Tomáš Kepka

On multiplication groups of loops

The concept of multiplication groups of quasigroups was introduced by Albert Cl] and the connection between quasigroups and corresponding multiplication groups has been studied by Bruck [6], Smith [20] and Ihringer [14, 151. While studying the multiplication group of a loop Q (a quasigroup with neutral element) a central role is played by the stabilizer of the neutral element. This subgroup 1(Q) of the multiplication group is called the inner mapping group of Q. If Q is a group then it is clear that r(Q) consists of the inner automorphisms of Q. We also know that a loop Q is an abelian group if and only if 1(Q) = I. In this paper we study some properties of the inner mapping group and we also give a partial answer to the question: What are the multiplication groups of loops? This question is closely connected to certain transversal conditions. Sections 2 and 3 are devoted to investigating these conditions and in Section 4 we characterize multiplication groups of loops with the aid of these conditions. In the same section we prove one of our main results: If Q is a finite loop whose inner mapping group is cyclic, then Q is an abelian group. Finally, in Section 5 we use the properties of the inner mapping group in order to show that certain groups are not multiplication groups of loops. We also give examples of groups which are multiplication groups of loops. Our notation is standard and for basic facts about groups and loops WC refer to [4,7, 133. 112

On the abelian inner permutation groups of loops

Let Q be a loop such that the inner permutation group I(Q) of Q is finite and abelian. The aim of the present note is to show that Q is then centrally nilpotent and no non-trivial primary component of I(Q) is cyclic. Notice that this is a generalization of [5, 2.1,2.4] and [7, 6.2,6.4], where the mentioned results are proved for finite loops.

The equational theory of paramedial cancellation groupoids

By a paramedial groupoid we mean a groupoid satisfying the equation xy · zu = uy · zx. As it is easy to see, the equational theory of paramedial groupoids, as well as the equational theory based on any balanced equation, is decidable. In this paper we are going to investigate the equational theory of paramedial cancellation groupoids; by this we mean the set of all equations satisfied by paramedial cancellation groupoids. (By a cancellation groupoid we mean a groupoid satisfying both xz = yz → x = y and zx = zy → x = y.) Clearly, the equational theory of paramedial cancellation groupoids is just the least cancellative equational theory containing the paramedial law. We will show that this equational theory is also decidable (Theorem 4.1), that it is a proper extension of the equational theory of paramedial groupoids (Theorem 4.3), and that whenever two terms are unrelated with respect to this equational theory, then their squares are also unrelated (Theorem 4.7). The results can be compared with those of [2] and [3] for medial groupoids.

Finitely generated algebraic structures with various divisibility conditions

Abstract. Infinite fields are not finitely generated rings. A similar question is considered for further algebraic structures, mainly commutative semirings. In this case, purely algebraic methods fail and topological properties of integral lattice points turn out to be useful. We prove that a commutative semiring that is a group with respect to multiplication can be two-generated only if it belongs to the subclass of additively idempotent semirings; this class is equivalent to

Modules Commuting (Via Hom) with Some Colimits

\ell

On conjugacy classes in finite loops

-groups.

Small finite trimedial quasigroups

AbstractFor every module M we have a natural monomorphism

Distributive groupoids and the finite basis property

\Psi :\coprod\limits_{i \in I} {{\text{Hom}}_R \left( {M,A_i } \right) \to {\text{Hom}}_R \left( {M,\coprod\limits_{i \in I} {A_i } } \right)}