Institutul de Matematică şi Informatică "Vladimir Andrunachievici"
Universitatea de Stat din MoldovaRO EN
Greedy quasigroups
Authors: T. A. Rice
– 0.28 Mb
Abstract
The paper investigates the quasigroup Qs constructed on the well-ordered set of natural numbers by placing a number $s$ known as the seed in the top left-hand corner of the body of the multiplication table, and then completing the Latin square using the greedy algorithm that chooses the least possible entry at each stage. The initial motivation comes from the theory of combinatorial games, where Q0 gives the usual nim sum, while Q1 gives the corresponding sums for positions in misere nim. The multiplication groups of these quasigroups are analyzed. The alternating group of the natural numbers is a subgroup of the multiplication groups. It is shown that these so-called greedy quasigroups Qs are mutually non-isomorphic. The quasigroup Q1 is subdirectly irreducible. For s>1, the greedy quasigroups Qs are simple, and for s>2 they are rigid, possessing no non-trivial automorphisms. Indeed in this case the endomorphism monoid contains just the identity and a single constant. The subquasigroup structures of the Qs are also determined. While Q0, Q1 have uncountably many subquasigroups, and Q2 has just one proper, non-trivial subquasigroup, Qs has none for s>2.Fulltext
– 0.28 Mb
Contents
- Fuzzy Lie ideals of Lie algebras with interval-valued membership functions
- Counting loops with the inverse property
- On middle translations of finite quasigroups
- Semigroup, monoid and group models of groupoid identities
- Direct product of quasigroups and generalized diagonal subquasigroup
- Algebraic properties of some varieties of central loops
- On reconstructing reducible n-ary quasigroups and switching subquasigroups
- Left almost semigroups defined by a free algebra
- Quasi union hyper K-algebras
- A note on a union hyper K-algebras
- On decomposition of Moufang groupoids
- Greedy quasigroups
