#noindex
결합법칙 만족
...

//eom
semigroup with identity

//wpko
[[항등원,identity_element]]을 갖는
[[결합법칙,associativity]]을 따르는
[[이항연산,binary_operation]]을 갖춘 [[대수구조,algebraic_structure]].

i.e.
"추상대수학에서 '''모노이드'''는 [[항등원,identity_element]]을 갖는, [[결합법칙,associativity]]을 따르는 [[이항연산,binary_operation]]을 갖춘 [[대수구조,algebraic_structure]]이다. [[군,group]]의 정의에서 [[역원,inverse_element]]의 존재를 생략하거나, [[반군,semigroup]]([[semigroup]])의 정의에서 [[항등원,identity_element]]의 존재를 추가하여 얻는다."
<<tableofcontents>>
= Sub =

== trivial monoid ==
trivial_monoid
WpEn:Trivial_monoid

via
 [[trivial_group]] - [[군,group]]

cmp
 [[trivial_group]]
 [[trivial_ring]]

Up: [[trivial]]

== free monoid ==
[[free_monoid]] =,free_monoid . free_monoid
{

[[WtEn:free_monoid]]
= https://en.wiktionary.org/wiki/free_monoid
 "(algebra) A monoid whose WtEn:underlying_set is the WtEn:Kleene_closure of some set of generator^^[[제너레이터,generator]]^^s, and whose operator^^[[연산자,operator]]^^ is concatenation^^[[concatenation]]^^."

[[WpEn:Free_monoid]]
= https://en.wikipedia.org/wiki/Free_monoid

https://encyclopediaofmath.org/wiki/Free_semi-group
 ''(Redirected from Free monoid)''

REL/MKL:
[[단어,word]]
[[알파벳,alphabet]]
[[문자열,string]]

}
== monoid object ==
[[모노이드대상,monoid_object]] - [[대상,object]]

== monoidal category ==
[[monoidal_category]] =,monoidal_category . monoidal_category
WtEn:monoidal_category
forkto [[모노이드범주,monoidal_category]]
[[범주,category]]

Sub: [[closed_monoidal_category]]

https://en.wiktionary.org/wiki/monoidal_category

https://en.wikipedia.org/wiki/Monoidal_category

Ndict:"monoidal category"
Ggl:"monoidal category"

=== closed monoidal category ===
closed_monoidal_category
Ggl:"closed monoidal category"

== submonoid ==
submonoid
=,submonoid . submonoid 
부분 모노이드
[[부분모노이드,submonoid]]
https://mathworld.wolfram.com/Submonoid.html
Ndict:submonoid 
Ggl:submonoid 

== history monoid ==
[[history_monoid]] =,history_monoid . history_monoid w

MKL
[[CCS,calculus_of_communicating_systems]]

WtEn:history_monoid x ([[Date(2023-08-16T15:59:02)]])
https://en.wikipedia.org/wiki/History_monoid
Ggl:"history monoid"

== syntactic monoid ==
[[syntactic_monoid]] =,syntactic_monoid . syntactic_monoid
{
'''syntactic monoid'''

https://en.wikipedia.org/wiki/Syntactic_monoid

WtEn:syntactic_monoid x 2024-03
} // syntactic monoid .... Naver:"syntactic monoid" Bing:"syntactic monoid" Ggl:"syntactic monoid"

== trace monoid ==
trace_monoid =,trace_monoid . trace_monoid

WtEn:trace_monoid x ([[Date(2023-08-16T15:59:02)]])
https://en.wikipedia.org/wiki/Trace_monoid
 trace는 [[문자열,string]]. =,trace . trace { Sub: [[trace_cache]] =,trace_cache =,trace_cache . trace_cache
 {
  (we)"In [[컴퓨터구조,computer_architecture]],
  a '''trace cache''' or '''execution trace cache'''
  is a specialized instruction_cache { [[instruction_cache]] WtEn:instruction_cache }
  which stores the dynamic stream of [[명령어,instruction]]s known as '''trace'''.
  .....
  A '''trace processor''' is an architecture designed around the '''trace cache''' and processes the instructions at trace level granularity.
  The formal mathematical theory of traces is described by '''trace monoid'''s."
trace cache
trace_cache
execution trace cache
execution_trace_cache
instruction cache
instruction_cache
trace processor
trace_processor
trace monoid
trace_monoid
[[명령어,instruction]] [[실행,execution]] [[트레이스,trace]]? [[캐시,cache]] [[프로세서,processor]] [[모노이드,monoid]]
https://en.wikipedia.org/wiki/Trace_cache
}

기타 trace / tracing 에 대해 더: =,trace =,tracing .

https://ko.wikipedia.org/wiki/트레이싱
 cmp event_logging

[[WpEn:Tracing_(software)]]
= https://en.wikipedia.org/wiki/Tracing_(software)
= https://en.wikipedia.org/wiki/Tracing_%28software%29

https://en.wikipedia.org/wiki/Branch_trace

https://en.wikipedia.org/wiki/Trace_theory
 concurrent_computation { https://en.wikipedia.org/wiki/Concurrent_computation }, process_calculus 

DTrace (FreeBSD and SmartOS) { https://ko.wikipedia.org/wiki/DTrace https://en.wikipedia.org/wiki/DTrace Bing:dtrace }
ftrace (Linux_kernel) https://en.wikipedia.org/wiki/Ftrace
WPP(?) (Windows) https://en.wikipedia.org/wiki/Windows_software_trace_preprocessor
''그리고 추가로 비슷한것들, via https://en.wikipedia.org/wiki/DTrace#See_also :''
eBPF (Linux_kernel) {
 BPF : Berkeley_Packet_Filter { '''Berkeley Packet Filter (BPF)''' https://en.wikipedia.org/wiki/Berkeley_Packet_Filter }
 https://en.wikipedia.org/wiki/EBPF ... Naver:"eBPF" Bing:"eBPF" Ggl:"eBPF" }
ktrace (BSD and macOS, kernel-program interaction tracer) { '''ktrace'''이게 생성한 것은 [[kdump]]로 읽는다. https://ko.wikipedia.org/wiki/Ktrace  https://en.wikipedia.org/wiki/Ktrace Naver:ktrace }
ltrace (Linux, userland application이 [[shared_library]] =,shared_library . shared_library { shared_library = shared_object .... https://en.wikipedia.org/wiki/Shared_library }호출하는 것을 보여주는 debugging utility) { (동적로딩 dynamic_loading 동적링킹 dynamic_linking linked to: https://ko.wikipedia.org/wiki/동적_적재 https://en.wikipedia.org/wiki/Dynamic_loading https://en.wikipedia.org/wiki/Dynamic_linker (Redirected from WpEn:Dynamic_linking) )시스템을 hooking ... shim 을 삽입 ... static_linking(rel static_library = statically-linked_library  .. linked to: https://ko.wikipedia.org/wiki/정적_라이브러리 https://en.wikipedia.org/wiki/Static_library ) 된 호출들은 trace할 수 없음. ...  https://ko.wikipedia.org/wiki/Ltrace https://en.wikipedia.org/wiki/Ltrace }
strace (Linux, system_call 및 signal 을 모니터링하는 debugging utility) https://en.wikipedia.org/wiki/Strace
LTT Linux Trace Toolkit (LTT) { https://en.wikipedia.org/wiki/Linux_Trace_Toolkit }
''and''
LTTng Linux Trace Toolkit Next Generation (LTTng)  {https://en.wikipedia.org/wiki/LTTng Ggl:LTTng }
SystemTap (Linux_kernel tracing tool) { '''SystemTap (stap)''' scripting_language and tool for dynamic instrumentation =,instrumentation . instrumentation  { instrumentation https://ko.wikipedia.org/wiki/인스트루먼테이션  [[WpEn:Instrumentation_(computer_programming)]] = https://en.wikipedia.org/wiki/Instrumentation_(computer_programming) =  https://en.wikipedia.org/wiki/Instrumentation_%28computer_programming%29  Ggl:trace+instrumentation } https://ko.wikipedia.org/wiki/SystemTap https://en.wikipedia.org/wiki/SystemTap  Naver:SystemTap  }
truss 
ProbeVue (IBM AIX lightweight dynamic tracing environment) { https://en.wikipedia.org/wiki/ProbeVue Naver:ProbeVue }
....
}
 [[process_calculus]] 등 [[concurrent_computation]]에 쓰임.
 [[history_monoid]]와 isomorphic([[동형사상,isomorphism]]).


........ADDHERE.........
= Topics: =

= MKL =
[[semigroup]] - [[반군,semigroup]](잠정적pagename)
[[항등원,identity_element]] // semigroup + 항등원 => '''모노이드'''

Cmp [[모나드,monad]]

= tmp bmks ko =
https://wikidocs.net/1747
위 링크는 https://en.wikibooks.org/wiki/Haskell/Monoids 의 번역

= tmp bmks en =
https://typeclasses.com/monoid

= tmp video en =
Monoids | Group theory episode 1 - YouTube / All Angles
https://www.youtube.com/watch?v=dYN8Q4Ms5U4
{
easy. 22m.

[[군론,group_theory]] series에서 첫번째 ep인 듯

가장 간단한 예는 (set of natural numbers) with (addition).
 $(N,+)$
여기서 $N=\{0,1,2,3,4,\ldots\}$
$N$ 은 closed under addition.
 neutral/identity element 포함. 저것은 $+$ 에 대한 효과가 없음(no effect).
$+$ 은 binary operation.
 associative. $(a+b)+c = a+(b+c)$

또 다른 예는 string(원소에 empty string 포함, 연산은 concatenation)
 * strings are closed under concatenation. ✔️
 * concatenation is associative. (ab)c=a(bc). ✔️
 * empty string is neutral/identity element. ✔️
따라서 monoid의 조건 만족.

원 색칠(파랑 노랑)으로 비유하는데 .... 왜 [[명령어,instruction]]들로 이루어진 subroutine의 [[실행,execution]]이 monoid에 비유되는지 알 수 있다. (semicolon ; is binary operation)
[[입출력,IO]]도 마찬가지.
for_loop 도.
암튼 PL의 [[스테이트먼트,statement]]s' 대수적algebraic [[추상화,abstraction]]에 쓰임.

[[인버스,inverse]]는 없다 ... 저게 포함되려면 [[군,group]]

}

...
YouTube:"monoid algebra"

= Inter =
https://artofproblemsolving.com/wiki/index.php/Monoid

https://mathworld.wolfram.com/Monoid.html

https://everything2.com/title/monoid
{
번역at[[Date(2023-09-25T19:50:29)]]
'''Monoid'''는 거의 [[군,group]]이다. 유일하게 군의 조건을 만족하지 못하는 것이 그 안의 모든 [[원소,element]]가 [[인버스,inverse]]''([[역,inverse]] [[역원,inverse_element]]?)''을 가질 필요가 없다는 점이다.
'''모노이드'''는 [[항등원,identity_element]]을 가진다(이 특징을 [[semigroup]]과 대조해 보라.) - TBW
Contrast: [[semigroup]]
}

https://groupprops.subwiki.org/wiki/Monoid

https://encyclopediaofmath.org/wiki/Monoid

[[Namu:모노이드]] = https://namu.wiki/w/모노이드

[[WpKo:모노이드]]
= https://ko.wikipedia.org/wiki/모노이드

[[WpEn:Monoid_(category_theory)]]
= https://en.wikipedia.org/wiki/Monoid_(category_theory)
= https://en.wikipedia.org/wiki/Monoid_%28category_theory%29

https://ncatlab.org/nlab/show/monoid

... Google:monoid Naver:monoid
... Google:모노이드 Naver:모노이드

Up:
[[추상대수,abstract_algebra]]