How exactly to adapt Brown’s collapse from monoids to algebras?

In The Geometry of Rewriting Systems, Brown describes a method to collapse the bar resolution of a monoid. Roughly:

  • Given a simplicial set $ X$ equipped with a collapsing scheme (a partition of the geometric realization $ \lvert X \rvert$ into essential, redundant and collapsible cells, which satisfy some properties) it is possible to collapse $ \lvert X \rvert$ into a smaller CW-complex with a cell for each essential cell of $ \lvert X \rvert$ .
  • A monoid $ M$ presented with a complete rewriting system (the set of relations $ R$ is terminating Church-Rosser) induces a collapsing scheme on the simplicial set $ BM$ .
  • The $ n$ -cells of $ \lvert BM \rvert$ are in correspondence with the generators of $ B_n$ in the normalized bar resolution of $ M$ . In particular, each $ B_n$ can be collapsed in a similar way that $ \lvert BM \rvert$ is.
  • If $ M$ has a good set of normal forms of finite type ($ M$ has a finite presentation $ (S,R)$ and $ R$ is terminating Church-Rosser), then the classifying space $ \lvert BM \rvert$ can be collapsed into a finite CW-complex
  • Under the assumptions on the last bullet, we can also collapse the bar resolution of $ M$ into one where all $ B_n$ are finitely generated. In particular, $ M$ is of type $ (FL)_{\infty}$ .

Brown states (emphasis mine):

The Method used in this section works, with no essential change, if the ring $ \mathbb{Z}[M]$ is replaced by an arbitrary augmented $ k$ -algebra $ A$ which comes equipped with a presentation satisfying the conditions of Bergman’s diamond lemma (The diamond lemma for ring theory, Theorem 1.2). Here $ k$ can be any commutative ring. One starts with the normalized bar resolution $ C$ of $ k$ over $ A$ , and one obtains a quotient resolution $ D$ , with one generator for each “essential” generator of the bar resolution. In particular, we recover Anick’s Theorem (On the homology of associative algebras, Theorem 1.4).

Let me state here the conditions of the diamond lemma:

Theorem: Let $ S$ be a reduction system for a free associative algebra $ k\langle X \rangle$ (a subset of $ \langle X \rangle$ \times k\langle X \rangle), and $ \leq$ a semigroup partial ordering on $ \langle X \rangle$ , compatible with $ S$ , and having descending chain condition. Then…

I can’t understand two aspects of this adaptation:

  • The conditions of the diamond lemma are stated for the free associative algebra $ k\langle X \rangle$ . What does it mean for the presentation of $ A$ to satisfy these conditions? If we assume $ A \cong k\langle X \rangle/I$ we can interpret Brown’s sentence as $ k\langle X \rangle$ satisfying the conditions, but what about the ideal $ I$ ? I assume it is related somehow to the reduction system, but how exactly?
  • What would be the essential generators of the normalized bar resolution? In the monoid setting, they originate from the essential cells of the classifying space of the monoid. In the algebra setting, we don’t have that tool anymore (unless we generalize classifying spaces for internal monoids over monoidal categories, which I don’t think is the case).

Can you reduce yourself, crawl into the Tarrasque’s airway, and try to collapse its lung by enlarging inside of it?

A player tried to use the spell enlarge/reduce on his PC while in the mouth of the Tarrasque to shrink himself, then go deeper into its airway and to try to collapse its lung by enlarging himself inside of it. I imagine there’s no real rule for this but I felt silly just denying it, because it seemed like a cool idea that you could only probably do with a creature as large as the Tarrasque.

Can you crawl into a gargantuan creature’s mouth or into its airway via reduce/enlarge and attack its brain or internal organs?

How do I circumvent protected sheets to expand or collapse groups and rows in Google Sheets?

A budget I am working on has each month on the same sheet and I would like to control which months are visible when comparing expenditure and income. In order to do so, I grouped the rows and columns according to category and month. The only cells which should be edited are the planned expense/income values. It makes sense to protect the sheet except for those cells. However, by protecting the sheet, I am unable to collapse or expand categories and months. Is there a workaround to retain the design and experience I would like to implement?

Bootstrap Date Picker working fine, but an issue with Collapse (Time picker) and Bootstrap Collapse

I’m able to get the date picker and I’ve called the required files as per the documentation but unable to collapse between date and time.

I have read the documentation and also followed the steps mentioned here -Bootstrap Datetime Picker not working with Bootstrap 4 Alpha 6

Here is a video sample –

Any help is greatly appreciated.


Problemas com Navbar Collapse

Opa, tudo bem??? Estou com problemas ao fazer o navbar collapse, qdo clico no ícone não aparece o menu abaixo.

Segue abaixo o código:

<div class="container">      <nav class="navbar navbar-expand-lg navbar-light bg-light">          <a class="navbar-brand" href="">O Rei do Bojo</a>            <button class="navbar-toggler" type="button" data-toggle="collapse" data-target="navbarMenu">             <span class="navbar-toggler-icon"></span>           </button>             <div class="navbar-collapse collapse" id="navbarMenu">           <ul class="navbar-nav">             <li><a class="nav-item nav-link active" href="">Home</a></li>             <li><a class="nav-link nav-link" href="">Sobre</a></li>             <li><a class="nav-link nav-link" href="">Produtos</a></li>             <li><a class="nav-link nav-link" href="">Contato</a></li>            </ul>         </div>      </nav><!-- /navbar --> </div> 

E abaixo segue os links referências para boostrap no Body e os scripts do js.

<link rel="stylesheet" href="assets/css/bootstrap.min.css">  <script type="text/javascript" src="assets/js/jquery-3.3.1.min.js"></script> <script type="text/javascript" src="assets/js/jquery/bootstrap.bundle.min.js"></script> 

Bootstrap помощь с collapse и якорем

хотел бы поинтересоваться, как сделать так, что-бы при нажатии на кнопки меню, к примеру на “Отзывы”, открывался коллапс соответствующего названия. Якорь уже есть и коллапс рабочий введите сюда описание изображения

<ul>   <li class="nav-item">           <a class="nav-link" href="#heading3">Отзывы</a>   </li> </ul>  <div class="card"> 	<div class="card-header" role="tab" id="heading3"> 		<a data-toggle="collapse" data-parent="#accordionEx" href="#collapse3" aria-expanded="true" aria-controls="collapse3"> 		    <h5 class="mb-0"> 			Отзывы 		</h5> 		</a> 	</div>  	<div id="collapse3" class="collapse" role="tabpanel" aria-labelledby="heading3" data-parent="#accordionEx"> 		<div class="card-body feedback"> 			TextTextTextTextText 		</div> 	</div> </div>

Bootstrap 4 navbar collapse is not collapsing

The problem is the bootstrap navbar. If I take the code and put it into a website where only the navbar is show it works perfectly but as soon as I try to implement it it looks horrible.

  1. It is not expanding the correct way. Normally it should appear from the top down but the site just stops for some time and then the navbar is there
  2. While collapsing the navbar shows a scrollbar on the right side

I’ve already tried to change the classes and get an animation running but it did not work thus I’ve changed everything back.

<nav class="navbar navbar-toggleable-md navbar-dark bg-secondary navbar-static-top">     <a class="navbar-brand text-light" href="index.html">EG-Mitfahrgelegenheit</a>     <h2 class="text-white mb-0 font-weight-bold">Übersicht</h2>     <button class="navbar-toggler navbar-toggler-right" type="button" data-toggle="collapse" data-target="#navbarSupportedContent" aria-controls="navbarSupportedContent" aria-expanded="false" aria-label="Toggle navigation">         <span class="navbar-toggler-icon"></span>     </button>     <div class="collapsing navbar-collapse" id="navbarSupportedContent">         <ul class="navbar-nav mr-auto">             <li class="nav-item disabled">                 <span class="navbar-text text-warning">Angemeldet als: Mensch</span>             </li>             <li class="nav-item">                 <a class="nav-link active text-light" href="overview.html">Übersicht</a>             </li>             <li class="nav-item">                 <a class="nav-link text-light" href="mRoutes.html">Meine Fahrten</a>             </li>             <li class="nav-item">                 <a class="nav-link text-light" href="createRoute.html">Fahrt erstellen</a>             </li>             <li class="nav-item">                 <a class="nav-link text-light" href="myProfile.html">Mein Profil</a>             </li>             <li class="nav-item">                     <a class="nav-link text-light" href="Hilfeblatt.pdf" target="_blank">Hilfe</a>             </li>             <li class="nav-item">                 <a class="nav-link text-light" href="#" id="logout">Abmelden</a>             </li>         </ul>     </div> </nav> 

This is my code for the navbar (tell me if you need something else).

Hope someone can help me, thanks in advance 🙂

Vertical navigation UX – Scroll or collapse

Which is preferred?

  • Scrollable (left) – 2nd-level always visible, scrolls if overflows
  • Toggleable Expansion (right) – 2nd level only visible if initiated by user (e.g. accordion or expand/collapse)

Assuming “depends” is a given, what are cases which one would be preferred over the other. Are there any alternatives to this style of vertical navigation where horizontal navigation is not an option (space is already utilized for another purpose).

Scrollable Nav Collapsable Nav

Expand / collapse details

I want to ask you for suggestions about expanding and collapsing some details from a document. I have a link for the prototype, just press show details/ hide details. Do you have any better idea for this action to hide and show details, instead of using a button? I already have 2 more buttons with reports and … screenshots also attached. enter image description here

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Link1: Link2: