Fractional dimension of partial orders

Given a partially ordered setP=(X, ≤), a collection of linear extensions {L1,L2,...,Lr} is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someLi (andy<x in someLj). For a positive integerk, we call a multiset {L1,L2,...,Lt} ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theLi's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask→∞. We prove various results about the fractional dimension of a poset.