On 2-ordered groups

Let h(x, y, z) denote the standard orientation of the plane R . Let M be a non-empty set, Z: M ^ {0, +1, -1}. If for every subset A of a set M, A! < 5, there exists a map ф: A ^ R , such that x,y, z e A implies Z(x, y, z) = п(ф(х), ф(у), ф(^), then (M, Z) is called a 2-ordered set and Z is called a 2-order function on M. If Z is a 2-order function on a group G such that for every x, y, z, a from the group G the equality Z(ax, ay, az) = Z(xa, ya, za) = Z(x, y, z) holds, then G is said to be a 2-ordered group. The paper contains new examples of 2-ordered groups. It is proved that every 2-ordered group contains only one involution or none. A criterion is formulated for a straight line in a 2-ordered group G to be a subgroup of G.

Keywords

Authors

References