PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 81(95) (2007), 111–117 DOI 102298/PIM0795111G

CONSTRUCTIONS OF (2, n)-VARIETIES OF GROUPOIDS FOR n = 7, 8, 9

Lidija Goračinova-Ilieva and Smile Markovski

Dedicated to Prof. Dr. Kazimierz Glazek

Abstract. Given positive integer n > 2, an algebra is said to be a (2, n)- algebra if any of its subalgebras generated by two distinct elements has n elements. A variety is called a (2, n)-variety if every algebra in that variety is a (2, n)-algebra. There are known only (2, 3)-, (2, 4)- and (2, 5)-varieties of groupoids, and there is no (2, 6)-variety. We present here (2, n)-varieties of groupoids for n = 7, 8, 9.

1. Introduction

The notion of variety of algebras having the property (k, n) was given in [4] and equationally defined classes of cancellative groupoids having the property (2, 4) and (2, 5) were considered there. This notion was qeneralized in [1], where it was shown that the condition of the cancellativity is superfluous, that is, any variety of groupoids with the property (2, n) is a variety of quasigroups.

Let k and n be two positive integers and k ⩽ n. An algebra A is said to have the property (k, n) if every subalgebra of A generated by k distinct elements has exactly n elements. We also say that A is a (k, n)-algebra. A class K of algebras is said to be a (k, n)-class if every algebra in K is a (k, n)-algebra. A variety is called a (k, n)-variety if it is a (k, n)-class of algebras.

Trivially, the variety of Steiner quasigroups (xx = x, xy = yx, x·xy = y) is a (2, 3)-variety. It is the unique variety of groupoids with the stated property, and the same holds for the (2, 4)-variety (x·xy = yx, xy·yx = x) given by Padmanabhan in [4]. He has also constructed two (2, 5)-varieties. One of them is commutative (xy = yx, x(y·xy) = y, x(x·xy) = y·xy), while the other one (x·xy = y, xy·y = yx) consists of anticommutative quasigroups. These two varieties together with the variety whose defining identities (x·xy = yx, xy·y = x) are dual to the identities of the preceeding variety are the only (2, 5)-varieties of groupoids. The non-existence of a (2, 6)-variety can be deduced from the correspondence between the (k, n)- varieties and Steiner systems S(k, n, v) [1].

2000 Mathematics Subject Classification: 03C05, 20N05.

Key words and phrases: (2, n)-algebra,quasigroup, variety.

Here we present (2, n)-varieties of groupoids for n = 7, 8 and 9. Their construction is given in Sections 2, 3 and 4 respectively. It is an open problem the existence of (2, n)-varieties for n ⩾ 10, as well as the answer of the question whether the set of integers {n | There exists a (2, n)-variety of groupoids} is finite.

2. A construction of (2,7)-variety of groupoids

We use the fact that any member of a (2, n)-variety of groupoids is a quasigroup, i.e., the choosing of the defining identities of the (2, 7)-variety V_{7} (as well as the varieties V_{8} and V_{9} in the next sections) is made in a manner that enables a variety of quasigroups to be obtained.

Theorem 2.1. Let V_{7} be a variety of groupoids, defined by the identities:

(1) xy = yx, (2) x(x·xy) = y, (3) xy·(y·xy) = y(x·xy)

Then V_{7} is a (2, 7)-variety of quasigroups.

Proof. Let (G,·) be arbitrary groupoid in V_{7} and a, b ∈ G. Since ab = ac ⟹ b = a(a·ab) = a(a·ac) = c, x = a·ab is the unique solution of the equation ax = b. By commutativity ax = xa we have that (G,·) is a quasigroup.

Next we show that the following identities also hold in (G,·). (The commutativity will not be pointed out when used.)

(4)  xx = x,  (8)  (x·xy)(y·xy) = xy,
(5) x·x(y·xy) = y(x·xy),  (9) (x·xy)·x(y·xy) = y·xy,
(6) x·y(x·xy) = y·xy,  (10) (x·xy)·y(x·xy) = x,
(7) xy·x(y·xy) = x,  (11) x(y·xy)·y(x·xy) = xy.

Namely, we have the following transformations:

xx =^{(2)} x(xx·(xx·(xx·x))) =^{(3)} x(xx·x(x·xx)) =^{(2)} x(xx·x) =^{(2)} x;
x·x(y·xy) =^{(3)} x(xy·(x·xy)) =^{(3)} (x·xy)·x(x·xy) =^{(2)} (x·xy)y;
x·y(x·xy) =^{(5)} x(x·x(y·xy)) =^{(2)} y·xy;
xy·x(y·xy) =^{(3)} xy·(xy·(x·xy)) =^{(2)} x;
(x·xy)(y·xy) =^{(2)} (x·xy)(x(x·xy)·xy) =^{(7)} xy;
(x·xy)·x(y·xy) =^{(3)} (x·xy)(xy·(x·xy)) =^{(3)} xy·x(x·xy) =^{(2)} xy·y;
(x·xy)·y(x·xy) =^{(5)} (x·xy)·x(x(y·xy)) =^{(3)} (x·xy)·x(xy·(x·xy)) =^{(7)} x;
x(y·xy)·y(x·xy) =^{(3)} (xy·(x·xy))(xy·(y·xy)) =^{(9)}

=^{(9)} (xy·(x·xy))(xy·((x·xy)·x(y·xy))) =^{(3)}

=^{(3)} (xy·(x·xy))(xy·((x·xy)(xy·(x·xy)))) =^{(7)} xy.

CONSTRUCTIONS OF (2, n)-VARIETIES OF GROUPOIDS FOR n = 7, 8, 9 113

Therefore, the multiplication table of any subquasigroup of a quasigroup in V_{7}, generated by the elements x and y (x ≠ y), is the following one:

x  y  xy  x·xy  y·xy x(y·xy) y(x·xy)
x  x  xy  x·xy  y  x(y·xy) y(x·xy) y·xy
y  xy  y  y·xy y(x·xy)  x  x·xy x(y·xy)
xy  x·xy  y·xy  xy  x(y·xy) y(x·xy)  x  y
x·xy  y  y(x·xy) x(y·xy) x·xy  xy  y·xy  x
y·xy x(y·xy)  x  y(x·xy)  xy  y·xy  y  x·xy
x(y·xy) y(x·xy) x·xy  x  y·xy  y  x(y·xy)  xy
y(x·xy) y·xy x(y·xy)  y  x  x·xy  xy  y(x·xy)

In order to complete the proof, it suffices to show that the elements x, y, xy, x·xy, y·xy, x(y·xy), y(x·xy) are distinct:

x = xy ⟹ xx = xy ⟹ x = y
x = x·xy ⟹ xx = x·xy ⟹ x = xy
x = y·xy ⟹ xy = (y·xy)y ⟹ xy = x
x = x(y·xy) ⟹ xx = x(y·xy) ⟹ x = y·xy
x = y(x·xy) ⟹ y(y·xy) = y(x·xy) ⟹ y·xy = x·xy ⟹ x = y
xy = x·xy ⟹ y = xy
xy = x(y·xy) ⟹ y = y·xy
x·xy = y·xy ⟹ x = y
x·xy = x(y·xy) ⟹ xy = y·xy
x·xy = y(x·xy) ⟹ x(x·xy) = x(y(x·xy)) ⟹ y = y·xy
x(y·xy) = y(x·xy) ⟹ x(y·xy) = xy·(y·xy) ⟹ x = xy.  □

3. A construction of (2,8)-variety of groupoids

Theorem 3.1. Let V_{8} be the variety of groupoids, defined by the identities:

(1) x·xy = xy·y, (2) x·yx = xy·x, (3) x(y·yx) = y.

Then V_{8} is a (2, 8)-variety of quasigroups.

Proof. First we show that the following identities are satisfied by any V_{8}- groupoid:

(4) x(x·xy) = yx,  (16)  (x·xy)·xy = yx,
(5)  xx = x,  (17)  xy·(yx·y) = yx,
(6)  xy·yx = x,  (18)  (y·yx)·xy = x,
(7) (xy·x)x = y,  (19)  (xy·x)·xy = y·yx,
(8) x(xy·x) = yx·y,  (20)  (yx·y)·xy = xy·x,

(9) x(yx·y) = y·yx,  (21) (x·xy)(y·yx) = xy·x,
(10) (x·xy)x = yx·y,  (22) (x·xy)(xy·x) = x,
(11) (y·yx)x = x·xy, (23) (x·xy)(yx·y) = xy,
(12) (yx·y)x = xy,  (24) (y·yx)(x·xy) = yx·y,
(13) xy·(x·xy) = y·yx,  (25) (yx·y)(x·xy) = y,
(14) xy·(y·yx) = yx·y,  (26) (xy·x)(yx·y) = x·xy,
(15) xy·(xy·x) = y.

Namely, we have:

x(x·xy) =^{(3)} y(x·xy)·(x·xy) =^{(1)} y·y(x·xy) =^{(3)} yx;
xx =^{(4)} x(x·xx) =^{(3)} x;
xy·yx =^{(4)} xy·(x·(x·xy)) =^{(3)} x;
(xy·x)x =^{(2)} (x·yx)x =^{(2)} x(yx·x) =^{(1)} x(y·yx) =^{(3)} y;
x(xy·x) =^{(2)} x(x·yx) =^{(1)} (x·yx)·yx =^{(4)} yx·(yx·(yx·(x·yx)))

=^{(2)} yx·(yx·((yx·x)·yx)) =^{(1)} yx·(yx·((y·yx)·yx))

=^{(2)} yx·((yx·(y·yx))·yx) =^{(2)} yx·(((yx·y)·yx)·yx) =^{(7)} yx·y;

x(yx·y) =^{(8)} x·x(xy·x) =^{(2)} x·x(x·yx) =^{(4)} yx·x =^{(1)} y·yx;
(x·xy)x =^{(2)} x(xy·x) =^{(8)} yx·y;
(y·yx)x =^{(4)} x(x·x(y·yx)) =^{(3)} x·xy;
(yx·y)x =^{(4)} x(x·x(yx·y)) =^{(9)} x·x(y·yx) =^{(3)} xy;
xy·(x·xy) =^{(1)} xy·(xy·y) =^{(1)} (xy·y)y =^{(1)} (x·xy)y =^{(11)} y·yx;
xy·(y·yx) =^{(4)} (y(y·yx))(y·yx) =^{(1)} y·y(y·yx) =^{(4)} yx·y;
xy·(xy·x) =^{(1)} (xy·x)x =^{(7)} y;
(x·xy)·xy =^{(1)} x(x·xy) =^{(4)} yx;
xy·(yx·y) =^{(2)} xy·(y·xy) =^{(2)} (xy·y)·xy =^{(1)} (x·xy)·xy =^{(16)} yx;
(y·yx)·xy =^{(13)} (xy·(x·xy))·xy =^{(2)} ((xy·x)·xy)·xy =^{(7)} x;
(xy·x)·xy =^{(2)} xy·(x·xy) =^{(13)} y·yx;
(yx·y)·xy =^{(14)} (xy·(y·yx))·xy =^{(2)} xy·((y·yx)·xy) =^{(18)} xy·x;
(x·xy)(y·yx) =^{(11)} ((y·yx)x)(y·yx) =^{(2)} (y·yx)·x(y·yx) =^{(3)} (y·yx)y =^{(10)} xy·x;

(x·xy)(xy·x) =^{(6)} x;
(x·xy)(yx·y) =^{(19)} ((yx·y)·yx)(yx·y) =^{(19)} y(y·yx) =^{(4)} xy;
(y·yx)(x·xy) =^{(13)} (y·yx)(yx·(y·yx)) =^{(2)} (y·yx)((yx·y)·yx) =^{(17)} yx·y;
(yx·y)(x·xy) =^{(10)} ((x·xy)x)(x·xy) =^{(19)} xy·(xy·x) =^{(15)} y;
(xy·x)(yx·y) =^{(8)} (xy·x)(x(xy·x)) =^{(2)} (xy·x)((x·xy)x) =^{(17)} x·xy.

In each groupoid in V_{8}, the equations ax = b and ya = b have solutions x = b·ba and y = ab·a. Moreover, the cancellation laws hold:

ac = ad ⟹ c = (ac·a)a = (ad·a)a = d,
ca = da ⟹ c = a(c·ca) = a(ca·a) = a(da·a) = a(d·da) = d.

Hence, V_{8} is a variety of quasigroups.

The multiplication table of the subquasigroup of a quasigroup in V_{8}, generated by the set {x, y}, is given as follows:

x  y  xy yx x·xy y·yx xy·x yx·y
x  x  xy x·xy xy·x yx  y yx·y y·yx
y  yx  y yx·y y·yx x  xy x·xy xy·x
xy xy·x x·xy xy  x y·yx yx·y y  yx
yx y·yx yx·y y  yx xy·x x·xy xy  x
x·xy yx·y y·yx yx  y x·xy xy·x x  xy
y·yx x·xy xy·x x  xy yx·y y·yx yx  y
xy·x y  yx y·yx yx·y xy  x xy·x x·xy
yx·y xy  x xy·x x·xy y  yx y·yx yx·y

All of the elements in the multiplication table are distinct:

x = xy ⟹ xx = xy ⟹ x = y;
x = x·xy ⟹ xx = x·xy ⟹ x = xy;
x = y·yx ⟹ xx = x(y·yx) ⟹ x = y;
x = xy·x ⟹ xx = xy·x ⟹ x = xy;
x = yx·y ⟹ xy = (yx·y)y ⟹ xy = x;
xy = yx ⟹ xy·yx = yx·yx ⟹ x = yx;
xy = x·xy ⟹ y = xy;
xy = y·yx ⟹ x·xy = x(y·yx) ⟹ x·xy = y;
xy = xy·x ⟹ xy·xy = xy·x ⟹ xy = x;
xy = yx·y ⟹ xy·y = (yx·y)y ⟹ x·xy = x;
x·xy = y·yx ⟹ x(x·xy) = x(y·yx) ⟹ yx = y;
x·xy = xy·x ⟹ x·xy = x·yx ⟹ xy = yx;
x·xy = yx·y ⟹ x·xy = y·xy ⟹ x = y;
xy·x = yx·y ⟹ (xy·x)x = (yx·y)x ⟹ y = xy.  □

4. A construction of (2,9)-variety of groupoids

Theorem 4.1. Let V_{9} be the variety of groupoids defined by the identities

(1) x·xy = yx, (2) xy·(y·xy) = x.

Then V_{9} is a (2, 9)-variety of quasigroups.

Proof. One can check that the identities (3)–(30), given below, are satisfied by any groupoid in V_{9}. We emphasis the identities that can be applied to the lefthand side of each equality in order to obtain its right-hand side.

Identity Left-hand side = Right-hand side Applyed identities
(3)  xy·x = x·yx  (1), (1)
(4)  xx = x  (1), (1), (3), (2)
(5)  (x·yx)·xy = xy·yx  (3), (3), (1)
(6)  xy·(x·yx) = yx  (3), (1), (1)
(7)  (xy·y)·xy = x  (3), (2)
(8)  (x·yx)·yx = y·xy  (1), (2), (3)
(9)  (x·yx)y = x  (2), (3), (7)
(10)  yx·x = xy·y  (1), (2), (3), (2)
(11)  (x·yx)x = yx·xy  (3), (10), (1)
(12)  x(xy·y) = yx·xy  (10), (3), (11)
(13)  xy·(yx·xy) = y  (1), (1), (2)
(14)  (xy·yx)·yx = xy·y  (1), (13), (10)
(15)  (xy·y)x = y·xy  (10), (10), (8)
(16)  (xy·y)(x·yx) = xy·yx  (15), (10), (1), (12)
(17)  x(yx·xy) = y·xy  (12), (1), (15)
(18)  x(y·xy) = xy·yx  (2), (10), (8), (5)
(19)  x(xy·yx) = y  (18), (1), (9)
(20)  (xy·yx)x = xy  (1), (19)
(21)  (x·yx)(xy·yx) = yx  (5), (1), (6)
(22)  (xy·yx)(x·yx) = y·xy  (1), (21), (8)
(23)  (xy·yx)y = yx·xy  (1), (17), (18)
(24) (xy·yx)(yx·xy) = x·yx  (23), (1), (17)
(25)  (x·yx)(xy·y) = xy  (10), (1), (3), (2)
(26)  (xy·yx)(y·xy) = yx  (11), (3), (2)
(27)  (x·yx)(yx·xy) = xy·y  (1), (26), (14), (10)
(28)  (xy·y)(xy·yx) = xy  (16), (1), (25)
(29)  (xy·yx)(xy·y) = x  (1), (28), (7)
(30)  (x·yx)(y·xy) = y  (8), (1), (3), (7)

Next, we show that every groupoid in V_{9} is a quasigroup. The equations ax = b and ya = b have solutions x = ab·ba and y = b·ab

respectively, and they are unique. Namely, if ac = b and da = b, we have that c = ca·(a·ca) = (a·ac)(ac·a) = ab·ba and d = da·(a·da) = b·ab.

By the above identities, we have that a subquasigroup generated by two distinct elements x and y is represented by the following table, and one can check that all of the elements are distinct.

x  y  xy  yx  x·yx y·xy xy·yx yx·xy xy·y
x  x  xy  yx  x·yx xy·y xy·yx  y  y·xy yx·xy
y  yx  y  y·xy  xy yx·xy xy·y x·yx  x  xy·yx
xy  x·yx xy·y  xy xy·yx yx  x  yx·xy  y  y·xy
yx  xy·y y·xy yx·xy yx  y  xy  x  xy·yx x·yx
x·yx yx·xy  x  xy·yx y·xy x·yx  y  yx  xy·y  xy
y·xy  y  xy·yx x·yx yx·xy  x  y·xy xy·y  xy  yx
xy·yx xy yx·xy  y  xy·y y·xy  yx xy·yx x·yx  x
yx·xy xy·yx yx  xy·y  x  xy  x·yx y·xy yx·xy  y
xy·y y·xy x·yx  x  y  xy·yx yx·xy xy  yx  xy·y
□

References

[1] L. Goračinova, S. Markovski, (2, n)-Quasigroups (preprint)

[2] J. Denes, A. D. Keedwell, Latin Squares and Their Applications, English Universities Press, London, 1974

[3] R. N. McKenzie, W. F. Taylor, G. F. McNulty, Algebras, Lattices, Varieties, Wadsworth and Brooks, Monterey, California, 1987

[4] R. Padmanabhan, Characterization of a class of groupoids, Algebra Universalis 1 (1972), 374–382.

Pedagogical faculty University “Goce Delčev” Štip Republic of Macedonia

lidija.goracinova@ugd.edu.mk

Institute of Informatics “Ss Cyril and Methodius” University, Faculty of Sciences Skopje Republic of Macedonia

smile@ii.edu.mk