Recursively presented group - Revision history

Vipul at 02:01, 5 November 2013

2013-11-05T02:01:37Z

← Older revision		Revision as of 02:01, 5 November 2013
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	# It possesses a [[defining ingredient::recursive presentation]], i.e., a [[defining ingredient::presentation of a group\|presentation]] where the set of generators is countably infinite (with an explicit enumeration) and the set of relations is recursively enumerable.		# It possesses a [[defining ingredient::recursive presentation]], i.e., a [[defining ingredient::presentation of a group\|presentation]] where the set of generators is countably infinite (with an explicit enumeration) and the set of relations is recursively enumerable.
	# It possesses a [[defining ingredient::recursive presentation]], i.e., a [[defining ingredient::presentation of a group\|presentation]] where the ~~number~~ of generators is ~~finite~~ and the set of relations is recursive.		# It possesses a [[defining ingredient::recursive presentation]], i.e., a [[defining ingredient::presentation of a group\|presentation]] where the set of generators is countably infinite (with an explicit enumeration) and the set of relations is recursive.
	# It is isomorphic to a subgroup of a [[defining ingredient::finitely presented group]].		# It is isomorphic to a subgroup of a [[defining ingredient::finitely presented group]].

Vipul at 02:01, 5 November 2013

2013-11-05T02:01:05Z

Vipul at 20:07, 26 May 2010

2010-05-26T20:07:39Z

← Older revision		Revision as of 20:07, 26 May 2010
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	==Definition==		==Definition==

	A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it ~~possesses a [[defining ingredient::recursive presentation]], viz a [[defining ingredient::presentation of a group\|presentation]] where~~:		A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it satisfies the following equivalent conditions:

	# ~~The~~ number of generators is finite		# It possesses a [[defining ingredient::recursive presentation]], viz a [[defining ingredient::presentation of a group\|presentation]] where the number of generators is finite and the set of relations is recursively enumerable.
	# ~~The~~ set of relations is ~~recursively enumerable~~		# It possesses a [[defining ingredient::recursive presentation]], viz a [[defining ingredient::presentation of a group\|presentation]] where the number of generators is finite and the set of relations is recursive.
			# It is a [[defining ingredient::finitely generated group]] that is isomorphic to a subgroup of a [[defining ingredient::finitely presented group]].

	==Relation with other properties==		==Relation with other properties==

Vipul: moved Recursively presentable group to Recursively presented group over redirect

2010-05-26T18:47:47Z

moved Recursively presentable group to Recursively presented group over redirect

← Older revision	Revision as of 18:47, 26 May 2010
(No difference)

Vipul at 00:42, 19 May 2010

2010-05-19T00:42:49Z

← Older revision		Revision as of 00:42, 19 May 2010
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	==Definition==		==Definition==

	A [[group]] is said to be '''recursively presented''' if it possesses a [[recursive presentation]], viz a presentation where:		A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it possesses a [[defining ingredient::recursive presentation]], viz a [[defining ingredient::presentation of a group\|presentation]] where:

	* The number of generators is finite		# The number of generators is finite
	* The set of relations is recursively enumerable		# The set of relations is recursively enumerable

	==Relation with other properties==		==Relation with other properties==
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	===Stronger properties===		===Stronger properties===

	* [[Finite group]]		{\| class="sortable" border="1"
	* [[Finitely ~~presented~~ group]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::Finite group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|recursively presentable group\|finite group}}
			\|-
			\| [[Weaker than::Finitely presentable group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|recursively presentable group\|finitely presentable group}}
			\|}

	===Weaker properties===		===Weaker properties===

	* [[Finitely generated group]]		{\| class="sortable" border="1"
	* [[Countable group]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::Finitely generated group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finitely generated group\|recursively presentable group}}
			\|-
			\| [[Stronger than::Countable group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|countable group\|recursively presentable group}}
			\|}

Vipul: moved Recursively presented group to Recursively presentable group

2010-05-19T00:23:13Z

moved Recursively presented group to Recursively presentable group

← Older revision	Revision as of 00:23, 19 May 2010
(No difference)

Vipul: 1 revision

2008-05-08T00:07:21Z

1 revision

← Older revision	Revision as of 00:07, 8 May 2008
(No difference)

Vipul at 07:30, 31 May 2007

2007-05-31T07:30:34Z

New page

{{group property}}

==Definition==

A [[group]] is said to be '''recursively presented''' if it possesses a [[recursive presentation]], viz a presentation where:

* The number of generators is finite
* The set of relations is recursively enumerable

==Relation with other properties==

===Stronger properties===

* [[Finite group]]
* [[Finitely presented group]]

===Weaker properties===

* [[Finitely generated group]]
* [[Countable group]]

← Older revision		Revision as of 02:01, 5 November 2013
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	A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it satisfies the following equivalent conditions:		A [[group]] is said to be '''recursively presentable''' or '''recursively presented''' if it satisfies the following equivalent conditions:

	# It possesses a [[defining ingredient::recursive presentation]], ~~viz~~ a [[defining ingredient::presentation of a group\|presentation]] where the ~~number~~ of generators is ~~finite~~ and the set of relations is recursively enumerable.		# It possesses a [[defining ingredient::recursive presentation]], i.e., a [[defining ingredient::presentation of a group\|presentation]] where the set of generators is countably infinite (with an explicit enumeration) and the set of relations is recursively enumerable.
	# It possesses a [[defining ingredient::recursive presentation]], ~~viz~~ a [[defining ingredient::presentation of a group\|presentation]] where the number of generators is finite and the set of relations is recursive.		# It possesses a [[defining ingredient::recursive presentation]], i.e., a [[defining ingredient::presentation of a group\|presentation]] where the number of generators is finite and the set of relations is recursive.
	# It ~~is a [[defining ingredient::finitely generated group]] that~~ is isomorphic to a subgroup of a [[defining ingredient::finitely presented group]].		# It is isomorphic to a subgroup of a [[defining ingredient::finitely presented group]].

	==Relation with other properties==		==Relation with other properties==
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::~~Finite~~ group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|recursively presentable group\|finite group}}		\| [[Weaker than::finite group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|recursively presentable group\|finite group}}
	\|-		\|-
	\| [[Weaker than::~~Finitely presentable~~ group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|recursively ~~presentable~~ group\|finitely ~~presentable~~ group}}		\| [[Weaker than::finitely presented group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|recursively presented group\|finitely presented group}}
	\|}		\|}

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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::~~Finitely generated group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|finitely generated group\|recursively presentable group}}~~		\| [[Stronger than::countable group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|countable group\|recursively presentable group}}
	\|-
	~~\| [[Stronger than::Countable~~ group]] \|\| \|\| \|\| \|\| {{intermediate notions short\|countable group\|recursively presentable group}}
	\|}		\|}