Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 2, 325-331.

On Subdirectly Irreducible Steiner Loops of Cardinality 2n

Magdi H. Armanious

El-Metawakel Ala Alah St. 3 El-Zeiton, P. C. 11321, Cairo, Egypt e-mail: m.armanious@excite.com

Abstract. Let L_{1} be a finite simple sloop of cardinality n or the 8-element sloop. In this paper, we construct a subdirectly irreducible (monolithic) sloop L = 2⊗_{α}L_{1} of cardinality 2n, for each n ≥ 8, with n ≡ 2 or 4 (mod 6), in which each proper homomorphic image is a Boolean sloop. Quackenbush [12] has proved that the variety V (L_{1}) generated by a finite simple planar sloop L_{1} covers the smallest nontrivial subvariety (the class of all Boolean sloops). For any finite planar sloop L_{1}, the variety V (L) generated by the constructed sloop L = 2⊗_{α} L_{1} covers the variety V (L_{1}). MSC 2000: 05B07 (primary); 20N05 (secondary)

1. Introduction

A Steiner loop (or sloop) is a groupoid S = (S, · , 1) with neutral element 1 satisfying the identities:

x · x = 1, x · y = y · x, x · (x · y) = y.

We use the abbreviation SL(n) for a sloop of cardinality n. If a sloop satisfies the associative law (x · y) · z = x · (y · z), then it will be a Boolean group that is also called a Boolean sloop. An extensive study of sloops can be found in [4], [8] and [12].

A Steiner triple system is a pair (P; B), where P is a set of points and B is a set of 3-element subsets of P called blocks such that for distinct points p_{1}, p_{2} ∈ P, there is a unique block b ∈ B with {p_{1}, p_{2}} ⊆ b.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

326 M. H. Armanious: On Subdirectly Irreducible Steiner Loops of Cardinality 2n

There is a one to one correspondence between the sloops and the Steiner triple systems [8], [12]. If the cardinality of the set of points P is equal to n, the Steiner triple system (P; B) will be denoted by STS(n). It is well known that a necessary and sufficient condition for the existence of an STS(n) is n ≡ 1 or 3 (mod 6 ).

Quackenbush [12] proved that the congruences of sloops are permutable, regular, and Lagrangian. A subsloop S of a sloop L is called normal iff (x · y) · S = x · (y · S) for all x, y ∈ L.

We have that the lattice of normal subsloops of a sloop L is isomorphic to the lattice of the congruence relation of L. Quackenbush [12] has also proved that if S is a subsloop of L and |L| = 2|S|, then S is normal.

There is a well known method for turning a Steiner triple system into another algebra called a Steiner quasigroup (or squag) [12].

In the comments and problems section of [12], Quackenbush has stated that there should be non-simple subdirectly irreducible sloops in which any proper homomorphic image must be a Boolean sloop. He stated that there should be non-simple subdirectly irreducible squags in which any proper homomorphic image must be a medial squag.

The author in [3] has given a construction of finite subdirectly irreducible squags in which all proper homomorphic images are medial squags.

In [1] and [2] the author has also given a construction of a subdirectly irreducible (monolithic) sloop of cardinality 2^{n}, in which the cardinality of the congruence class of the unique atom of its congruence lattice is equal to 2 (the minimal possible size of a proper normal subsloop).

In this paper, we construct a subdirectly irreducible sloop of cardinality 2n, for each n ≥ 8, with n ≡ 2 or 4 (mod 6), in which its congruence lattice is a chain of length 2 and its proper homomorphic image is the 2-element Boolean group. Moreover, the cardinality of the congruence class of its unique atom is equal to n (the maximal possible size of a proper normal subsloop).

We will use in this article some basic concepts of universal algebra [9] and other concepts of graph theory [10].

2. Construction of 2 ⊗_{α} L_{1}

Let (P_{1}^{∗}; B_{1}) be an STS(n − 1) and its corresponding sloop L_{1} = (P_{1}; · , 1), where P_{1}^{∗} = {a_{0}, a_{1}, …, a_{n−2}} and P_{1} = P_{1}^{∗} ∪ {1}. Consider the set of 1-factors on P_{1} defined by F_{i} = {a_{i}a_{k}: a_{i} · a_{k} = a_{i}}, then the class F = {F_{0}, F_{1}, …, F_{n−2}} forms a 1-factorization of the complete graph K_{n} on the set of vertices P_{1}.

By taking the set P_{2} = {b, b_{0}, b_{1}, …, b_{n−2}} with P_{1} ∩ P_{2} = ∅ and G_{i} = {bb_{i}} ∪ {b_{l}b_{k}: a_{l} · a_{k} = a_{i} for i ∉ {l, k}}, then the class of 1-factors G = {G_{0}, G_{1}, …, G_{n−2}} forms a 1-factorization of the complete graph K_{n} on the set of vertices P_{2}. There is a well known construction of an STS(2n − 1) = (P^{∗}; B) [11], where P^{∗} = P_{1}^{∗} ∪ P_{2} and the set of triples B = B_{1} ∪ {{b_{l}, b_{k}, a_{i}}: b_{l}b_{k} ∈ G_{α(i)}} for any permutation α on the set {0, 1, …, n − 2}.

The corresponding sloop SL(2n) of the STS(2n − 1) = (P^{∗}; B) will be denoted by L = 2 ⊗_{α} L_{1} = (P; · , 1) where P = P_{1} ∪ P_{2} and P^{∗} = P − {1}.

If we choose the permutation α equal to the identity map on the set {0, 1, …, n − 2}, then the constructed sloop L = 2⊗_{α}L_{1} is isomorphic to the direct product of SL(n) = L_{1} and the 2-element sloop SL(2). We observe that L_{1} is a subsloop of 2 ⊗_{α} L_{1} for any permutation α.

In the following section, we choose a simple sloop L_{1} of cardinality n and a suitable permutation α to construct a subdirectly irreducible sloop of cardinality 2n.

2.1. Subdirectly irreducible sloops SL(2n)

An STS is planar if it is generated by every triangle and contains a triangle. A planar STS(n) exits for each n ≥ 7 and n ≡ 1 or 3 (mod 6) [6]. Quackenbush [12] proved in the next theorem that almost all planar SL(n)’s are simple.

Theorem 1. [12] Let (P^{∗}; B) be a planar STS(n−1) and (P; · , 1) be its corresponding sloop, then either (P; · , 1) is simple or n = 8.

Accordingly, we may say that for any n > 8 with n ≡ 2 or 4 (mod 6) there is a simple SL(n).

Lemma 2. Let F be a 1-factorization of the complete graph K_{n}. For any two distinct 1factors F_{1} and F_{2} of F, there is always a 1-factor F_{3} of F satisfying that the three factors F_{1}, F_{2}, and F_{3} do not contain any sub 1-factorization of the complete graph K_{4}.

Proof. The number of edges of a 1-factor F_{i} of F is n/2. Then the maximum number of sub 1-factorizations on K_{4} of F with sub 1-factors f_{1} ⊆ F_{1} and f_{2} ⊆ F_{2} on a 4-element subset of vertices is [n/4] (the greatest integer in n/4).

For any possible 4-element subset of vertices {x, y, z, w}, if f_{1} = {xy, zw} and f_{2} = {xz, yw}, then there is at most only one 1-factor F_{i} of F containing the third sub 1-factor f_{i} = {xw, yz} of the sub 1-factorization of K_{4}.

If the 1-factorization F contains the maximum number of sub 1-factorizations of K_{4} with sub 1-factors f_{1} ⊆ F_{1} and f_{2} ⊆ F_{2} on a 4-element subset of vertices, then there are at most [n/4] distinct 1-factors of F – {F_{1}, F_{2}}, each containing the third sub 1-factor f_{i} of a sub 1-factorization of K_{4}.

Since |F| = n − 1 and n − 3 > [\frac{1}{4}n] for n > 4, then we may say that for n > 4 there is always at least one 1-factor F_{j} not containing the third sub 1-factor {xw, yz} for all possible 4-element subset of vertices {x, y, z, w}. This completes the proof of the lemma. □

Now, we are ready to construct a sloop L of cardinality 2n having only one proper congruence relation ϕ, in which its homomorphic image L/ϕ is the 2-element Boolean group.

Theorem 3. Let L_{1} be a simple sloop of cardinality n > 8, then there is a permutation α on the set {0, 1, …, n − 2} such that the construction 2 ⊗_{α} L_{1} will be a subdirectly irreducible sloop of cardinality 2n, in which each proper homomorphic image is Boolean.

Proof. Without loss of generality, we may assume that a_{0} · a_{1} = a_{2} in L_{1}; then

1a_{0}, a_{1}a_{2} ∈ F_{0}, 1a_{1}, a_{0}a_{2} ∈ F_{1} and 1a_{2}, a_{0}a_{1} ∈ F_{2}.

And according to the definition of the 1-factorization G in the construction 2 ⊗_{α} L_{1}, there is a sub 1-factorization of G on K_{4} namely:

bb_{0}, b_{1}b_{2} ∈ G_{0}, bb_{1}, b_{0}b_{2} ∈ G_{1} and bb_{2}, b_{0}b_{1} ∈ G_{2}.

By Lemma 2, we may also say that there is a 1-factor G_{i}; i ≠ 0, 1, 2 such that the three 1-factors G_{0}, G_{1}, G_{i} do not contain any sub 1-factorization of the complete graph K_{4}.

By choosing the permutation α = (2 i) on the set {0, 1, …, n − 2}, we will prove that the constructed sloop L = 2 ⊗_{α} L_{1} is a subdirectly irreducible sloop, in which each proper homomorphic image is Boolean.

Since L_{1} is simple and |L| = 2|L_{1}|, then for any permutation α the constructed sloop L = 2⊗_{α} L_{1} contains the subsloop L_{1} as the unique subsloop of cardinality n; L_{1} is necessary normal.

If there is an isomorphism f between L and the direct product of L_{1} and SL(2): = ({0, 1}; + , 0), then it must be f(P_{1}) = P_{1} × {0} and f(P_{2}) = P_{2} × {1}. But any 4-element subsloop {1, a_{i}, a_{j}, a_{k}}×{0} of P_{1}×{0} lies in an 8-element subsloop Y = {1, a_{i}, a_{j}, a_{k}}×{0, 1} of the direct product L_{1} × {0, 1}. Since the image f^{−1}(Y) is a subsloop of L, hence we may say that if L ≅ L_{1} × {0, 1}, then any 4-element subsloop U = {1, a_{i}, a_{j}, a_{k}} of P_{1} lies in an 8-element subsloop X = {1, a_{i}, a_{j}, a_{k}, b′, b_{i′}, b_{j′}, b_{k′}} of L with X − U ⊆ P_{2}.

Accordingly, to prove that L is not isomorphic to the direct product of L_{1} and the 2element sloop, it is enough to show that there is no subsloop X of L of cardinality 8 containing the subsloop U = {1, a_{0}, a_{1}, a_{2}} with X − U ⊆ P_{2}.

Assume there is a subsloop X of cardinality 8 containing U = {1, a_{0}, a_{1}, a_{2}} with X −U = {b_{i}, b_{j}, b_{l}, b_{k}} ⊆ P_{2}. We have α(0) = 0, so {a_{0}, b_{l}, b_{k}} is a block of B iff b_{l}b_{k} ∈ G_{0}. Hence, there is a sub-1-factor {b_{i}b_{j}, b_{l}b_{k}} of G_{0} related with a_{0}. And we have α(1) = 1, so there is also a sub-1-factor, say {b_{i}b_{l}, b_{j}b_{k}} of G_{1} , related with a_{1}. If the set X = {1, a_{0}, a_{1}, a_{2}, b_{i}, b_{j}, b_{l}, b_{k}} forms a subsloop, then the operation ”·” is associative on X. Hence a_{2} = a_{0} · a_{1} = (b_{i} · b_{j}) · (b_{i} · b_{l}) = b_{j} · b_{l} and so a_{2} = a_{0} · a_{1} = (b_{l} · b_{k}) · (b_{i} · b_{l}) = b_{i} · b_{k}. Moreover, we have α(2) = i ; this means that a_{2} is related with the 1-factor G_{i} (i.e. {a_{2}, b_{l}, b_{k}} is a block of B iff b_{l}b_{k} ∈ G_{i}). This implies that G_{i} contains the sub 1-factor = {b_{j}b_{l}, b_{i}b_{k}}, contradicting the assumption that G_{0}, G_{1}, G_{i} does not contain any sub 1-factorization of K_{4}.

Accordingly, the constructed sloop L = 2 ⊗_{α} L_{1} is not isomorphic to the direct product of L_{1} and the 2-element sloop SL(2).

Next, we will show that the constructed sloop L = 2⊗_{α}L_{1} has only one proper congruence ϕ satisfying L/ϕ is Boolean.

Since the constructed sloop L = 2 ⊗_{α} L_{1} has L_{1} as a subsloop, then L_{1} is a normal subsloop of L. This means that there is a congruence relation ϕ of L defined by [1]ϕ = L_{1}.

Let θ be a non-trivial congruence relation of L and θ ≠ ϕ; then [1]θ ∩L_{1} = {1}, otherwise θ_{L_{1}} (θ restricted on L_{1}) is a non-trivial congruence on L_{1}, contradicting the assumption that L_{1} is simple. Consequently, |[1]θ| = 2; then we may assume that [1]θ = {1, b_{i}} for b_{i} ∈ P_{2}. This means that θ ∩ ϕ = Δ (the diagonal relation) and θ ∘ ϕ = ∇ (the largest congruence). This implies that L is isomorphic to the direct product of L_{1} and SL(2), contradicting the preceding result that L = 2 ⊗_{α} L_{1} ≅ L̸_{1} × SL(2).

This means that the constructed sloop L has no non-trivial congruence θ with θ ≠ ϕ. This completes the proof of the theorem. □

In fact, a sloop SL(m) is simple if m can not be factored into (6n_{1} +i)(6n_{2} +j) for some n_{1}, n_{2} and some i, j ∈ {2, 4}. In particular, there are only simple SL(m)’s, if m is not divisible by 4.

If m ≡ 2 or 4 (mod 6), then m = 2n. Moreover, if n ≢ 2 or 4 (mod 6), then there are only simple sloops of cardinality m.

Let m ≡ 2 or 4 (mod 6) and m = 2n with n > 8. For n ≡ 2 or 4 (mod 6), so by Theorem 1, there is a simple sloop L_{1} of cardinality n. And by Theorem 3, there is a subdirectly irreducible sloop L = 2 ⊗_{α} L_{1} of cardinality m with only one proper congruence ϕ satisfying that L/ϕ is isomorphic to the 2-element Boolean sloop.

For m ⩽ 16, there are only Boolean sloops SL(m) for m = 2, 4, 8 and only simple sloops SL(m) for m = 10, 14. To complete the result of Theorem 3 we will show that there is a subdirectly irreducible L =SL(16) having only one proper congruence ϕ with |L/ϕ| = 2 as follows:

In the catalogue of all 80 STS(15)’s, see [5], choose one which has exactly one 7-element subsystem. The corresponding sloop has exactly one 8-element subsloop. It has no other proper non-trivial normal subsloops, since otherwise one could construct a second 7-element subsystem. Finally, we may say that:

For each n > 4 with n≡ 2 or 4 (mod 6), there is a subdirectly irreducible (monolithic) sloop L of cardinality 2n having only one proper non-trivial congruence relation ϕ; furthermore L/ϕ is the 2-element Boolean sloop.

Quackenbush [12] has proved that the variety V (L_{1}) generated by a simple planar sloop L_{1} has only two subdirectly irreducible sloops L_{1} and the 2-element sloop SL(2) and then V (L_{1}) covers the smallest nontrivial subvariety (the class of all Boolean sloops). Quackenbush [12] has also showed that the variety V (L_{1}, L_{2}, …, L_{m}) generated by pairwise non-isomorphic finite simple planar sloops L_{1}, L_{2}, …, L_{m} equal to P_{s}(L_{1}, L_{2}, …, L_{m}, SL(2)). This implies that the variety V (L) generated by the constructed sloop SL(2n) = L, is not subvariety of the variety V (L_{1}, L_{2}, …, L_{m}) generated by any set of pairwise non-isomorphic finite simple planar sloops L_{1}, L_{2}, …, L_{m}.

On the other hand, let L_{1} be a planar sloop of cardinality n and L = 2 ⊗_{α} L_{1} be the constructed sloop SL(2n). For any subsloop S other than L_{1} of L with |S| > 4, one can easily prove that |L_{1} ∩ S| = \frac{1}{2}|S|. Accordingly, if L_{1} is planar, then the class of all proper subsloops of L = 2 ⊗_{α} L_{1} are exactly L_{1} and SL(n) for n = 8, 4, 2. This means that the variety V (L) generated by L properly contains the subvariety V (L_{1}).

Finally, we want to sharpen the previous result by proving that the variety V (L) covers the variety V (L_{1}). For it we need some concepts and theorems on the congruence modular varieties due to R. Freese and R. McKenzie [7] and H. Werner [13].

In [12] Quackenbush has proved that any finite simple sloop L_{1} with |L_{1}| > 2 is functionally complete; i.e. L_{1}^{n} has no skew congruence for any n ≥ 2. He also proved that L_{1}^{n} × C_{2}^{m} has no skew congruence for any positive integers n and m, where C_{2} denotes to the 2-element sloop SL(2).

Theorem 4. [13] Let K be a permutable variety with A_{1}, A_{2}, …, A_{n} ∈ K. Then A_{1} × A_{2} × ⋯ × A_{n} has a skew congruence iff for some i ≠ j; A_{i} × A_{j} has a skew congruence.

Accordingly, we may say that L^{r} × L_{1}^{n} × C_{2}^{m} has a skew congruence iff L × L_{1}, L^{2}, L × C_{2} or C_{2}^{2} has a skew congruence.

We want to prove that if L^{2}, L×C_{2} or C_{2}^{2} has a skew congruence ϕ, then the homomorphic image of any of them by ϕ is isomorphic to C_{2}.

Theorem 5. [13] Let K be a permutable variety with A, B ∈ K. Then A × B has a skew congruence iff there are homomorphic images A′ of A and B′ of B and a 1-1 map μ with dom(μ) ⊆ A′, |dom(μ)| > 1, range(μ) ⊆ B′ such that {(a, μ(a)): a ∈ dom(μ)} is a congruence class on A′ × B′.

Theorem 6. [13] Let K be a permutable variety with A, B ∈ K. If {(a_{i}, b_{i}): i ∈ I} is a congruence class of A × B, then {a_{i}: i ∈ I} is a congruence class of A (i.e. the projection of a congruence class is a congruence class).

Theorem 7. Let L_{1} be a finite simple sloop and L = 2 ⊗_{α} L_{1} be the constructed sloop. Then L × L_{1} has no skew congruence and each of L^{2}, L × C_{2} and C_{2}^{2} has only one skew congruence ψ in which the homomorphic image of any of them by ψ is isomorphic to C_{2}.

Proof. Let A and B be two finite non-trivial sloops. By applying the previous two theorems then we may say that: A×B has a skew congruence iff there are A′ ∈ H(A), B′ ∈ H(B) and a 1-1 map μ: A′_{1} → B′ for a subsloop A′_{1} of A′ with | A′_{1} |> 1 such that (A′_{1}, μ(A′_{1})): = {(a, μ(a)): a ∈ A′_{1} = dom(μ)} is a congruence class of A′×B′. We observe that A′_{1} and μ(A′_{1}) are also congruence classes of A′ and B′ respectively. Moreover, (A′_{1}, μ(A′_{1})) is a non-trivial congruence class of A′_{1} × B′.

This means that if A = L and B = L_{1}, since L_{1} is simple, then μ(A′_{1}) must be equal to L_{1}. Thus A′ = L and A′_{1} must be equal to L_{1}. Hence, if there is a skew congruence on L × L_{1}, then A′_{1} × B′ = L_{1} × L_{1} has also a skew congruence, contradicting the fact that L_{1} is functional complete. Therefore, L × L_{1} has no a skew congruence.

Let A = L and B = L, so if A′ = L and A′_{1} = L_{1}, then μ(A′_{1}) = L_{1}. In this case, if L × L has a skew congruence, then (A′_{1}, μ(A′_{1})) = (L_{1}, L_{1}) is a congruence class of a skew congruence of A′_{1} × B′ = L_{1} × L_{1}. Which is impossible, for the same reason given above. This means that if A = L and B = L, then A′_{1} must be a congruence class of A′ = L/θ ≅ C_{2}, hence A′_{1} = A′ ≅ C_{2} and μ(A′_{1}) ≅ C_{2}. Since μ(A′_{1}) is a congruence class of the homomorphic image B′, then B′ ≅ C_{2}. Which implies that L × L has only one skew congruence ψ, if ψ/θ × θ is a skew congruence of L/θ × L/θ = A′ × B′. Therefore, the only skew congruence ψ of L × L satisfies that (L × L)/ψ ≅ C_{2}.

Let A = L and B = C_{2} , then A′_{1} must be equal to C_{2}. Hence L × C_{2} has a skew congruence iff (A′_{1}, μ(A′_{1})) ≅ (C_{2}, C_{2}) as a congruence class of a skew congruence of A′ × B′ = C_{2} × C_{2}. Which implies that L × C_{2} has only one skew congruence ψ satisfying that (L × C_{2})/ψ ≅ C_{2}. This completes the proof of the theorem. □

Quackenbush [12] has proved that HSP_{f}(L_{1}) = P_{f}{L_{1}, C_{2}}. Similarly and according to the previous discussion, one may show that HSP_{f}(L) = P_{f}{L, L_{1}, C_{2}}. Consequently, we may say that L, L_{1}, C_{2} are the only subdirectly irreducible (monolithic) sloops in the variety V (L), then V (L) = P_{s}{L, L_{1}, C_{2}}. Which implies that the variety V (L) covers the variety V (L_{1}).

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Received August 18, 2000