Definitions: Flock, Parallel Lines, Star Flock,Monomial Flock, Dual Setting, Semi-Field Type Flock


Proposition 11.2.4: Let F = F(ta, t, tb) be a monomial semi-field type flock in PG(3,q) with q = pn for some prime p, a = pi, b = pj with 0 < i < j < n. In the dual setting, the relation of parallelism on the lines containing at least two points of DF is an equivalence relation.

Pf: Let m1, m2 and m3 be three lines of PG(3,q), each containing at least two points of DF, with m1 || m2 and m2 || m3 in their respective affine planes. In the completion of the affine plane containing m1 and m2, these lines meet at a point P of W'=0. In the completion of the affine plane containing m2 and m3, these lines must also meet at a point of W'=0. Since this point is on m2, it must also be P. Thus, m1 and m3 meet at P. In the affine plane determined by m1 and m3, these lines are parallel since they meet at P on W'=0. Therefore, the relation of parallelism is transitive, and so, is an equivalence relation.


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