UDC 519.123

MATHEMATICS

I. Sh. o. ALIEV

SYMMETRIC ALGEBRAS AND STEINER SYSTEMS

(Presented by Academician A. I. Mal'cev, 15 VII 1966)

In this paper certain combinatorial designs are considered, i.e., combinations of elements of a given set possessing specified properties, and an algebraic interpretation of these designs is given. Such designs are Steiner triples and quadruples—they correspond to \(TS\)-quasigroups \((^1)\) and \(S\)-algebras \((^2)\). On the set of finite \(S\)-algebras a binary operation is defined, distinct from the direct product.

Let a Steiner triple system \(M\) be given on a set \(A\), i.e., a system of triples \((x,y,z)\) of pairwise distinct elements of \(A\), such that any pair \(x,y \in A\) \((x \ne y)\) occurs in one and only one triple of \(M\). Adjoin to the system \(M\) the trivial triples \((x,x,x)\); then we obtain the system \(\overline{M}\). Define multiplication on \(A\) by putting \(xy=z \Longleftrightarrow (x,y,z)\in \overline{M}\). Then \(A\) becomes a \(TS\)-quasigroup \((^1)\), i.e., a quasigroup with the identities

\[ x^2=x,\qquad (xy)y=x,\qquad x(xy)=y. \]

Obviously the converse is also true: if \(A\) is a \(TS\)-quasigroup, then on the set \(A\) one can define a Steiner triple system by putting \((x,y,z)\in M \Longleftrightarrow xy=z\).

Adjoin to the \(TS\)-quasigroup \(A\) an element \(e\) and define on the set \(\bar A=A\cup\{e\}\) a new operation \((\cdot)\), putting, for arbitrary \(x,y\in A\) \((x\ne y)\),
\(x\cdot y=xy,\ x\cdot e=e\cdot x=x,\ x\cdot x=e,\ e\cdot e=e\). Then \(\bar A\) becomes a \(TS\)-loop with identity \(e\), i.e., a loop with the identities \(x^2=e,\ (xy)y=x,\ x(xy)=y\). The correspondence obtained between \(TS\)-quasigroups and \(TS\)-loops is, obviously, invertible; thus the following assertion is true.

Proposition 1. There exists a one-to-one correspondence between the sets: I—Steiner triple systems of order \(n\); II—\(TS\)-quasigroups of order \(n\); III—\(TS\)-loops of order \(n+1\).

Let a Steiner quadruple system \(N\) be given on a set \(B\), i.e., a system of quadruples \((x,y,z,t)\) of pairwise distinct elements of \(B\), such that any triple \(x,y,z\in A\) \((x\ne y\ne z\ne x)\) occurs in one and only one quadruple of \(N\). Adjoin to the system all possible quadruples of the form \((x,x,y,y)\); then we obtain the system \(\overline{N}\).

Define on \(B\) a ternary operation \(xyzt\), putting

\[ xyzt=t \Longleftrightarrow (x,y,z,t)\in \overline{N}. \]

With respect to this operation \(B\) is an \(S\)-algebra in the sense of \((^2)\), i.e., an algebra with one basic ternary operation \(xyzt\), satisfying the identities

\[ xxyt=y,\qquad xyzyzt=x,\qquad xxyztzt=y,\qquad xyxyzt\tau=z. \tag{1} \]

We note that the identities (1) may also be written in the form of conditional identities

\[ x=y\Rightarrow xyzt=z,\qquad xyzt=t\Rightarrow tyzt=x\ \&\ xtzt=y\ \&\ xytt=z \]

and imply the commutativity of the basic operation

\[ xyzt=xzyt=zxy\tau. \]

It is easy to see that to each \(S\)-algebra \(B\) there corresponds a certain Steiner system of quadruples on the set \(B\). Thus, the following assertion is true.

Proposition 2. There exists a one-to-one correspondence between the set of Steiner systems of quadruples of order \(n\) and the set of \(S\)-algebras of order \(n\).

The class of \(S\)-algebras is determined by the identities (1); consequently, this class is a variety and, in particular, is closed under direct (Cartesian) products. On this class we shall define another operation, which we shall call the product of \(S\)-algebras with an identified element (or the matrix product). The construction of this product will be set out for the case of finite \(S\)-algebras, but it is applicable without any changes also to infinite \(S\)-algebras.

Let \(A\) be an \(S\)-algebra of order \(t_1+1\), and \(B\) an \(S\)-algebra of order \(t_2+1\). Choose one element in each of \(A\) and \(B\) and identify them; denote this common element by \(\varepsilon\). Denote the remaining elements of \(A\) by \(a_i\), and the elements of \(B\) by \(b_i\) \((1 \le i \le t_1,\ 1 \le j \le t_2)\). Let
\[ C=\{\varepsilon\}\cup\{c_{ij},\ 1\le i\le t_1,\ 1\le j\le t_2\},\quad \bar C=\|c_{ij}\|. \]
On the set \(C\) we shall define a Steiner system of quadruples \(P\), with which the construction of an \(S\)-algebra on \(C\) is associated; the resulting \(S\)-algebra of order \(t_1t_2+1\) will be the desired one. The construction of the system \(P\) is divided into three stages.

I. \(t_1=t_2=3\).

a) The quadruples from \(P\) containing \(\varepsilon\) will be the quadruples of the form
\[ (\varepsilon,\ c_{i1},\ c_{i2},\ c_{i3}),\quad (\varepsilon,\ c_{1j},\ c_{2j},\ c_{3j})\quad (i,j=1,2,3), \]
as well as quadruples of the form
\[ (\varepsilon,\ c_{i_1j_1},\ c_{i_2j_2},\ c_{i_3j_3}), \]
where \((i_1,i_2,i_3)\), \((j_1,j_2,j_3)\) are permutations of the numbers \(1,2,3\); the number of such quadruples is 12.

b) Combine into one quadruple the elements of any minor of order 2 of the matrix \(\bar C\); then we obtain another 9 quadruples.

c) To each element \(c_{ij}\) assign the quadruple of elements located in the \(i\)-th row and in the \(j\)-th column of the matrix \(\bar C\) and distinct from \(c_{ij}\); this rule gives another 9 quadruples.

Thus, 30 quadruples are obtained, and they form the complete system \(P\); we shall denote this system below by \(P_3\).

Remark. Every Steiner system of quadruples of order 10 is isomorphic to \(P_3\).

II. \(t_1>3,\ t_2=t>3\).

a) Let \((j_1,j_2,j_3)\) be an arbitrary triple of numbers for which
\[ b_{j_1}b_{j_2}b_{j_3}\tau=\varepsilon. \]
Then consider the minor of order \((3\times3)\) situated in the columns of the matrix \(\bar C\) with numbers \(j_1,j_2,j_3\). Associate with this minor (and with the element \(\varepsilon\)) the system \(P_3\); then we obtain a certain set of quadruples, in particular all quadruples from \(P\) containing \(\varepsilon\).

b) Let \((j_1,j_2,j_3,j_4)\) be a quadruple of numbers for which
\[ b_{j_1}b_{j_2}b_{j_3}\tau=b_{j_4}. \]
Then consider the minor of order \((3\times4)\) situated in the columns of the matrix \(\bar C\) with numbers \(j_1,j_2,j_3,j_4\). Since the numbers \(j_k\) \((k=1,2,3,4)\) are distinct, we may, without loss of generality, regard them as ordered increasingly:
\[ j_1<j_2<j_3<j_4. \]
From the elements of the indicated minor we construct quadruples of the form
\[ (c_{i_1j_1},\ c_{i_2j_2},\ c_{i_3j_3},\ c_{i_4j_4}), \]
where
\[ (i_1,i_2,i_3,i_4)=(i,j,j,i),\ (i,j,i,j)\quad (i,j=1,2,3); \]
\[ (i,i,j,k),\ (i,j,k,k)=(j_1,j_2,j_3,j_4), \]
where \((i,j,k)\) is a permutation of the numbers \(1,2,3\).

Rules a), b) give the complete Steiner system of quadruples over \(C\); denote it by \(P_t\). Analogously, the Steiner quadruples are constructed in the case \(t_1=t>3,\ t_2=3\); we shall denote this system by \(P_t'\) \((P_3=P_3')\).

Remark. The construction of the system \(P_t(P_t')\) depends on the structure of the algebra \(B(A)\), and therefore one ought to write \(P_t(B)\), \(P_t'(A)\). We use the abbreviated notation, since from the context it is clear which algebras are meant.

III. \(t_1,t_3>3\).

a) Let \((i_1,i_2,i_3)\) be an arbitrary triple of numbers for which
\[ a_{i_1}a_{i_2}a_{i_3}\tau=\varepsilon. \]
Consider the minor of order \(3\times t_2\), situated in the rows

the matrix \(\bar C\) with numbers \(i_1, i_2, i_3\). Associate with this minor (and with the element \(\varepsilon\)) the Steiner system \(P_{t_2}\). Similarly, if \((j_1, j_2, j_3)\) is a triple of numbers for which \(b_{j_1} b_{j_2} b_{j_3} t = \varepsilon\), associate with the minor of the matrix \(\bar C\) situated in the columns with numbers \(j_1, j_2, j_3\) the system of quadruples \(P'_{t_1}\).

b) Let \((i_1, i_2, i_3, i_4)\), \((j_1, j_2, j_3, j_4)\) be quadruples of pairwise distinct numbers for which
\[ a_{i_1} a_{i_2} a_{i_3} \tau = a_{i_4}, \qquad b_{j_1} b_{j_2} b_{j_3} \tau = b_{j_4}. \]
Then put
\[ (c_{i_1 j_1}, c_{i_2 j_2}, c_{i_3 j_3}, c_{i_4 j_4}) \in P . \]

Rules a), b) exhaust the construction of the Steiner system of quadruples. In each of the cases I–III, denote by \(A \circ B\) the \(S\)-algebra corresponding to the constructed system of quadruples. We summarize the results of the construction described above in the following theorem.

Theorem. Let \(A_i\) \((i=1,2)\) be an \(S\)-algebra of order \(t_i + 1\), \(t_i \ge 3\). Then there exists an \(S\)-algebra of order \(t_1 t_2 + 1\), containing \(t_1\) subalgebras isomorphic to \(A_2\), \(t_2\) subalgebras isomorphic to \(A_1\), \(t_1(t_1 - 1)/6\) subalgebras isomorphic to \(P_{t_2}\), \(t_2(t_2 - 1)/6\) subalgebras isomorphic to \(P'_{t_1}\), and
\[ t_1 t_2 (t_1 - 1)(t_2 - 1)/36 \]
subalgebras of order \(10\).

To prove the theorem it is enough to consider the product \(A_1 \circ A_2\) of the \(S\)-algebras \(A_1, A_2\) with identified element. Note that the \(S\)-algebras \(A_1 \circ A_2\) and \(A_2 \circ A_1\) are isomorphic. However, the question of whether the product \(A_1 \circ A_2\) depends on the choice of the elements in \(A_1, A_2\) subject to identification remains open. In this connection, the question of whether the \(TS\)-loops obtained from a given \(S\)-algebra \(A\) by introducing the derived operation \(x \cdot y = xy\alpha\), as the element varies, will be isomorphic to one another is also of interest.

The author expresses his gratitude to E. N. Kuzmin for discussion of the results.

Novosibirsk State University

Received
11 VII 1966

REFERENCES

1. R. H. Bruck, Trans. Am. Math. Soc., 60, 245 (1946).
2. I. Sh. o. Aliev, Algebra i Logika, 4, no. 5, 5 (1965).