A regular left-order on finitely generated group G is a total,
left-multiplication invariant order on G whose corresponding positive cone is
the image of a regular language over the generating set of the group under the
evaluation map. We show that admitting regular left-orders is stable under
extensions and wreath products and give a classification of the groups all
whose left-orders are regular left-orders. In addition, we prove that solvable
Baumslag-Solitar groups B(1,n) admits a regular left-order if and only if
n≥−1. Finally, Hermiller and Sunic showed that no free product admits a
regular left-order, however we show that if A and B are groups with regular
left-orders, then (A∗B)×Z admits a regular left-order.Comment: 41 pages,9 figure