*(Background, may be skipped by the knowledgeable reader: A Galois-Tukey connection between two partial orders $(P,\le)$ and $(Q,\le)$ is a pair of maps $\varphi^+:P\to Q$ and $\varphi^-:Q\to P$ satisfying
$$ \forall p\in P \ \forall q\in Q: \left[ \varphi^-(q)\le p \Rightarrow q\le \varphi^+(p)\right]$$
for all $p\in P$, $q\in Q$. (Note: Only "$\Rightarrow$", not "$\Leftrightarrow$" as in Galois connections. Also, it's $\le$ on both sides, and the maps are not necessarily monotone.)*


*If there is such a Galois-Tukey connection, then $cf(P)\ge cf(Q)$, where 
$cf(P)$ is the smallest cardinality of a cofinal (or: "dominating") set in $P$. This is easy to see (and well known). END OF BACKGROUND)*


I am looking for a reference and/or a name for the following generalisation: 


Let $P$, $Q$ be partial orders, $P\_0\subseteq P$ and $Q\_0\subseteq Q$. 
A pair of maps $\varphi^+:P\to Q$ and $\varphi^-: Q\_0\to P\_0$ is called a **BLANK**, if we have 
$$ \forall p\in P \ \forall q\in Q\_0: \left[ \varphi^-(q)\le p \Rightarrow q\le \varphi^+(p)\right]$$


Is there a better name than "strong GT connection", or perhaps even a well-established one? 


My motivation is this: If we define (as in Bartoszyński-Judah 2.1.3.) the cardinal $cf(P\_0, P)$ as the smallest size of a subset $D$ of $P$ which "dominates" $P\_0$ (i.e., $\forall p\_0\in P\_0\ \exists d\in D: p\_0\le d$), and dually the notion $add(P\_0,P)$, then it is easy to see that the existence of a strong GT function as above will imply $add(P\_0,P)\le add(Q\_0,Q)$ and $cf(P\_0,P)\ge cf(Q\_0,Q)$. 




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Edit: Thanks to Peter Vojtas and Andreas Blass for pointing out that 


1. a (generalized) Galois-Tukey relation (or "morphism") does not have to be between two orders; any two relations (subsets of $P\_0\times P$ and $Q\_0\times Q$, 
where no inclusion relation is required between $P\_0$ and $P$, or between $Q\_0$ and $Q$) will do;
2. Letting $\le'\_P$, $\le'\_Q$ be the restrictions of $\le\_P$, $\le\_Q$ to $P\_0\times P$ and $Q\_0\times Q$, respectively, my "strong GT relation" is just the usual generalized GT relation/morphism between $\le'\_P$ and $\le'\_Q$. Both relations happen to be partial orders, but that is irrelevant.



A "strong" Galois-Tukey connection between orders with suborders
Martin, first definition is just J.W.Tukey cited in my <http://www.ksi.mff.cuni.cz/~vojtas/MathPub/1993_GeneralizedGaloisTukey.pdf> . We have to be careful the implication should not be true in a void sense (false implies \* is true) - so some restrictions on $P\_0, \phi^- , ...$ are necessary. In my Galois-Tukey relations can be arbitrary binary ones - so $\mathord\le\cap (P\_0\times P)$ is OK. So what you ask is something between Tukey connections and Galois-Tukey (let us call it strong Tukey). 


It seems, that if $Q\_0, P\_0$ are dominating in $Q, P$, then one can extend $\phi^-$ to whole Q. So, it is interesting when there is a strong Tukey between $\mathord\le\cap (P\_0\times P)$ and $\mathord\le\cap (Q\_0\times Q)$ and no Tukey connection between $P$ an $Q$. 


