Abstract: Following [1], we suggest two ways to define an ideal of a poset. Within the frames of the first approach, a topology defined on a set plays the principal role; it defines a partial order on this set (the specialization order). There are at least two ways to embed an arbitrary topological T0-space into a space which is a join semilattice (and even a lattice) with respect to its specialization order—embedding into an injective space and embedding into its own essential completion. Then ideals are defined as restrictions of those of join semilattices on the original space. An inner characterization of ideals obtained in this way is presented. Along with that, sufficient conditions are found for two extensions of a topological space to be isomorphic. The second, a more general, approach does not establish such a tight connection of partial order with topology, allows nonetheless to obtain generalizations of some results from [2], where they were obtained for join semilattices. For example, the Hofmann–Mislove theorem holds also in case of posets. Apart from that, we provide a characterization of [almost] sober spaces as spectra of posets with topology (or, equivalently, semitopological posets) and give a description of essential completions of posets with topology. All the main ideas which we use go back to [2]. Adapting those ideas in case of arbitrary posets involves the definition of ideal in posets given in [3]. Both authors were supported by the fundamental research program of the Siberian Branch of the Russian Academy of Sciences I.1.1, project 0314-2019-0002, and by RFBF, project 18-01-00624a. References [1] Ershov Yu. L., Schwidefsky M. V. To the spectral theory of partially ordered sets // Siberian Math. J. 60, no. 3 (2019), 450–463. [2] Ershov Yu. L. The spectral theory of semitopological semilattices // Siberian Math. J. 44, no. 5 (2003), 791–806. [3] Batueva C., Semenova M. Ideals in distributive posets // Cent. Eur. J. Math. 9, no. 6 (2011), 1380–1388. Sobolev Institute of Mathematics SB RAS, Novosibirsk E-mail: ershov@math.nsc.ru, semenova@math.nsc.r

Cite: Ershov Y.L. , Schwidefsky M.V.
To the spectral theory of posets
In compilation Международная конференция «Мальцевские чтения», 19-23 августа 2019 г. Тезисы докладов.. – ИМ СО РАН, НГУ., 2019. – C.198.

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