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| Snis Pilbor |
 Posted: Fri Oct 20, 2006 3:56 pm |
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Guest
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Hello :)
Burris and Sankappanavar define an "equational class" to be a class A
of algebras such that A is precisely the class of algebras of some type
F satisfying a set Sigma of identities of type F.
My question is, is there a special name for an equational class which
is precisely the class of algebras of some type F satisfying a _finite_
set Sigma of identities of type F? Or in English, the algebras that
can be axiomatized by finitely many identities?
Thank you very much =) |
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| Arturo Magidin |
| Posted: Fri Oct 20, 2006 3:56 pm |
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Guest
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In article <1161370602.184339.216560@e3g2000cwe.googlegroups.com>,
Snis Pilbor <snispilbor@yahoo.com> wrote:
[quote:312c772c24]Hello :)
Burris and Sankappanavar define an "equational class" to be a class A
of algebras such that A is precisely the class of algebras of some type
F satisfying a set Sigma of identities of type F.
My question is, is there a special name for an equational class which
is precisely the class of algebras of some type F satisfying a _finite_
set Sigma of identities of type F? Or in English, the algebras that
can be axiomatized by finitely many identities?
Thank you very much =)
[/quote:312c772c24]
Such equational classes are said to be "finitely based".
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org |
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| Snis Pilbor |
| Posted: Fri Oct 20, 2006 9:11 pm |
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Guest
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Arturo Magidin wrote:
[quote:95c4fce122]In article <1161370602.184339.216560@e3g2000cwe.googlegroups.com>,
Snis Pilbor <snispilbor@yahoo.com> wrote:
Hello :)
Burris and Sankappanavar define an "equational class" to be a class A
of algebras such that A is precisely the class of algebras of some type
F satisfying a set Sigma of identities of type F.
My question is, is there a special name for an equational class which
is precisely the class of algebras of some type F satisfying a _finite_
set Sigma of identities of type F? Or in English, the algebras that
can be axiomatized by finitely many identities?
Thank you very much =)
Such equational classes are said to be "finitely based".
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
[/quote:95c4fce122]
Thank you very much Arturo Magidin =) As always, you are the
unchallenged master of UA :)
One more question. If we have a variety, by Birkhoff, it is an
equational class. If it happily turns out to be a finitely based
equational class, is it proper to refer to it as a "finitely based
variety"??
Thank you very much =) |
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| Arturo Magidin |
| Posted: Sat Oct 21, 2006 8:13 am |
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In article <1161389505.554263.276710@i3g2000cwc.googlegroups.com>,
Snis Pilbor <snispilbor@yahoo.com> wrote:
[quote:2278b40632]
Arturo Magidin wrote:
In article <1161370602.184339.216560@e3g2000cwe.googlegroups.com>,
Snis Pilbor <snispilbor@yahoo.com> wrote:
Hello :)
Burris and Sankappanavar define an "equational class" to be a class A
of algebras such that A is precisely the class of algebras of some type
F satisfying a set Sigma of identities of type F.
My question is, is there a special name for an equational class which
is precisely the class of algebras of some type F satisfying a _finite_
set Sigma of identities of type F? Or in English, the algebras that
can be axiomatized by finitely many identities?
Thank you very much =)
Such equational classes are said to be "finitely based".
Thank you very much Arturo Magidin =) As always, you are the
unchallenged master of UA 
[/quote:2278b40632]
I wouldn't go that far, though I do seem to know a bit more about it
than the average sci.math reader.
[quote:2278b40632]One more question. If we have a variety, by Birkhoff, it is an
equational class. If it happily turns out to be a finitely based
equational class, is it proper to refer to it as a "finitely based
variety"??
[/quote:2278b40632]
Yes; that is the usual terminology.
(For example, a conjecture that was long-standing was whether every
variety of groups was finitely based; the study of varieties of groups
was inagurated by Bernhard Neumann, and he, Hanna Neumann, and others
proved that the semigroup of varieties of groups (under variety
multiplication given by the Mal'cev product) was a cancellation
semi-group with zero and identity. It was proven that a number of
varieties were indeed finitely based, and one of the neatest theorems
was that of Powell and Oates, who proved that the identities satisfied
by a finite group are always finitely based. Alas, in the early 70s
it was shown that not every variety of groups is finitely based, and
in fact that there are uncountably many distinct non-finitely based
varieties of solvable groups).
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org |
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