Steiner Loops Research Articles

The centre of a Steiner loop and the maxi-Pasch problem

A binary operation ``$\cdot$'' which satisfies the identities $x\cdot e = x$, $x \cdot x = e$, $(x \cdot y) \cdot x = y$ and $x \cdot y = y \cdot x$ is called a Steiner loop. This paper revisits the proof of the necessary and sufficient conditions for the existence of a Steiner loop of order $n$ with centre of order $m$ and discusses the connection of this problem to the question of the maximum number of Pasch configurations which can occur in a Steiner triple system (STS) of a given order. An STS which attains this maximum for a given order is said to be {\it maxi-Pasch}. We show that loop factorization preserves the maxi-Pasch property and find that the Steiner loops of all currently known maxi-Pasch Steiner triple systems have centre of maximum possible order.

Non-Moufang variety of Steiner loops satisfying Moufang’s theorem

We construct a non-Moufang variety of Steiner loops H such that every loop from H satisfies Moufang’s theorem. This example solves Rajah’s problem on the existence of such varieties.

Steiner Loops of Affine Type

Steiner loops of affine type are associated to arbitrary Steiner triple systems. They behave to elementary abelian 3-groups as arbitrary Steiner Triple Systems behave to affine geometries over {mathrm {GF}}(3). We investigate algebraic and geometric properties of these loops often in connection to configurations. Steiner loops of affine type, as extensions of normal subloops by factor loops, are studied. We prove that the multiplication group of every Steiner loop of affine type with n elements is contained in the alternating group A_n and we give conditions for those loops having A_n as their multiplication groups (and hence for the loops being simple).

A variety of Steiner loops satisfying Moufang’s theorem: a solution to Rajah’s Problem

A loop $X$ is said to satisfy Moufang's theorem if for every $x,y,z\in X$ such that $x(yz)=(xy)z$ the subloop generated by $x$, $y$, $z$ is a group. We prove that the variety $V$ of Steiner loops satisfying the identity $(xz)(((xy)z)(yz)) = ((xz)((xy)z))(yz)$ is not contained in the variety of Moufang loops, yet every loop in $V$ satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah.

Nilpotent Steiner loops of class 2

We describe Steiner loops of nilpotency class 2 and establish the classification of finite 3-generated nilpotent Steiner loops of nilpotency class 2.

Semiautomorphic inverse property loops

ABSTRACTWe define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for Moufang and Steiner loops. That is, the commutant is a subloop and if a is in the commutant, then a2 is a Moufang element, a3 is a c-element and a6 is in the center. Finally, we give two constructions for semiautomorphic inverse property loops based on Chein’s and de Barros and Juriaans’ doubling constructions.

Effective codescent morphisms in the varieties determined by convergent term rewriting systems

It is shown that the elements of amalgamated free products in a variety of universal algebras have unique normal forms if the variety is represented by a confluent term rewriting system satisfying some additional requirements for its signature and rules. Applying this fact it is proved that any codescent morphism is effective in such varieties. In particular, this is the case for the variety of Mal'tsev algebras, the varieties of magmas with unit and two-sided inverses, idempotent quasigroups, unipotent quasigroups, left Steiner loops, and right Steiner loops.

Free Steiner triple systems and their automorphism groups

The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is not finitely generated if the loop is more than 3-generated. For the free Steiner loop with three generators we describe the generator elements of the automorphism group and some relations between them.

Oriented Steiner loops

We study oriented Steiner loops L which are strictly related to oriented Steiner triple systems and discuss thoroughly the interplay between the structure of these Steiner triple systems and the properties of L. We describe the groups generated by the left, right and all translations of L and show how the group of automorphisms of L is related to the automorphism group and to the blocking subsystems of the corresponding Steiner triple system. In the final section we classify the non-rigid oriented Steiner loops of order 16 and 20.

Extensions of Groups by Weighted Steiner Loops

Solving functional equations given in Nagy and Strambach (Czechoslov. Math. J. 58(133):759–786, 2008) for extensions of a group A by a weighted Steiner loop S we obtain concrete description for all loops with interesting weak associativity properties if the Steiner loop S induces only the trivial automorphism on A. We show that the restricted Fischer groups and their geometry play an important role for loop extension with right alternative property. Also the automorphism groups of these extensions as well as the conditions for isomorphisms between two extensions are studied.

On Parity Vectors of Latin Squares

The parity vectors of two Latin squares of the same side n provide a necessary condition for the two squares to be biembeddable in an orientable surface. We investigate constraints on the parity vector of a Latin square resulting from structural properties of the square, and show how the parity vector of a direct product may be obtained from the parity vectors of the constituent factors. Parity vectors for Cayley tables of all Abelian groups, some non-Abelian groups, Steiner quasigroups and Steiner loops are determined. Finally, we give a lower bound on the number of main classes of Latin squares of side n that admit no self-embeddings.

Primary Decompositions in Varieties of Commutative Diassociative Loops

The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth powers are central, for a fixed n. For n = 2, we get precisely commutative C loops. For n = 3, a prominent variety is that of commutative Moufang loops. Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M 0(Q) such that Q/M 0(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C 0(Q) such that Q/C 0(Q) is a Moufang loop of exponent 3. Since squares (resp., cubes) are central in commutative C (resp., Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2-loop, a Moufang 3-loop, and an abelian group with each element of order prime to 6. We also discuss the definition of Moufang elements and the quasigroups associated with commutative RIF loops.

Translation groups of Steiner loops

If the order of any product of two different translations of a finite Steiner quasigroup of size n > 3 is odd, then the group G generated by the translations of the corresponding Steiner loop of order n + 1 contains the alternating group of degree n + 1 .

On semi-planar Steiner quasigroups

A Steiner triple system (briefly ST) is in 1–1 correspondence with a Steiner quasigroup or squag (briefly SQ) [B. Ganter, H. Werner, Co-ordinatizing Steiner systems, Ann. Discrete Math. 7 (1980) 3–24; C.C. Lindner, A. Rosa, Steiner quadruple systems: A survey, Discrete Math. 21 (1979) 147–181]. It is well known that for each n ≡ 1 or 3 (mod 6) there is a planar squag of cardinality n [J. Doyen, Sur la structure de certains systems triples de Steiner, Math. Z. 111 (1969) 289–300]. Quackenbush expected that there should also be semi-planar squags [R.W. Quackenbush, Varieties of Steiner loops and Steiner quasigroups, Canad. J. Math. 28 (1976) 1187–1198]. A simple squag is semi-planar if every triangle either generates the whole squag or the 9-element squag. The first author has constructed a semi-planar squag of cardinality 3 n for all n > 3 and n ≡ 1 or 3 (mod 6) [M.H. Armanious, Semi-planar Steiner quasigroups of cardinality 3 n , Australas. J. Combin. 27 (2003) 13–27]. In fact, this construction supplies us with semi-planar squags having only nontrivial subsquags of cardinality 9. Our aim in this article is to give a recursive construction as n → 3 n for semi-planar squags. This construction permits us to construct semi-planar squags having nontrivial subsquags of cardinality >9. Consequently, we may say that there are semi-planar SQ ( 3 m n ) s (or semi-planar ST ( 3 m n ) s ) for each positive integer m and each n ≡ 1 or 3 (mod 6) with n > 3 having only medial subsquags at most of cardinality 3 ν (sub- ST ( 3 ) ν ) for each ν ∈ { 1 , 2 , … , m + 1 } .

C-LOOPS: EXTENSIONS AND CONSTRUCTIONS

C-loops are loops satisfying the identity x(y · yz) = (xy · y)z. We develop the theory of extensions of C-loops, and characterize all nuclear extensions provided the nucleus is an abelian group. C-loops with central squares have very transparent extensions; they can be built from small blocks arising from the underlying Steiner triple system. Using these extensions, we decide for which abelian groups K and Steiner loops Q there is a nonflexible C-loop C with center K such that C/K is isomorphic to Q. We discuss possible orders of associators in C-loops. Finally, we show that the loops of signed basis elements in the standard real Cayley–Dickson algebras are C-loops.

On Steiner loops and power associativity

In this paper we investiagte Steiner loops introduced by N.S. Mendel- sohn (Aeq. Math. 6 (1991), 228-230) and provide six (seven) equivalent iden- tities to characterize it. We also prove the power associativity of Bol loops by using closure (Hexagonal) conditions.

A Generalization of Moufang and Steiner Loops

We study a variety of loops, RIF, which arise naturally from considering inner mapping groups, and a somewhat larger variety, ARIF. All Steiner and Moufang loops are RIF, and all flexible C-loops are ARIF. All ARIF loops are diassociative.

Nilpotent Steiner skeins with nilpotent derived Steiner loops

Armanious and Guelzow obtained the structure theorem of finite nilpotent Steiner skeins. Guelzow gave a construction of a Steiner skein of nilpotence class n with all its derived Steiner loops of nilpotence class 1. Armanious gave a construction for Steiner skeins of nilpotence class n with all its derived Steiner loops of nilpotence class n. In this article we survey the main results on nilpotent Steiner skeins and give a new and simple construction, in the form of polynomials, for Steiner skeins of nilpotence class n with all its derived Steiner loops of nilpotence class n. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 232–238, 2000

The smallest non-derived steiner triple system is simple as a loop

The smallest non-derived triple system is simple as a loop. THEOREM.If A, B are Steiner loops, and f:A→B is a homomorphism, then if B and f −1 (1) are derivable from Steiner quadruple systems, then so is A.

A single groupoid identity for Steiner loops

A loop which satisfies the identitiesx2 =e, xe = ex = x, andx(yx) = (xy) x = y is called a generalized Steiner loop. In this paper it is shown that a generalized Steiner loop is a groupoid with a single lawx(((yy) z) x) = z.