Přemysl Jedlička

Constructions of Commutative Automorphic Loops

A loop whose inner mappings are automorphisms is an automorphic loop (or A-loop). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loops of small orders and also of order p 3, where p is a prime.

A Combinatorial Construction of the Weak Order of a Coxeter Group

ABSTRACT Let W be a (finite or infinite) Coxeter group and W X be a proper standard parabolic subgroup of W. We show that the semilattice made up by W equipped with the weak order is a semidirect product of two smaller semilattices associated with W X . #Communicated by J. Aleu.

NUCLEAR SEMIDIRECT PRODUCT OF COMMUTATIVE AUTOMORPHIC LOOPS

Automorphic loops are loops where all inner mappings are automorphisms. We study when a semidirect product of two abelian groups yields a commutative automorphic loop such that the normal subgroup lies in the middle nucleus. With this description at hand we give some examples of such semidirect products.

On commutative A-loops of order PQ

We study a construction introduced by Drapal, giving rise to commutative A-loops of order kn where k and n are odd numbers. We show which combinations of k and n are possible if the construction is based on a field or on a cyclic group. We conclude that if p and q are odd primes, there exists a non-associative commutative A-loop of order pq if and only if p divides q2 - 1 and such a loop is most probably unique.

On a partial syntactical criterion for the left distributivity and the idempotency

We study here so called cuts of terms and their classes modulo the identities of the left distributivity and the idempotency. We give an inductive definition of such classes and this gives us a criterion that decides in some cases whether two terms are equivalent modulo both identities.

Distributive and trimedial quasigroups of order 243

We enumerate three classes of non-medial quasigroups of order 243 = 3 5 up to isomorphism. There are 17 004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, Beneteau and Lacaze), 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Nźmec), and 6 non-medial distributive Mendelsohn quasigroups of order 243 (extending the work of Donovan, Griggs, McCourt, Oprsal and Stanovský).The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang loops, and on computer calculations with the LOOPS package in GAP.