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In mathematics, a ranked poset is a partially ordered set in which one of the following (non-equivalent) conditions hold: it is a graded poset, or; a poset with the property that for every element x, all maximal chains among those with x as greatest element have the same finite length, or; a poset in which all maximal chains have the same ...
The rank is consistent with the covering relation of the ordering, meaning that for all x and y, if y covers x then ρ(y) = ρ(x) + 1. The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset.
A convex set in a poset P is a subset I of P with the property that, for any x and y in I and any z in P, if x ≤ z ≤ y, then z is also in I. This definition generalizes the definition of intervals of real numbers. When there is possible confusion with convex sets of geometry, one uses order-convex instead of "convex".
An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler . Eulerian lattices generalize face lattices of convex polytopes and much recent research has been devoted to extending known results from polyhedral combinatorics , such as various restrictions on f -vectors of convex simplicial polytopes ...
Comparison of user features of messaging platforms refers to a comparison of all the various user features of various electronic instant messaging platforms. This includes a wide variety of resources; it includes standalone apps, platforms within websites, computer software, and various internal functions available on specific devices, such as iMessage for iPhones.
App Publisher Category Similarweb top 50 apps ranking (As of December 2024) [1] Store Rank Google Chrome: Google LLC Communication 1 () () Samsung One UI Home: Samsung Electronics Co., Ltd. Personalization 2 () () Google: Google LLC Tools 3 () () Facebook: Meta Platforms, Inc. Social 4 () 37 (10) Google Messages: Google LLC Communication 5 ...
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The app supports chat history syncing and voice input (using Whisper, OpenAI's speech recognition model). In July 2023, OpenAI unveiled an Android app, initially rolling it out in Bangladesh, Brazil, India, and the U.S. [70] [71] The app later became available worldwide. OpenAI is working on integrating ChatGPT with Android's assistant APIs. [72]