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IMI/Publicaţii/QRS/Ediţii/QRS v.16, n.1 (19), 2008/

Greedy quasigroups

Authors: T. A. Rice

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.

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