A "strong" Galois-Tukey connection between orders with suborders

Question

(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.

Answer 1

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 $Q$.

Answer 2

As indicated in Peter Vojtas's answer, this notion is a special case of what he called generalized Galois-Tukey connections (in the paper he linked to) and what I later called morphisms (in my chapter of the Handbook of Set Theory). Specifically, what you describe is a morphism from $(P_0,P,\leq)$ to $(Q_0,Q,\leq)$. (In the terminology of generalized G-T connections, "from" and "to" would be interchanged.) In general, the domain and codomain of a morphism could be any triples of the form $(A_-,A_+,R)$ where $R\subseteq A_-\times A_+$, i.e., any two sets and a relation between them. What's special in your situation seems to be just that (1) $P_0\subseteq P$, (2) the $\leq$ relation that you actually use (between $P_0$ and $P$) is the restriction of a partial order on all of $P$, and (3,4) the same for $Q_0$ and $Q$.