Sertel, MuratSlinko, Arkadii2009-08-282009-08-282002-06Department of Mathematics - Research Reports-482 (2002)1173-0889http://hdl.handle.net/2292/5146We investigate the ways in which a linear order on a finite set $A$ can be consistently extended to a linear order on a set $P_k(A)$ of multisets on $A$ of cardinality $k$. We show that, when $card(A)=3$, all linear orders on $P_k(A)$ are additive and classify them by means of Farey fractions. For $card(A)ge 4$ we show that there are non-additive consistent linear orders on $P_k(A)$, we prove that they cannot be extended to a consistent linear order on $P_K(A)$ for sufficiently large $K$. We give the lower bounds for the number of consistent linear orders on $P_2(A)$ and for the total number of consistent linear orders on $P_2(A)$.https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmTechnical ReportFields of Research::230000 Mathematical Sciences::230100 MathematicsThe author(s)