# Fixed points and universal maps for posets

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.

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

• Very nice & natural questions! Do you have an intuition as to Q1? – Dominic van der Zypen Nov 18 '14 at 15:53
• @dominiczypen, thank you for kind words. I'd need to go back to my topic of the universal maps and universal morphisms. It's been half a century ago :-) – Włodzimierz Holsztyński Nov 19 '14 at 22:38
• I was wondering, is there a universal property in "category theory language" that captures the essence of a universal map? - I guess, I better make this a real question (with a category-theory tag). – Dominic van der Zypen Dec 10 '14 at 12:51
• The full categorical definition is given in Włodzimierz Holsztyński, 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\$ is universal if 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\$ is stable (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 a real question :-) – Włodzimierz Holsztyński Dec 10 '14 at 16:28
• In the case of topological category one should consider the category of non-empty spaces. In the case of Banach algebras one should consider the notion of the co-universal morphism. – Włodzimierz Holsztyński Dec 10 '14 at 16:32