Linear invertible algebras and their nucleus Текст научной статьи по специальности «Математика»

УДК 512.577

LINEAR INVERTIBLE ALGEBRAS AND THEIR NUCLEUS

Davidov S.

Yerevan State University E-mail: davidov@ysu.am

In this paper, the left (right) h -nucleus of an invertible algebra is defined and its connection with regular permutations of the invertible algebra is investigated. Using the notions of the h -nucleus, we have obtained the characterizations of linear invertible algebras and those for left (right) linear invertible algebras by the second-order formulas. Keywords: quasigroup, invertible algebra, algebra linear over a group, left nucleus, right nucleus, second-order formula

Introduction

A binary algebra (Q,Z) is called invertible algebra or system of quasigroups, if each operation from Z is a quasigroup operation. The invertible algebras first were considered by Shaufler [1] in connection with coding theory. He pointed out that the resulting message would be more difficult to decode by unauthorized receiver that in the case when a single operation is used for calculation. Later such algebras were investigated many authors (see [2; 3; 4] and citing their literatures).

There are some notions of right and left nuclei in quasigroups. Let Q(-) be a groupoid. The following set: Nl = {a e Q | ax ■ y = a ■ xy, for all x,y e Q} is called left nucleus. The right nucleus is defined similarly: Nr ={a e Q | x ■ ya = xy ■ a, for all x, y e Q} [5] (see also [6]). The nuclei Nl and Nr play an important role in the theory of loops

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(quasigroups with identity element). It holds for the quasigroup (Q; ■), that N, ^ 0 (Nr ^ 0) iff (Q; ■) has a left (right) identity element [5].

In [5] the concepts of the left and right nucleus in quasigroup Q(-) with respect to a fixed element h e Q is considered. Namely, a

left (right) nucleus is the following set: {a eQ | ax■ y = a■ Ll(hx■ y)}

( {a eQ I x ■ ya = L^(x ■ yh) ■ a}), where Lhx = hx (Rhx = xh ).

The more general concept of the left (right) h -nucleus were introduced by G.B.Belyavskaya in [7] and it is closely connected with linear quasigroups. The left (right) h -nucleus is called the maximal set Nj(h) (Nr(h)) of such elements a e Q for which:

ax ■ y = aeh ■ Lhl(hx ■ y) ( x ■ ya = Rl(x ■ yh) ■ fha ) for all x, y e Q and

Nj (h) ■ eh = Nj (h) (fh ■ Nr(h) = Nr(h))), where eh and fh are the right

and left local identity elements of h e Q, i.e. h ■ eh = h and fh ■ h = h .

When passing from quasigroups to invertible algebras it becomes necessary to consider various hyperidentities of associativity (for hyperidentities see [8]). According to [8], if a non-trivial hyperidentity of associativity holds in a functional non-trivial invertible algebra, then it can be of the functional rank 2 and in one of the following forms:

In the present paper the right Nr(h) and left Nj(h) h -nuclei of the invertible algebra (Q;£) are defined with respect to hyperidentity

X ( x,Y ( z)) = Y ( X ( x, y), z), X ( x,Y ( y, z)) = X (Y ( x, y), z), X ( x, X ( y, z)) = Y (Y ( x, y), z).

(1) (2) (3)

(3).

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Regular Permutations

Let (Q;Z) be a binary algebra. We denote by LAa and RAa the left and right translations

L4 a : x i—» A(a, x) (R4 A(x, a)).

If the algebra (Q; Z) is an invertible algebra, then the translations La and Ra are bijections for all a eQ and all AeZ. The group generated by all translations of the invertible algebra (Q;Z) is called multiplication group of the invertible algebra (Q;Z) and is denoted by

G = MulQ.

The permutation aeMulQ is called inner mapping relative to the element h e Q, if ah = h [5]. All inner mappings relative to the element h of the invertible algebra (Q;Z) form a group, which is called inner mapping group Ih of (Q;Z) . We give the following definition.

Definition 1 Let (Q; Z) be an invertible algebra. A permutation A(p) of the set Q is called left (right) regular permutation of (Q; Z), if for each operation X eZ there exists a permutation Xx(p*x) (in the general case depending on X ) of the set Q such that

AX ( x, y ') = X (Xxx, y),

(PX (x y ) = x (x pXy %

for all x,y e Q and XeZ.

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Obviously, for any XeS the permutation Xx is defined uniquely and is called conjugate permutation of X with respect to the operation X . From the definition of regular permutations immediately follows that

for all x e Q and X eS.

The set L(R) of all left (right) regular permutations of (Q;S) form a group, moreover L(R) is a subgroup of the multiplication group MulQ, and this follows from the equalities (4) and (5).

The set of all conjugate permutations of left (right) permutations of (Q;S) with respect to the operation XeS we denote by L*X (RX). L* (R* ) also forms a group, moreover it is a subgroup of the multiplication group MujQ .

Note that if each operation XeS has the identity element eX, then X = XX and p = p*X for all X.

Consider the mapping fX : L ^ L*X (gx : R ^ Rx) which is defined in the following way: fXX = XX (gXp = p*X). Obviously, fx(gX) is an isomorphism between the groups L and L*X (R and RX). Hence, the groups L*X and LLY (RX and RY) are isomorphic for all

Let L^L (R^R). We say that L (R) is invariant with respect to fx (gx), if fx L = L (gx R = R) Suppose, that

(4)

(5)

X,Y e Z.

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for every X eS. The set

Mv={LcL|/vL = L}, Nx={R^R\fxR = R}

La- = [J LGMX,

is the maximal subset in Mx, since

/ <■' »= / <U, , L) = U, , / <■') = Ц , L = L Similarly,

Rx = [J RgNx,

is the maximal subset in Nx, since

= ¿mU , *) = U= Uk , R R

If L^L (R^R) is invariant with respect to fx(Sx), ie. fx L = L c: L*v (gx R = R c: R*), then from (4) and (5) we have

Ь А',ЬЛ\. /• /• (6)

Д", Li

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K /• •.,!*/.•.', A' A' (7)

J,R.v

for all X eZ.

Definition 2 ([9]) The invertible algebra (Q;Z) is called left (right) linear over the group (Q;+), if every operation AeZ has the form

A(x y) = Pax+PAy (A(x y) = aAx + wAy\

where [3A (aA ) is a permutation of the set Q, and pA (yA ) is an authomorphism of the group Q(+).

Definition 3 We say that a binary algebra (Q; Z) is principally isotopic to the groupoid (Q;-), if each operation from Z is principally isotopic to the groupoid (Q;-), i.e. for every operation AeZ there exist permutations aA ,fiA of the set Q such that

A(x y) = aAx -fiAy,

for all x, y e Q.

Definition 4 Let (Q;-) be a groupoid. A permutation A (p) of the set Q is called left (right) regular permutation of (Q;-), if Ax- y = A( x - y) (x- p = p( x - y)) for all x, y e Q.

All regular permutations of the groupoid form a group [5].

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Lemma 5 If an invertible algebra (Q; Z) is principally isotopic to the loop (Q;-), then L = L = axLxax, for all XeZ, where L is a group of the left regular permutations of Q(-).

Proof. Let Ae L, i.e. for every operation XeZ there exists permutation AAX such that

AX (x, y ') = X (AxX, y).

We have

A(axx ■ ßXy) = ax-AXx ■ ßxy, A( x ■ y ) = axAXaX1 x ■ y.

If we take y = 1 (the identity element of the loop Q(-)) in the last equality then we get: A = axAAx<, thus A = <xxAAx< e L, i.e. AeaxL*xax. Hence, axLxax e L and L^ L for all XeZ. Conversely, let AeL, i.e. A(x ■ y) = Ax ■ y . Then AX(ax x, ßx y) = X(axAx, ßx y) or AX(x, y) = X(axAaxx, y). Thus, for all XeZ there exists the permutation AX = aX^AaX such that AX(x,y) = X(Axx,y), i.e. Ae L. Therefore L ^ L and L = L. On the other hand we have axAaxe Lx or axLax e Lx, i.e. L <^axLxax , hence axLxax = L.

It follows from the proof of the last lemma that axLxax = aYL*<1 for all X,Y e Z.

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Lemma 6 Let (Q; S) be a left linear invertible algebra and let L be a group of the left regular permutations of the group Q(+). Then L = Lx = L = Lx for all X e S.

Proof. By Lemma 5, for all XeS we have L = (pxLx( . L*X = (X:L(X . Since (X is an automorphism of the group Q(+), then (pxL(x = L. Further, since fx(L) = Lx = L, then L = Lx .

Left and Right h -Nuclei of Invertible Algebras.

Consider the following equation

X(X(a, b),c) = Y (a,Y (b, x)). (8)

in a binary invertible algebra (Q;S). This equation for any X,Y eS and a, b e Q has a unique solution that depends on X,Y, a, b, c . Thus, we can assume that

* = rUfc,

where yfj^ is a permutation of the set Q .

The expression Y(b^fc) is denoted by (3)Yf (b, c), where yffc is a solution of the equation (8). Thus we have

X (X (a, b), c) = Y (a,(3) YaX (b, c)). From the equality (8), we get

LX,X(a,b)c = L,aL,bx , x = L,b^Y,aLX,X(a,b)c .

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Therefore,

X ,Y _ T-1 T-1 T

7a,b LY ,bLY ,aLX ,X (a,b)'

Hence, we obtain:

(3)Ya (b, c) = LY,aLX,X(a,b)C-

In the same way, if we consider the equation

X (a, X (b, c)) = Y (Y (x, b),c),

we get a new operation X Y (a,b) on Q , which depends on X ,Y and c, such that

xcY(3){a, b) - Ry]cRx^x(bcC)a.

Fix the element h e Q and consider the following equality X (X (h, a),b) = Y (h,(3) YhX (a, b)), where a,b e Q, X,Y e Z . From the above equality we obtain

Rx,bRx,ah - RYX3)Yhx (a,b)h;

or

Rr IyX (abb)RXbRX,ah - h.

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Thus, the following permutation

T?x y = j? j? j?

(3)Ra,b ,(3(a,b)Rxa,

is an inner permutation with respect to the element h for any a,be Q and X,Y eZ.

Similarly, from the following equality

Y (XY(3)(a, b),h) = X (a, X (b, h)), we obtain that for any a,b e Q and X,Y e Z the following permutation

jxY = r-1 L L

(3) ja,b JY Y (a,b)jx aJx ,b '

is inner with respect to the element h .

Consider now the equation X(h,a) = Y(x,h). If X,Y and h are fixed, then this equation has the unique solution, which depends on a , i.e.

X(h, a) = Y (of ,Ya, h), (9)

or Lx^ha = R, haX'Ya. Hence, &XlJ = RY \ Lx^h. From (9) follows that

Rx ah = L y7h, or L 1X7 Rx ah = h, i.e. the permutation

, y, yx ' ^a '

Tx ,Y — T n

Ta JY ,0х YaRxa '

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is an inner mapping with respect to the element h .

The following theorem is proved in the same way as Theorem 4.4 for quasigroups [5].

Theorem 7 The inner mapping group Ih of an invertible algebra (Q; S) is generated by permutations Tf,Y ,(3) R^^ ,(3) Lx Y, where x, y e Q and X,Y eS.

The element exa (fX) of an invertible algebra (Q;S) is called right (left) local identity element of an element a with respect to the operation X, if X(a,eX) = a (X(ff,a) = a).

Definition 8 A left h -nucleus N(3)(h) of an invertible algebra (Q; S) is called the maximal subset H of the set Q such that:

1. (3)Kia = Y (a> eYh ), for all X ,Y eS, x, y e Q, a e H;

2. X(H,eX) = H for all XeS.

Condition 1 in the above definition means that the element a e N(3)(h) satisfies the following equality

X(X(a, x),y) = Y(Y(a, £ ), L^X(X(h, x),y)), (10) for all x,y eQ and all X,Y eS.

Definition 9 A right h -nucleus N(3\h) of an invertible algebra (Q; S) is called the maximal subset H of the set Q such that:

1. (sLX^a = Y (fY, a), for all X ,Y eS, x, y e Q, a e H;

2. X(fh,H) = H for all XeS.

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Condition 1 in the above definition means that the element a e N(f>(h) satisfies the following equality

X (x, X (y, a)) = Y ( Ry ¿X (x, X( y, h)),Y f, a)), (11)

for all x,y eQ and all X,Y eS .

If the invertible algebra (Q;S) has an identity element e (an identity element of (Q; S) such that X(e, x) = X(x, e) = x for all x e Q and X eS), then:

N(3) (e) = {a | X(a, X(x, y)) = Y (Y (a, x), y), for all X ,Y eS, x, y e Q}, Nf (e) = {a IX (X (x, y), a) = Y (x, Y (y, a)), for all X ,Y eS, x, y e Q} .

Lemma 10 Let (Q; S) be an invertible algebra and let h be an arbitrary fixed element of Q. Then N(3\h) ^ LXh, for all X eS.

Proof. Let Nf\h) = H and a e H. We have for all X,Y eS,

x, y e Q

X (X (a, x), y) = Y (Y (a, eh ), L ]hX (X(h, x), y))

or

x (lx ,ax y) = lyj{ajh) lr,hx (lx .h^ y i

Let us fixe the operation Y and take Lxhx instead of x in the last equality, then we obtain

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X(Lx,aL~X,hx'y) - LYJ(a/h )lr,hX(x y).

Thus, Л - L y\Ly\ is a left regular permutation of (Q;T), for which

Y Y (a ,eY )

ЛХ LX,aLX ,h , i'e-

fX (Lrr(aeh )Lh,h) LX ,aLX,h . (12)

By the definition of nuclei, from the last equality we obtain

fx (Ly ,HLY^,h ) LX,HLX,h .

(13)

On the other hand, according to the equality (10), we have

X(LX,ax y) = L y\LY,hX(LX.h^ yI

or

X (LX,aLX,hX' У) = Lr,Ha,Jh ) Lhl.hX ( X' У ).

Since the operations X and Y are arbitrary, we get

L Y\t?h = L zLz\ for all operations Y,Z eZ. Therefore, by the

Y ,Y (a,eh ) , Z,Z(a,ek ) ,

definition of nuclei we have LhL\ = LZHL~zh, and by equality (13) we obtain

fx ( Ly )- Ly

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Thus, the set Lt Ly ¡jLy is invariant with respect to fX. Hence,

Ly c: Lx . But, we have Lr h = LrHLxhh = LY He\ = Y(H, e\ ) = H. Therefore, we obtain H e Lxh .

Lemma 11 Let (Q; Z) be a left linear invertible algebra and let

h be an arbitrary fixed element of Q. Then Lxh ^ N(f)(h).

Proof. By Lemma 6 we have: L = L*x = L = Lx . Since fxL = L for all XeZ, then from (6) for all xeQ and all XeZ we obtain

T = T T 1

Particularly, for h = x we have

T = T T 1

T TX,LhTXhh ■

From the last equality follows

Lh = LxML~lhh = X(Lh, lLx hh) = X(Lh, exh ). As L is the maximal subset invariant with respect to fx, we have:

L = fX (L) = fX (LY,LhLhh,h ) = LY,LhLhh,h

and since the left nucleus H is the maximal subset satisfying (13), we obtain that Lh ^ H.

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Corollary 12 Let (Q; Z) be a left linear invertible algebra and let h be an arbitrary fixed element of Q. Then Lh = N(3)(h).

The last corollary shows that the left h -nucleus of a left invertible algebra is an orbit of the element h with respect to a group L of regular permutations.

Proposition 13 Let (Q; Z) be a left linear invertible algebra, then N(3)(h) = Q for an arbitrary fixed element h e Q.

Proof. By Lemma 6, we have: L = Lx = L. Hence, according to Corollary 12 we obtain: N(\h) = Lh = Lh = Q (since L is a group of left regular permutations of the group Q(+), and the left regular permutations of Q(+) are translations of Q(+)).

In the same way we can proof the following proposition.

Proposition 14 Let (Q; Z) be a right linear invertible algebra,

then Nf\h) = Q for an arbitrary fixed element h e Q.

Main Results

Theorem 15 An invertible algebra (Q; Z) is left linear if and only if N(3)(h) = Q for some h e Q.

Proof. Let N(3\h) = Q for some h e Q. Then the equality (10) is valid for all a, x, y e Q and all X,Y eZ, i.e. we have the following

X(X(x, y), z) = Y(Y(x, erh),L\hX(X(K y), z)) (14)

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for all x, y, z e Q and for all X,Y eZ. Let us fix the operations X,Y in equality (14), then we obtain

X ( X ( x, y), z) = A( x, B( y, z)\

where A(x,y) = Y(R rx,Lz\y),B(x,y) = X(Lxhx,y)■ According to the

Y ,eh ' '

last equality and Belousov's theorem about four quasigroups which are connected with the associative law [10], all the operations XeZ are isotopic to the same group (Q;*).

For every XeZ let us define the operations + x in the following

way

x + xy = x (RzIx L^^yX

where a, b are some elements of Q. These operations are loops with the identity element 0X = X(b,a) and they are isotopic to the group (Q;*). Hence by Albert's theorem ([5]) they are groups for every X eZ.

Let us rewrite equality (14) in terms of the operations + x and +Y , and take y = Lxb0x, then we obtain

RX,aRX,ax +X LX,bz = Rh,a (Rh,ax +Y LY,beh ) +Y LY,b^Y,h (RX,aRX,ah +X LX,bz),

or

x +Xz - Rha (Rh,aR2aX +Y Lh A ) +Y Lh,bL]h (RX,aRX,ah +X ZX

i.e.

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x z = Ax +7 jz, (15)

where

Ax = Rh a (RhRax +Y Lh A ^ / = Lh Lh (RX,aRX,ah +x z)

are permutations of the set Q . We have

X(x y) = RX,ax +X LX,by. (16)

According to (15) and (16) we obtain

x(x y) = rnxax +y L^

or

X(x, y) = 6xx +y d2y, (17)

where dx and 62 are permutations of the set Q .

Let us fix the operation +Y and rewrite equality (14) in terms of the operation +Y , and using (16) and (17):

&i(Ry,ax +yLy,by) +Y #2z =

= Rya(Rxax +x Lx X) +y Ly a(#(Ry,ah +y Ly^ +y #2z)-We take z = d2l0h in the last equality, then we obtain

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(ry ax +y ly ,by) = ry a (rx,ax +x lx,bef ) +y ly l]a (ry ah +y ly

or

A (x +Y y) = ccx +Y py,

where a,p are permutations of the set Q. Therefore, 0 is a quasiautomorphism of the group (+Y) . According to Lemma 2.5 in [5] we have: dxx = (pXx +Y cx, where (px e Aut(Q(+Y)), and cx is some element of the set Q .

Hence, from (17) it follows that

X(x, y) = (xx +y Cx +Y Ay■

Since the operation X is arbitrary, we obtain that the invertible algebra (Q;Z) is left linear over the group Q(+Y).

The converse statement follows from Proposition 13. In the same way we can proof the following theorem. Theorem 16 An invertible algebra (Q; Z) is right linear if and

only if N(3\h) = Q for some h e Q.

Definition 17 ([11]) An invertible algebra (Q; Z) is called linear over the group (Q;+) if every operation AeZ has the form

A( x y ) = (ax + ta + ¥aУ,

where (A, yA are automorphisms of (Q;+) for all AeZ, and tA are fixed elements of Q .

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Theorem 18 An invertible algebra (Q; Z) is linear if and only if NfXh) = N(3)(h) = Q for some h e Q.

Proof. The proofs follows from Theorem 15 and Theorem 16, taking into account that a left and right linear invertible algebra is a linear invertible algebra.

Corollary 19 The class of all left invertible algebras is characterized by the following second-order formula

X(X(x, X 1 (u, y), z) = Y(Y(x,Y 1 (u, u)), Y 1 (u, X(y, z))), (18)

for all x, y, u, z e Q and all X,Y eZ.

Proof. By Theorem 15, an invertible algebra (Q;Z) is left linear if and only if it satisfies the following second order formula

X (X (x, y), z) = Y (Y ( x, eh ), L\hX (X (h, y), z))

for all x,y,u,z e Q and all X,Y eZ . We have: eh = Y l(u,u), L^ux = Y~l(u, x) and if in the last equality we take y = X~l(u, y), then we obtain

X(X(x,X \u,y)),z) = Y(Y(x,Y~\u,u)),L\uX(X(u,X \u,y)),z)),

or

X(X(x, X-1 (u, y), z) = Y (Y (x, Y(u, u)), Y(u, X(y, z))).

Corollary 20 The class of all right linear invertible algebras is characterized by the following second-order formula

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X(x, XC X(y, u), z)) = YCY(X(x,y), u), Y(~lY(u, u), z)), (19)

for all x, y, u, z e Q and all X,Y eZ.

Corollary 21 The class of all linear invertible algebras is characterized by the second-order formulas (18) and (19).

Corollary 22 A quasigroup (Q;-,/,\) is a left (right) linear quasigroup if and only if the following identity

(x(u \ y))z = (x(u \ u))(u \ yz) (x((y/u) z) = (xy/u)((u/u) z))

is valid in (Q;-,/,\) . Conclussion

In the present paper the right Nr(h) and left Nj(h) h -nuclei of the invertible algebra (Q;Z) are defined with respect to hyperidentity (3), and their connection with regular permutations of the invertible algebra is investigated. It is proved that Nj(h) = Q (Nr(h) = Q) iff the invertible algebra (Q;Z) is left linear (right linear) over a group. As corollaries we obtain the characterizations of the left (right) linear and the linear invertible algebras by second-order formula.

In [12] the right and left h -nuclei of the invertible algebra are defined with respect to hyperidentity (1).

References

1. R. Schauffler Die Assoziativitat im Ganzen Besonders bei Quasigruppen., Math. Zeitscher., 67(5)(1957), 428-435.

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2. W. A. Dudek, V. S. Trokhimenko Algebras of Multiplace Functions (de Gruyter, Versita, 2012).

3. G. B. Belyavskaya, Successively Orthogonal Systems of k-ary Operations, Quasigroups Realted Systems, 22(2014), 165-178.

4. A. Ehsani, A. Krapez, Yu. Movsisyan, Algebras with Parastrophically Uncancellable Quasigroup Equations, Bul. Acad. Stiinte Repub. Mold. Mat., 1(2016), 41-63.

5. V.D. Belousov Foundations of the Theory of Quasigroups and Loops (M., Nauka, 1967) (Russian).

6. P. T. Nagy, I. Stuhl, Rigth nuclei of quasigroups extensions., Comm. Algebr. 40(2012), 1893-1900.

7. G. B. Belyavskaya, Nuclei and Center of a Quasigroup, Research of Operations and Quasigroups, Math Issled., Kishinev, Shtiinta, 102(1988), 37-52 (Russian).

8. Yu. M. Movsisyan, Introduction to the Theory of Algebras with Hyperidentities (Yerevan State University press, 1986) (Russian).

9. S. S. Davidov, A characterization of binary invertible algebras of various types of linearity, Quasigroups Related Systems, 20(2012), 169-176.

10. V. D. Belousov, Globaly associative systems of quasigroups, Mat. Sb., 55(97)(1961), 221-236 (Russian)

11. S. S. Davidov, A characterization of binary invertible algebras linear over group, Quasigroups Realted Systems, 19(2011), 207-222.

12. S. S. Davidov, A characterization of invertible algebras linear over a group by second-order formulas, Journal of Algebra and Its Applications, 16(1 )(2017), to appear. DOI: 10.1142/S02019498817502383.

ОБРАТИМЫЕ ЛИНЕЙНЫЕ АЛГЕБРЫ И ИХ ЯДРА Давидов С.

В работе определяются левые (правые) ядра обратимой алгебры и изучается их связь с регулярными подстановками обратимой алгебры.

Ключевые слова: квазигруппа, обратимая алгебра, алгебра линейная над группой, левое ядро, правое ядро, формула второго порядка.

Дата поступления 10.12.2016.

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