arXiv subjects
Publications and source records attributed to Michał Kukieła.
A poset P is called reversible if every order preserving bijective self map of P is an order automorphism. P is called hereditarily reversible if every subposet of P is reversible. We give a complete characterization of hereditarily reversible posets in terms of forbidden subsets. A similar result is stated also for preordered sets. As a corollary we extend the list of known examples of hereditarily reversible topological spaces.
We prove an infinite analogue of the main theorem of discrete Morse theory formulated in terms of discrete Morse matchings. Our theorem holds under the assumption that the given Morse matching induces finitely many equivalence classes of infinite directed simple paths. A homological version of the theorem is also given.
We generalise some results of R. E. Stong concerning finite spaces to wider subclasses of Alexandroff spaces. These include theorems on function spaces, cores and homotopy type. In particular, we characterize pairs of spaces X,Y such that the compact-open topology on C(X,Y) is Alexandroff, introduce the classes of finite-paths and bounded-paths spaces and show that every bounded-paths space and every countable finite-paths space has a core as its strong deformation retract. Moreover, two bounded-paths or countable finite-paths spaces are homotopy equivalent if and only if their cores are homeomorphic. Some results are proved concerning cores and homotopy type of locally finite spaces and spaces of height 1. We also discuss a mistake found in an article of F.G. Arenas on Alexandroff spaces. It is noted that some theorems of G. Minian and J. Barmak concerning the weak homotopy type of finite spaces and the results of R. E. Stong on finite H-spaces and maps from compact polyhedrons to finite spaces do hold for wider classes of Alexandroff spaces. Since the category of T_0 Alexandroff spaces is equivalent to the category of posets, our results may lead to a deeper understanding of the notion of a core of an infinite poset.