Sets and Ordered Structures

Applications

Author(s): Sergiu Rudeanu

Pp: 211-240 (30)

DOI: 10.2174/978160805338411201010211

* (Excluding Mailing and Handling)

Abstract

We have selected in five chapters what we believe to be the core of sets and order in mathematics. Of course,we are aware that this was an ambitious plan and that our selection could not be infallible. Yet we did our best and we conclude the eBook with several examples which illustrate the applications of lattices to topology (§§1-3), to universal algebra (§4), to a new field of applied mathematics called formal concept analysis (§5) and to logic (§6). Besides the references to the literature given in the sections of this chapter, we recommend a nice paper by Birkhoff [1970], with the suggestive title “What can lattices do for you?”.


Keywords: Closure algebra, topological closure, almost discrete space, Heyting algebra, algebraic lattice, equivalence lattice, congruence lattice, formal context, formal concept, Lindenbaum-Tarski algebra, completeness theorem.

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      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 1004 and `b`.`banner_from_date` <= 1740497154 and `b`.`banner_to_date` >= 1740497154 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.5msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 1004
      • 3: 1740497154
      • 4: 1740497154
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='39b7b4cac44aece3ffe8ac1ccac7acc4'
      280μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    11Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        sYkU5xBxHarfUabUl6nUpPHkrA1O5be9NkxjKR5G
        uc_cart_id
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        _flash
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        PHPDEBUGBAR_STACK_DATA
        []
        path_info
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        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
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        request_request
        []
        
        request_headers
        0 of 0
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        request_cookies
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        ClearShow all
        Date ↕MethodURLData
        #12025-02-25 15:25:54GET/chapter/31931121813