I am studying preference relations in fuzzy logic with regards to quantifying consistency of preferences. In particular, how to quantify the consistency of somebodies preferences between three options, $i, j, k$.
My preference between any two variables can be any value in the domain $0\leq r \leq 1$
If I like $i$ more than $j$, then $r_{ij}$ is > 0.5
If I like $j$ less than $k$, then $r_{jk}$ is < 0.5
If I can't choose between $i$ and $j$ then $r_{ij} = 0.5$
Also, $r_{ij} = 1-r_{ji}$
If $r_{ij} = 0.7$ (arbitrary example) and $r_{jk} = 0.4$ then there is some function $f$ which determines my preference $r_{ik}$, if I can be modelled as perfectly rational.
$$r_{ik} = f(r_{ij}, r_{jk})\forall i,j,k$$ Where $f$ is a binary operator of the form: $$f:[0,1]\times[0,1]\to[0,1]$$
I keep seeing these functions described as "unique up to a positive linear transformation".
My issue is that I don't really have any idea what this means. I am a mechanical engineer, have done A-level maths, and a little university level maths as part of my engineering degree. Is it possible to explain roughly what this means, In a way that I would be able to understand. Appreciate any help anyone can offer.
This question already has an answer in other forms on this site, under the guise of "What does the phrase 'up to [...]' mean in mathematics?" To answer you directly, in general when we say that an object $f$ in mathematics is unique up to $x$ where $x$ depends on the context, we mean that $f$ is uniquely determined, except possibly for the addition of some $x$.
Examples help:
For instance, if $f'$ is the derivative of some function $f$ on $[a,b]$, then we can say that $f$ is unique up to a constant because if $f' = g'$, then $f = g + C$ for some constant $C$.
If we are classifying groups of order $4$, then we would say that there are exactly two groups of order $4$, up to isomorphism. So, we can make a list $G,H$ of two nonisomorphic groups of order $4$, such that if any other group $N$ has order $4$, then there is an isomorphism $\varphi$ such that $\varphi:N\cong G$ or $\varphi:N\cong H$, but not both.
In your example, for a function $f$ to be "unique up to a positive linear transformation," it likely means that $f$ is only determined up to the addition of some positive linear map $T$, so talking about $f$ itself is sort of meaningless because we can only talk about $f$ "up to $T$". Just like saying "the antiderivative of $f'$ is $f$" is meaningless, because we can only talk about the antiderivative of $f'$ up to a constant $C$.