle loop which is a group isotope and let (1) be its canonical decomposition defined by 0 . Since the group isotope is commutative, then X = (. Moreover,
0 = 0 o o = a(0) + a + /?(0) = a. The equality -Y o _y = 0 means a(x) + a(x) = 0 . Replacing a(x) with A' , we obtain x + x = 0 which means that the group (Q; +, 0) has exponent two.
A length of a subterm is called the quantity of functional symbols in it. An identity is called quadratic, if in it each variable appears twice.
Theorem 2. [7] Let a quasigroup satisfy a quadratic identity in three variables satisfying the conditions: the sets of variables of subterms of the length one has two variables and these sets are pairwise different, the set of variables of a subterm of the length two has three elements. Then the quasigroup is isotopic to a group.
An algebra ° (T f) =G f ' f = f (in brief, (Q; f) is called a ternary quasigroup, if it satisfies the following identities:
f((14 )f(x, y, z), y, z) = x, (14 )f(f(x, y, z), y, z) = x, f(x,( 24 )f(x,y,z),z) = y, ( 24)f(x,f(x,y,z),z) = y , (2)
f(x , y,(34)f(x, y, z)) = z, (34)f(x, y, f(x, y, z)) = z .
The operation f is called invertible. A ternary quasigroup is a universal loop, if each element is neutral, i.e.,
f ( y,x,x ) = f (x,y,x ) = f (x,x,y )= y (3)
A ternary totally symmetric universal loop is called an SQS-skein which is also called Steiner Ternar idem-potent totally symmetric 3-quasigroup, or Steiner 3-quasigroup [1,2,8].
A G -parastrophe of an invertible operation f is called an operation G f defined by
rTf(xia,x2a,x3a) = x4g ^ f(xi,x2,x3) = ^ G£ S4,
where S4 denotes the group of all bijections of the set {1,2,3,4} . Therefore in general, every invertible
operation has 24 parastrophes. Some of them can coincide. If all parastrophes coincide, the quasigroup is called totally symmetric. Since parastrophes of a quasigroup satisfy the equalities
G( *f) =~f and 'f = f, (4)
then the symmetric group S4 defines an action on the set A3 of all ternary invertible operations defined on the same carrier. In particular, the fact implies that the number of different parastrophes of an invertible operation is a factor of 24. More precisely, it is equal to 24/ | Ps( f ) \, where Ps( f) denotes a stabilizer group of f under this action which is called parastrophic symmetry group of the operation f . Consequently, a totally symmetric quasigroup is a quasigroup whose parastrophic symmetry group is S4. If the parastrophic symmetry group
of a ternary quasigroup is trivial, then the quasigroup has 24 different parastrophes and it is called asymmetric.
Thus, a quadratic identity of the length three is an identity whose individual variables appear twice in it and it has three functional symbols.
For example, all of the identities
Gf( Tf(x, y, z), x, u) =vf(y, z, u) where G,T,V& S4 .
are ternary quadratic identities of the length three.
The main result
Theorem 3. In the variety U of all ternary universally loops, each quadratic identity of the length three is equivalent to exactly one of the following identities:
f (z,x,f (x, y, y)) = f ( z,u,u), (5)
f (x, u f (y, u>z)) = f (x, y,z), (6)
f (f (^ zX ^ u) = f (^ u), (7)
f (^ ^ f (^ z, u))=f (x, u, z), (8)
f (f (x, ^ z), u, y) = f (z, u, x). (9)
The identity (6) is equivalent to f (z, x, f (x, y, y)) = f (z, u, u) which is obtained in [2].
Theorem 4. In the variety of all universal loops U, the identities (5) - (9) define the following subvarieties:
1. (5) defines the variety U;
2. (6) defines the variety HB of boolean skeins, i.e., the class of all ternary quasigroups (Q; f) such that f (x, y, z) = x + y + z for some boolean group (Q; +);
3. (7) the variety A of all ternary quasigroups (Q', f) such that /(\*, y7 z) = (x 0 ^y) 0 z for some com-
mutative middle loop (0', 0) ;
4. (8) defines the variety ^^ A of all ternary quasigroups _/") such that f(x, y, z) = (z 0 y^ 0 x for
some commutative middle loop (0', 0) ;
5. (9) defines the variety ('2Y> A of all ternary quasigroups v' = CO' such that f(x, y, z) = (x 0 z) 0 y for
some commutative middle loop (O', o) .
The obtained varieties are solutions of A. Krapez's functional equation F(x, x, y ) = y and thus it is a partial solution of his problem.
Functional equations
Let A3 be the set of all invertible ternary functions defined on a carrier Q. The relationships (2) and (4) are true for all functions from A . In other words, the following hyperidentities are true over the set A :
T(TF) =TTF, F =F, (14)F(F(x, y, z), y, z) = x; (24) F (x, F (x, y, z), z) = y ;
(34) F (x, y, F (x, y, z)) = z ; (10)
Fx ^ x3 ) =aF(xla, x2^ xsJ , T e S3 .
The hyperidentities are called primary.
Two quasigroup functional equations are called: equivalent over a set Q if they have the same solution set over the carrier; equivalent if they are equivalent over each set.
Definition 1. [5] Two functional equations are called parastrophically primarily equivalent if one can be obtained from the other in a finite number of the following steps: 1) replacing the equation sides; 2) renaming the functional variables; 3) renaming the individual variables; 4) applying the hyperidentities (10).
Lemma 1. Let V = G and V = G be generalized ternary functional equations of the length three. If they
are parastrophically primarily equivalent, then there exists a bijection T in S3 and bijections (1(2,(3 in S4 such that for an arbitrary solution (f, f, f ) of V = G the sequence
(VlT, T2 f2T, f ) (11)
is a solution of the equation V = G .
In this case, (t, T1, T2, T3) is called a defining bijection system of the equations V = G and V = G
Theorem 5. [3] Every generalized quadratic ternary quasigroup functional equation of the length three is parastrophically primarily equivalent to exactly one of the following equations:
Fi( z, x, F,(x, y, y)) = F3( z, u, u), (12)
F(Fi(x, y, y), z, z) = F3(x, u, u), (13)
F(Fi(x, y, z), u, u) = F3(x, y, z) , (14)
F (F (x, y, z), x, u) = F3( y, z, u) . (15)
A functional equation is an identity if all functional variables are parastrophic. Each quasigroup identity defines a variety of quasigroups. Considering the identity as a functional equation means that the solution of the identity is its variety. Two identities are called equivalent, if they define the same variety. Two identities are parastrophically equivalent, if they define parastrophic varieties. Therefore, if identities are parastrophically primarily equivalent, then they are parastrophically equivalent.
Theorem 6. [3] A triplet (fl, _/2, f3) of ternary invertible operations defined on set Q is a solution of the functional equation (15) if and only if there exist binary invertible operations o, * , 0 on Q such that
fi(y,x,u) = (xOy)*u , (a)
r
f2(x,y,z) = x<Xyoz),Q3) (16)
f3(y,z,u) = (yoz)*u.(c)
6 The scientific heritage No 59 (2021)
Functional equations on universal loops
A ternary quasigroup is universal if only if its operation is universally neutral. It is easy to see that each triplet of universally neutral operations is a solution of each of the functional equations (12), (13), (14). The solution of the equation (15) is given in the following theorem.
Theorem 7. A triplet (f, f, f ) of ternary universally neutral invertible operations defined on set Q is a solution of the functional equation (15) if and only if f = f and there exists a binary commutative middle loop (Qз °) such ^at
f1(x,y,u) = (x°y)°u,(a)
r
/2{х,у,г) = хо{уог).ф) (17)
Proof. Let a triplet (fl, f2,f3) of ternary universally neutral invertible operations defined on set Q be a
solution of the functional equation (15). According to Theorem 6 there exist binary invertible operations о, * , 0 on Q such that (16) hold.
Let X := u in (16,a); X := y = u, y := z in (16,c) and X := u = z in (16,c):
y = (xOy)*x, y = (xoy)*x, y = (y°x)*x.
These identities imply the following equalities:
(xOy) * X = (x о у) * X , (x О у) * JC = (у о х) * X
which mean that the operation (°)is commutative and ( 0 ) coincides with (o). Moreover, the equality у = (у о x) * x implies that the operation (*) equals (o). Thus, j\ = f} and (17) hold. Put x:=y in (17, a):
(xox)o U = U .
Consequently, the term X о x is a left neutral element of the operation (o) for all X of Q. Since (Q', °) is a quasigroup, then X о x is a constant, i.e., X о x = e for some element e from Q . It means that the quasigroup ('Q; °) is a commutative middle loop.
Vice versa, let (Q; e) be a binary commutative middle loop. We have to prove that the operations j\ and f2 defined by (17) are universally neutral, i.e., both satisfy the identities (3). Indeed,
fx{x,x,u) = (xox)ou = e°u = u, f1(x,u,x) = (xou)ox = (uox)ox = u , /г(и, X, x) = (и О x) О JC = U.
r r r
f2(x,x,z) = x°(x°z) = z, f2(x,z,x) = x°(z°x) = x°(xoz) = z,
r r
f2(z,x,x) = z°(xox) = z°e = z.
Quadratic identities
A functional equation is called an identity in quasigroups, if all its functional variables are pairwise parastrophic. For example,
F ^ F ^ Xi ,X2 ^, Xi ^— F ^ X2 ^ , (18)
where <J, T eS4, is a quadratic identity of the length 3. This identity is said to be true in a quasigroup (Q; f), if the equality
f ( f (Xi,X2,X3),Xi,X4)= f (X2,X3,X4), (19)
is true for all Xx,..., X6 eQ . It is called an identity in the quasigroup (Q; / ).
{1,2} e {{2a, 3a}, {la, 4a}} n {{if, 2t} , {3t, 4t}} . (20)
Theorem 8.
1. Let the condition (20) be true. Then (19) is the identity in a ternary universal loop (Q; f) if an only if there exists a commutative middle loop (Q; 0) such that
f(x,y,z) = (xoy)oZ; (21)
2. Let the condition (20) be false. Then the identity (19) is true in a ternary universal loop (Q; f) if and
only if there exists a group (Q; +, 0) of exponent two such that
f (x,y,z) = x + y + z . Note, totally symmetric groups are exactly groups of exponent two.
Proof. The identity (19) is true if universal loop (Q; f) means that the triple (f, T f,T f) is a solution of the functional equation (15). Since each parastrophe of a universal loop is also a universal loop, then according to Theorem 7, it is equivalent to the existence a binary commutative middle loop {Q; 0) such that
f(xl,x2,x3) = (xlox2)ox3,
r
f
These equalities are equivalent to
af(xl,x2,x3) = xlo(x2°x3),
!
T f (xl, X2, X3 ) = (xl O JC2 ) O x, .
t
f (xj,x2,x3) = x4 <=> (xj o jc2)ojc3 = x4, (a)
r
a f (x1, x2, x3) = x4 <=> x1 O (x2 O JC3 ) = x4, (b) (23)
T f (jCj, x2, x3) = x4 -G> (jCj ojc2)ojc3 =X4. (c)
%2 j i — >—-r l 2 / 3 — •
Replace xi with x( in (23, b) and xi with xiT in (23, c) for all i = 1,2,3,4 :
f ( x1, x2 ,x3
r
"f (Xla - ,*3<t)= X4a <=> ° {X2a °X3a ) = =
Y - >X3r) = X4r <=> °X2r)°X3r=X4r-
Using the definition of parastrophes of a ternary invertible operation, we obtain
f (Xj,x2,x3) = x4 <=> (jCj o jc2)ojc3 =x4 , (a)
r
f(x1,x2,x3) = x4 <^\a°{x2a°x3a) = x4a,^o) (24)
f(x1,x2,x3) = x4o(xlT °x2t)ox3t =x4t.(c)
Apply (24, a) to (24, b) and (24, c):
f(xl,x2,x3) = (xlox2)ox3,
i ,
t
(XXOX2)OX3=X40(XITOX2T)OX3T =X4t.
By definition of £ - and r-parastrophes of binary invertible operations, the following relationships are true:
f(x1,x2,x3) = (x1ox2)ox3, (a)
ox2 = x3 ox4 ox4CT = X2CT °X3ct, (b) (25)
oX2 =X3 ox, XlT oX2r =X3r ox4r . (c)
Thus, the identity (19) is equivalent to the existence of a commutative middle loop (Q; 0) such that (25) holds. Consider items 1 and 2 of this theorem.
1. If (19) is true, then (24, b) and (24, c) follow from commutativity of the operation (oj . Therefore, item 1 has been proved.
2. Let there exist a group (Q; +, 0) of exponent two such that (°) = (+) and (22) holds, then (25) can be written as follows
f (x, x^, x3) — x ^ x^ ^ x3,
xi + x2 = x3 + x4 ^ xiG + x4g = x2G + x3G , xi + x2 = x3 + x4 ^ xir + x2r = x3r + x4r .
These relationships are true for all g, t because the group (Q; + ,0) is totally symmetric. Consequently, the identity (19) is true in the universal loop (Q; f).
Vice versa, let the identity (18) be true in a universal loop (Q; f). Therefore, the relationships (25) are true
for some commutative middle loop (Q; 0) . Substitute (x, °x2)ox3 for A*4 in the right parts of the relationships (25, b) and (25, c). Considering on T and G, we obtain eight possible identities:
° X2 ) ° X3 ^ ° X4a = X2a ° ° ° X2 ) ° X3 ^ = X2a ° X3a >
e
X1 a ° X4a = ° X2 ) ° X3 J ° ° X4a = X2a ° ° X2 )
° X2 ) ° X3 ^ ° X2. = X3. ° X4. . Xlr ° ° X2 ) ° X3 j = X3. ° X4. =
*1T ° X2. = ° X2 ) ° X3 ^ ° X4. . *1T ° X2. = X3. ° ° X2 ) ° X3 j •
Since (20) is false, then at least one of them satisfies the condition of the Theorem 2. Consequently, the commutative middle loop (Q; 0) is isotopic to a group. By Proposition 1, there exists a group (Q; ©, 0) of
exponent two and a self bijection a of the set Q such that a 0 = 0 and Xoy = a (x) © a (>'). Therefore,
x°y = a1 (x © a (j)) and so
f (x, y, z) = (x o j) o z = a1 (a (x) © a (y) © a (z)) = x + y + z ,
where x + y:=a 1 (a (x) © a (y )). It is easy to see, that the loop (Q; +, 0) is isomorphic to the group
(Q; ©, 0) . Therefore, (Q; +, 0) is a group of exponent two and (22) holds. Proofs of Theorem 3 and Theorem 4
Since each parastrophe of a universal loop is also universal loop and each triple of universally neutral functions is a solution of the equations (12), (13), (14), then each identity being parastrophically equivalent to the identity of the form (12), (13), (14) is true in any universal loop. That is why such identities are equivalent and define the subvariety Uof all ternary universal loops in U. One of this identities is (5) and so the item 1 of Theorem 4 has been proved. It remains to analyze identities of the form (15).
Let A denote the variety of universal loops (Q; f ) defined by (21). Let (20) be false. Then according to
the item 2 of Theorem 8 there is only one variety, namely the variety of all boolean skeins. Consequently, all such identities are equivalent and (6) is one of them. Thus, the item 2 of Theorem 4 has also been proved.
Let (20) be true. According to Theorem 5 these identities parastrophically equivalent to the identity (22). Remember that for parastrophic symmetry orbit and parastrophic symmetry group of A the relationships
Po(A):={°A |aeS4}, Ps(A):={aeS3 |aA = A }, \Po(A)|-|Ps(A)| = 24. (26)
To determine all varieties being parastrophic to A , we'll prove the following lemma.
Lemma 2. The parastrophic symmetry group of the variety A is a dihedral subgroup of , namely
Ps(A) = Da := {/,(12),(34),(12)(34),(13)(24),(1324),(14)(23),(1423)}.
D is a subgroup of the parastrophic symmetry group Ps( f) of any universally loop (Q; f ) from A .
Proof. Let v be any element of the parastrophic symmetry group Ps(A) , i.e., VA - A . By Theorem 8, for any commutative middle loop (0>'°) the v -parastrophc of the loop (0,'/) defined by (21) belongs to A . Consequently, there exists a commutative middle loop (Q; *) such that
f (xi, x2 , x3 ) = (xi * X2 ) * x3 (26)
for all x^,x2,x3 e Q . In other words,
^ ^ , X2 j ^ ^ — Ji^ ^ ? ^ ^ JC2 ^ ^ — JC^ .
Replace xi with Xw for all i = i, 2,3,4 :
f (Xlv ' X2v > X3v ) = X4v (Xlv * X2v ) * X3v = X4v '
Using the definitions of parastrophe of ternary and binary quasigroups, we obtain
f( f(x, У, z), u, y) = f(z, u, x) .
Applying (21), we have
oj2)oi3 = x4 xlv *x2v = x3v *x4v . (28)
Substitute (x1 o a';) o a', = a4 for X4 in the right part of the relationship. Since 4 e {1v, Iv, 3v, 4vj , then
we consider four cases.
Let 4 = iV, then we obtain the identity
(jXl 0 X2 ) '0 •X3 j * X2v = X3v * X4v • (29)
If {l, 2} ^ {3i/, 4v ), then by Theorem 2 the loop (Q; o) is isotopic to a group. It contradicts to the assumption. Therefore, {i, 2} = {3V, 4v} and 2v = 3 and so there are two possible values for V :
(i 2 3 4^ , x (i 2 3 4^ , w v
14 3 , 2J=(i423) and [4 3 2 1 J=(i4)(23). <30)
In both cases the identity (29) is equivalent to
i
((Xoy)oZ)*Z = X*y, (31)
i.e.
(29)(x*y)*z = (xoy)oz ,
which means V f = f . By Theorem 8, it is true for all universal loop (Q;f) from A , then the bijections (30) belong to Ps(A) and Ps(f) .
Let 2v = 4, then (28) is equivalent to the identity
Xlv*^XlOX2>X3^=X3v*X4v- (32)
Theorem 2 implies {i, 2} = {3v, 4v} and iv = 3. Therefore, v has two values:
(i 2 3 4^ , w x (i 2 3 4^ , x
(3 4 3 ^(i3)(24), [3 4 33 4 J =(i3 24). (33)
In both cases, the identity (32) is equivalent to (31) and so the permutations (33) belong to Ps(A) and
Ps(f).
Let 3v = 4 , then we obtain
Xlv *X2v =f(VX2^°X3^*X4v (34)
Theorem 2 implies the equality {iv, 2v} = {i, 2} . Consequently, there are two values for v : (i 2 3 4^ , . (i 2 3 4N
12 4 3
= (34)' 01 » J = (12)(34)
V y V
In both cases (34) imply (31).
2 1 4 3
Therefore, these self bigections belong to Ps( A ).
If {4v} = 4, then the self bigections l and (12) belong to Ps( A ).
Since we considered all possible cases, then Ps( A) = D4 .
By Lemma 2, v f = f for all v e D4 , then the equality
S4 = D4 u (13D u (23D (35)
implies that it enough to consider (19) when t,t e{i,(13),(23)} . But (20) is true only if t = (13) and t = l. That is why there is exactly one variety which is defined by quadratic identities of the length three satisfying the condition (20) and one of these identities is
f( 3f(x1 ,x2,x3),x1 ,x4) = f(x2,x3,x4) .
We obtain it from (19) putting T = (13) and t = l. By definition of parastrophy of ternary operations
f(f(x3,x2,x1),x1 ,x4) = f(x2,x3,x4) . (36)
Replacing these variables, we get
f(f(x,y,z),z,u) = f(y,x,u) (37)
Since Ps( A) = D4 , then Pof A) consists of | S4 \/\D4 |= 24 / 8 = 3 varieties. The equality (35) implies that
Po(A) = {A, (13) A,(23) A} .
To find identities which defined the varieties (13) A and (23) A , it is enough to respectively replace f with (13)f and f with (23)f in (36):
( 3 f( 3f(x3, X2 ,xi),x! ,x4) — ( 3f(x2 ,x3 >x<\) ,
(233f((2Vf(x3,x2,xl),xl,x4) — (233f(x2,x3,x4). By definition of parastrophy of ternary quasigroups, these identities are equivalent to
f(x4, xi ,f(x\ ,x2, x3)) — f(x 4 ,x3 ,x2 ) Replacing the variables, we obtain
f(x, y,f(У Z u)) — f(x, u, z)
Thus, Theorems 3 and 4 have been proved.
f(f(x3 ,xi ,x2 ),x4 ,xi) — f(x2 ,x 4 ,x3 )
f( f(x, У, u, У) — f(z, u, x) .
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