Order-preserving map

The concept of order preserving maps plays an important role in applications of Hasse Diagram Technique (HDT). For an introduction, this concept will be exemplified by the so-called level construction (see Fig. 6). 

Fig. 6. The Hasse diagram (left side (a)) is mapped onto the Hasse diagram (b). All order relations of the domain set, and order relations (left side) are preserved in the range of the mapping < > (right side). Finally an order-preserving map rj) F - F is applied to obtain a linear order (diagram (c)).

Extensions are order-preserving maps from the ground set E into the ground set E see Davey Priestley 1990. Linear extensions (LEX(E ), <) are order-preserving maps from E to E, which assign to (E, <) a linear order. 

The application of weighting schemes as performed by METEOR is not the only way to get linear orders. Another possibility was found by (Winkler 1982) and worked out by Lerche and Sorensen (2003), Briigge-mann et al. (2004). The principle to get a linear order is first to find all order preserving maps of an empirical poset. Hereby a set of linear extensions, LE is found, where each element of this set is a single linear order preserving all <-relations of the empirical poset. The set LE can be very large. A very crude upper estimation of LT, the number of all linear extensions of an empirical poset is n , with n the number of all elements of the quotient set. The set LE of all linear extensions can be interpreted as probability space Let us assume that the rank of an object x, found for one specific linear extension, rk(x) has a certain value, Rk. Then the probability of x to get this value Rk is the number of linear extensions where rk(x) = Rk, L(rk(x) = Rk), divided by LT (see chapter by Briiggemann and Carl-sen, p. 86). We write... 

For example, C could be the category of topological spaces, or abstract simplicial complexes, or posets, in which case the structure-preserving maps would respectively be continuous maps, or simplicial maps, or order-preserving maps (see Definition 10.3). 

In the context of acyclic categories, the role of order-preserving maps is played by functors, see Chapter 4 for their definition. It is an important and very useful fact that all acyclic categories together with all possible functors between them form a full subcategory of Cat we call that category AC. 

We can also consider all posets, and all possible functors between them. This is also a full subcategory of Cat, which we shall call Posets. The functorial properties provide one explanation for the wide occurrence and accepted usefulness of order-preserving maps, as opposed to, for example, order-reversing ones. 

An example of a functor between acyclic categories and an associated trisp map between their nerves is shown in Figures 10.3 and 10.4. For posets, Proposition 10.6 translates into saying that an order-preserving map between posets P and Q (these are the morphisms in Posets) induces a simphcial map between the corresponding order complexes A P) and A(Q), which additionally preserves edge orientations. 

Just like A, this is a functor from the category of regular CW complexes to the category of posets, since a cellular map between regular CW complexes will induce an order-preserving map between their face posets. 

Assume that p P Q is an order-preserving map, and assume that we have acyclic matchings on subposets p (g), for all q Q. Then the union of these matchings is itself an acyclic matching on P. 

Definition 13.11. An order-preserving map from a poset P to itself is... 

Proof. Set Q =. F(DGn). Define the map f Q Q, taking each graph to its transitive closure. It is easily checked that f is an order-preserving map, that ifP = if, and that G < f G), for any graph G. We conclude that 99 is an ascending closure operator. 

Let ip P Q be an order-preserving map such that for any x (E Q the complex is contractible. Then the induced map between the simpli-... 

By Theorem 18.3 we can replace the neighborhood complexes M KGnk) by the Horn complexes Bip KG k) as far as the homotopy type is concerned. Again, we define an order-preserving map... 

Again, using composition we can define an order-preserving map... 

Intuitively, one can think of the map g as the pullback map. It is important to remark that if tp is not injective, it may happen that dim g r]) > dim 77. Since g is an order-preserving map, the induced map between abstract simphcial complexes A g) Bd (Horn (G, K)) — Bd (Horn (T, K)) is simplicial and gives the corresponding map of topological spaces, which we denote by V k- It is important to notice that the map g does not always come from a cellular map. 

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