In a recent post about f.p.p. for poset products, @M.Mirabi brought back an old-standing problem about the fixed point property of the product of two arbitrary posets which already enjoy the fixed point property. Thus it may be useful to shake up the topic by asking related questions.

The notion of a universal mapping applies to posets too. Thus in the context of posets, given two posets $\ X\ Y,\ $ and their order-preserving function $\ u : X\rightarrow Y,\ $ function $\ u\ $ is universal $\ \Leftarrow:\Rightarrow\ $ for every order-preserving $\ f:X\rightarrow Y\ $ there exists $\ x\in X\ $ such that $\ f(x) = u(x)$.

We see that $\ X\ $ has the fixed point property $\ \Leftrightarrow\ $ identity $I_X:X\rightarrow X$ is universal. Also, if $\ u:X\rightarrow Y\ $ is universal then $\ Y\ $ has the fixed point property. More generally, if $\ v\circ u\ $ is universal then so is $\ v$.

**QUESTIONS:**

**Q1.** Given universal order-preserving $\ f : X\rightarrow Y\ $ and $\ g : Y\rightarrow Z$, is the composition $\ g\circ f:X\rightarrow Z\ $ universal too?

**Q2.** Given universal order-preserving $\ f : X\rightarrow Y\ $ and $\ f' : X'\rightarrow Y'$, is the cartesian product $\ f\times f':X\times X' \rightarrow Y\times Y'\ $ universal too?

REMARK: If the composition of a finite sequence of mappings is not universal then also their product is not universal.

Problems about fixed point property for general objects can be sometimes (often) approximated by, and reduced to finite situations (like finite polyhedra in the topological case).

**TERMINOLOGY:** The order in a cartesian product $\ X\times Y$ of two posets is meant in the sense of the categorical direct product, i.e.

$$(x\ y)\ \le\ (x'\ y')\quad\Leftarrow:\Rightarrow\quad (x\le x'\ \ \&\ \ y\le y')$$

Universal Mappings and Fixed Points Theorems, Bull Acad Polon Sci, v.XV, No 7, 1967, pp.433-438. Quote: DEFINITION. A morphism $\ f:Y\rightarrow X\ $ in category $\ K\ $ isuniversalif for any morphism $\ g: Y\rightarrow X\ $ there is in $\ K\ $ an object $\ Z\ $ and a morphism $\ h:Z\rightarrow Y\ $ such that $\ f\circ h = g\circ h.\ $ Object $\ X\ $ isstable(it has the property analogous to fixed-point property) if the identity $\ i_X:X\rightarrow X\ $ is a universal morphism. No need to ask areal question:-) $\endgroup$ – Włodzimierz Holsztyński Dec 10 '14 at 16:28co-universal morphism. $\endgroup$ – Włodzimierz Holsztyński Dec 10 '14 at 16:325more comments