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is listed under "Algebra-like structures". Shouldn't it be listed under "Module-like structures"? An inner product space has V × V → F, but not V × V → V. — Preceding unsigned comment added by Herbmuell ( talk • contribs) 06:23, 5 December 2013 (UTC)
are described in the article as "composite systems defined over two sets, a ring R and a free R module M. Counting the two ring operations and the single module operation, this can be viewed as a system with three binary operations." Aren't you just describing a module here? I thought an algebra always has a multiplication, which makes 4 binary operations (2 in R and 2 in M). - Apart from that, many thanks to all the people who have contributed to this very readable article. — Preceding unsigned comment added by Herbmuell ( talk • contribs) 05:36, 4 December 2013 (UTC)
Would it be correct to say that "field" and "group" are algebraic structures, or that "the field of reals" and "the group of integers under addition" are algebraic structures? The article flips back and forth between the two in a confusing fashion. (The definition at the top of the article implies that it is specific instances which are algebraic structures; but the last paragraph implies that "group" is an algebraic structure.) Cwitty 03:18, 8 Nov 2003 (UTC)
Division should be disambiguated in some way on this page. It is not correct to implie that "inverse element" and "division is always posible" are equivalent - see the short definition of group. This is concrete thinking in an abstract world. There is no such thing as division that is understood as an operator on tables, chairs, cars or bags - yet all these things can be seen as sets and with apropriate operators made part of a group. Without defining a division operator, that is. What I am trying to say is that "inverse element" is NOT to be thought of as "one divided by the element". If a is the element then 1/a is NOT the inverse element, unless we are dealing with numbers and the groups composition rule is multiplication. What is 1/a if the element a is a car? Still, it is perfectly posible to define an inverse to the element a! (213.112.153.244)
Okay, I've separated out different senses of algebraic structure. The article still needs help, though! Melchoir 02:53, 16 November 2005 (UTC)
I've done a great deal of work on this entry in recent weeks, because doing so has given me an opportunity of advance a dream I've had for decades, namely to do for math what the periodic table does for chemistry. As of this writing, the entry defines, however briefly, 54 structures. I've read that it is believed that circa 200 structures have been discussed in the literature as of the 1990s.
I draw your collective attention to the following points:
I continue working on this deeply fascinating yet frustratingly hard entry. It now touches on more than 60 structures. The entry now mentions varieties, and makes clear that the main way fields, vector spaces, and other interesting structures are not varieties is the requirement that S be nontrivial. I have recently added plain old linear algebra as a species of associative algebra with matrices as the multivectors, but am not confident that doing so is correct. Only today did I read in Birkhoff and MacLane that modules have bases, but that a module cannot have an orthonormal basis because it lacks an inner product. I've concluded that relation algebras are proper extensions of interior algebras but no printed source mentions that; am I mistaken? I've chanced on Weyl algebras but don't know how to describe them concisely. 132.181.160.42 08:17, 14 June 2006 (UTC)
for introducing me to interior algebras by slipping into this entry a mention thereof. I have shifted that mention a bit.
I am a largely self-taught amateur logician and mathematician, living in the southern hemisphere. I earn my living teaching something other than mathematics and philosophy, my true loves, for financial reasons, as you aptly put it. I also love Physics.
I have time for classical and Biblical history. I am a bad Catholic and occasional Anglican. I do not know what "maximal" and "minimal" mean in the context of the Bible. I think up tunes all the time, but do not bother writing them down. I like classical music and pre-1970 jazz. Like you, I have little time for deconstructionists and post-modernists. The world is close to the point where anyone who wants an honest education in the history of our civilisation and its ideas, will have to acquire that education on his own.
I have moved most of the content of this entry to a new entry titled List of algebraic structures and continue to edit and expand that list; it contains about 70 items. I've only recently discovered the existence of Wiki entries giving lists of mathematical topics; the value of such lists is evident.
I propose that the scope of this entry be drastically cut back to two things: a careful definition of the term "algebraic structure," and a bit of friendly talk re category theory (seen as a close and healthy rival of universal algebra). I should add that Burris of Burris and Sankappanavar (1981) tells me he is quite unhappy with the definition of algebraic structure set out in the entry. Fortunately, section 2.1 of B&S contains ample material for an improved definition. 132.181.160.42 06:16, 12 July 2006 (UTC)
To say that "a group is a monoid with unary operation, inverse, giving rise to an inverse element equal to the identity element" makes no sense. It could have meant "giving rise to an inverse element which when binop'ed upon with the original element is equal to the identity element" OR "giving rise to an inverse element and by the way the result of this unop on the identity element is equal to the identity element", but neither of those statements should be put here for different reasons (namely the first is just repeating the definition of inverse element for one thing and the second could/should go in a 'simple deductions' section)-- Netrapt 12:58, 10 February 2007 (UTC)
This article references the term positive definite, which is a disambiguation page. Please review this usage and determine which of the articles at the disambiguation is intended and adjust as appropriate. Chromaticity 02:21, 7 May 2007 (UTC)
This article seems to attract people from many different backgrounds. Among other things, it appears that it tries to play the role of an article on algebras in the sense of universal algebra, but IMHO it does so very poorly.
The first paragraph defines the general setting of the article as universal algebra. What happens if we take this seriously, encouraged by later references to universal algebra?
The second paragraph seems to claim that an algebra is the same thing as a variety (universal algebra); which is just nonsense. The confusion between individual structures/algebras and classes of structures which (i.e. the classes) are defined by a certain set of axioms continues throughout the entire article.
The explanation is that the universal algebra content was added relatively late to this article. In particular, the roots of the second paragraph are older than the first paragraph: http://en.wikipedia.org/?title=Algebraic_structure&oldid=50029042 . Note that the original setting was essentially exactly the opposite of the current one: "In higher mathematics, "algebraic structure" is a loosely-defined phrase [...]". In this original setting of classical mathematics and its fuzzy use of language, structure has a completely different meaning, and the entire article suddenly makes sense.
Perhaps someone with more Wikipedia experience can replace the first paragraph by something more appropriate? -- Hans Adler 16:48, 13 November 2007 (UTC)
Is it me or this is a contradiction? On the first line, you read, "an algebraic structure consists of one or more sets closed under one or more operations," while some lines below, it is said that the set is a degenerate algebraic structure having no operations. What exactly means degenerate? Saying a set is an algebraic structure is a strong affirmation to me. Bogdanno ( talk) 23:17, 1 July 2009 (UTC)
The intro paragraph is good. I also found the algebraic structure table and the examples useful. But that whole listing of structures is too long and replicates an existing article List of algebraic structures. The first paragraph is also strange as well because its tone is completely different from the intro paragraph. Angry bee ( talk) 22:38, 14 April 2011 (UTC)
This article seems to contradict the article Algebra over a field, on the number of binary operations. The current article places Algebra over a field under the subsection "Four binary operations." But the referenced article says, "An algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field." The way I count, multiplication, addition, and scalar multiplication are only three binary operations. What is the fourth binary operation? —Preceding unsigned comment added by 134.79.192.112 ( talk) 22:09, 13 October 2010 (UTC)
Several editors have complained about the confusing notation that uses the word "structure" in two senses:
I have now boldly reworded (hopefully) all references to this second concept. A structure in the sense of this article is now a "single" structure -- a set (possibly many-sorted) together with operations.
-- Aleph4 ( talk) 15:20, 31 October 2011 (UTC)
Hello: it recently came to the attention of the math wikiproject how much work was needed on this page. There will probably be huge changes to deal with the organization of the page, which seems to be entirely from a universal algebra standpoint, and is not accessable to laypeople. Some of the content is fine, but the major task is to avoid the "identities" "not by identities" dichotomy, and to limit repetition. There are also some really questionable organizational issues like discussing domains and fields and ringlike structures separately... that could just be me though.
I've recommended the following tasks:
I'll begin work on #2, but #1 and #3 are best explained by someone other than me :) Rschwieb ( talk) 14:18, 1 February 2012 (UTC)
Should we mention that constants such as 0 and 1 are " nullary operations", by the analogy to situation with logical connectives? Incnis Mrsi ( talk) 10:16, 12 March 2012 (UTC)
I would like to call the "no binary operation" section into question. Brief consultation with my common sense says that a set alone is not an algebraic object. I feel the same way about a pointed set. A case might be made for keeping a set with a unary operation, but we would need more evidence. Overall this little subsection seems a little shaky, so it should be talked about. Anyone have references which would lend credibility to a set with unary operation (without binary operations) as being an algebraic structure? Rschwieb ( talk) 13:17, 12 March 2012 (UTC)
From what I can tell, a lot of the confusion here comes from the fact that some editors are coming from a model theory background, where a variety is just the olde-fashioned name for an equational theory, while other editors are coming from a category theory background, and want to see everything as a class. Most readers probably don't have a background in either, and so get stumped by the statement: "an algebraic structure is a set with binary operators". But what is that set? So, for example: consider the set V={x,y,z,...} whose elements are interpreted as variables. Then (V, 0, +) would seem to be an "algebraic structure" according to this article. But other readers imagine a set Z={0,1,2,...} and so imagine that (Z,+) is an "algebraic structure". I think this confusion needs to be clarified first. It seems that Structure (mathematical logic) provides the correct definition of an algebraic structure, but this article does not seem to link to that one... or, at least, doesn't underscore that an algebraic structure is a special case of a structure.
This article also evokes other, related ideas that serve only to confuse, so for example, the relationship between a representation (say, of a group) and the abstract group itself (both have the same "algebraic structure"(?) but are not the same). Similarly, the relationship between the interpretation (model theory) and the model itself. That is, since this article implies that "algebraic structures" are like universal algebra, but with binary operators only, this really just implies that algebraic structures are just a special case of an equational theory. So there should be at least a tiny hint, in this article, that there is a model, and that there is a theory of all of the valid interpretations of the model. Otherwise, one risks, potentially, of having "algebraic structures" which have interpretations that don't fit the theory, yeah? (I'm hand-waving here, but see the point?). And... on a completely different vein, the relationship between this and standard category theory should be clarified. Anyway, all these should be given at least a nod, so as to draw a sharper distinction. linas ( talk) 15:20, 13 April 2012 (UTC)
Reply, part 2: I have three books that define algebraic structures: Paul Cohen, "Universal algebra", page 48ff Baader & Nipkow, "Term rewriting" page 44ff and Wilifred Hodges, "A shorter model theory" starting on page 2. The 2011 version of this article Some earlier version of this article, that I can no longer find, defined something that, at least vaguely, if incompletely, resembled what these books say. The current article does not do even that.
This is not an appropriate place to perform "original research" as to the definition of an "algebraic structure". Many authors have already defined this for us. We need merely reproduce what they have written. An "algebraic structure" is not supposed to be a vague, intuitive thing: its supposed to be a crisp definition; one is supposed to be able to go off and state theorems about them, etc. linas ( talk) 00:09, 14 April 2012 (UTC)
An algebraic structure on a set A is essentially a collection of finitary operations on A...
Just some suggestions:
-- J58660 ( talk) 04:06, 1 November 2012 (UTC)
A. G . Kurosch and Birkhoff use the expression Algebraic system . Bourbaki use Algebraic structure.-- Tarpuq ( talk) 17:58, 31 December 2013 (UTC)
![]() | This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
is listed under "Algebra-like structures". Shouldn't it be listed under "Module-like structures"? An inner product space has V × V → F, but not V × V → V. — Preceding unsigned comment added by Herbmuell ( talk • contribs) 06:23, 5 December 2013 (UTC)
are described in the article as "composite systems defined over two sets, a ring R and a free R module M. Counting the two ring operations and the single module operation, this can be viewed as a system with three binary operations." Aren't you just describing a module here? I thought an algebra always has a multiplication, which makes 4 binary operations (2 in R and 2 in M). - Apart from that, many thanks to all the people who have contributed to this very readable article. — Preceding unsigned comment added by Herbmuell ( talk • contribs) 05:36, 4 December 2013 (UTC)
Would it be correct to say that "field" and "group" are algebraic structures, or that "the field of reals" and "the group of integers under addition" are algebraic structures? The article flips back and forth between the two in a confusing fashion. (The definition at the top of the article implies that it is specific instances which are algebraic structures; but the last paragraph implies that "group" is an algebraic structure.) Cwitty 03:18, 8 Nov 2003 (UTC)
Division should be disambiguated in some way on this page. It is not correct to implie that "inverse element" and "division is always posible" are equivalent - see the short definition of group. This is concrete thinking in an abstract world. There is no such thing as division that is understood as an operator on tables, chairs, cars or bags - yet all these things can be seen as sets and with apropriate operators made part of a group. Without defining a division operator, that is. What I am trying to say is that "inverse element" is NOT to be thought of as "one divided by the element". If a is the element then 1/a is NOT the inverse element, unless we are dealing with numbers and the groups composition rule is multiplication. What is 1/a if the element a is a car? Still, it is perfectly posible to define an inverse to the element a! (213.112.153.244)
Okay, I've separated out different senses of algebraic structure. The article still needs help, though! Melchoir 02:53, 16 November 2005 (UTC)
I've done a great deal of work on this entry in recent weeks, because doing so has given me an opportunity of advance a dream I've had for decades, namely to do for math what the periodic table does for chemistry. As of this writing, the entry defines, however briefly, 54 structures. I've read that it is believed that circa 200 structures have been discussed in the literature as of the 1990s.
I draw your collective attention to the following points:
I continue working on this deeply fascinating yet frustratingly hard entry. It now touches on more than 60 structures. The entry now mentions varieties, and makes clear that the main way fields, vector spaces, and other interesting structures are not varieties is the requirement that S be nontrivial. I have recently added plain old linear algebra as a species of associative algebra with matrices as the multivectors, but am not confident that doing so is correct. Only today did I read in Birkhoff and MacLane that modules have bases, but that a module cannot have an orthonormal basis because it lacks an inner product. I've concluded that relation algebras are proper extensions of interior algebras but no printed source mentions that; am I mistaken? I've chanced on Weyl algebras but don't know how to describe them concisely. 132.181.160.42 08:17, 14 June 2006 (UTC)
for introducing me to interior algebras by slipping into this entry a mention thereof. I have shifted that mention a bit.
I am a largely self-taught amateur logician and mathematician, living in the southern hemisphere. I earn my living teaching something other than mathematics and philosophy, my true loves, for financial reasons, as you aptly put it. I also love Physics.
I have time for classical and Biblical history. I am a bad Catholic and occasional Anglican. I do not know what "maximal" and "minimal" mean in the context of the Bible. I think up tunes all the time, but do not bother writing them down. I like classical music and pre-1970 jazz. Like you, I have little time for deconstructionists and post-modernists. The world is close to the point where anyone who wants an honest education in the history of our civilisation and its ideas, will have to acquire that education on his own.
I have moved most of the content of this entry to a new entry titled List of algebraic structures and continue to edit and expand that list; it contains about 70 items. I've only recently discovered the existence of Wiki entries giving lists of mathematical topics; the value of such lists is evident.
I propose that the scope of this entry be drastically cut back to two things: a careful definition of the term "algebraic structure," and a bit of friendly talk re category theory (seen as a close and healthy rival of universal algebra). I should add that Burris of Burris and Sankappanavar (1981) tells me he is quite unhappy with the definition of algebraic structure set out in the entry. Fortunately, section 2.1 of B&S contains ample material for an improved definition. 132.181.160.42 06:16, 12 July 2006 (UTC)
To say that "a group is a monoid with unary operation, inverse, giving rise to an inverse element equal to the identity element" makes no sense. It could have meant "giving rise to an inverse element which when binop'ed upon with the original element is equal to the identity element" OR "giving rise to an inverse element and by the way the result of this unop on the identity element is equal to the identity element", but neither of those statements should be put here for different reasons (namely the first is just repeating the definition of inverse element for one thing and the second could/should go in a 'simple deductions' section)-- Netrapt 12:58, 10 February 2007 (UTC)
This article references the term positive definite, which is a disambiguation page. Please review this usage and determine which of the articles at the disambiguation is intended and adjust as appropriate. Chromaticity 02:21, 7 May 2007 (UTC)
This article seems to attract people from many different backgrounds. Among other things, it appears that it tries to play the role of an article on algebras in the sense of universal algebra, but IMHO it does so very poorly.
The first paragraph defines the general setting of the article as universal algebra. What happens if we take this seriously, encouraged by later references to universal algebra?
The second paragraph seems to claim that an algebra is the same thing as a variety (universal algebra); which is just nonsense. The confusion between individual structures/algebras and classes of structures which (i.e. the classes) are defined by a certain set of axioms continues throughout the entire article.
The explanation is that the universal algebra content was added relatively late to this article. In particular, the roots of the second paragraph are older than the first paragraph: http://en.wikipedia.org/?title=Algebraic_structure&oldid=50029042 . Note that the original setting was essentially exactly the opposite of the current one: "In higher mathematics, "algebraic structure" is a loosely-defined phrase [...]". In this original setting of classical mathematics and its fuzzy use of language, structure has a completely different meaning, and the entire article suddenly makes sense.
Perhaps someone with more Wikipedia experience can replace the first paragraph by something more appropriate? -- Hans Adler 16:48, 13 November 2007 (UTC)
Is it me or this is a contradiction? On the first line, you read, "an algebraic structure consists of one or more sets closed under one or more operations," while some lines below, it is said that the set is a degenerate algebraic structure having no operations. What exactly means degenerate? Saying a set is an algebraic structure is a strong affirmation to me. Bogdanno ( talk) 23:17, 1 July 2009 (UTC)
The intro paragraph is good. I also found the algebraic structure table and the examples useful. But that whole listing of structures is too long and replicates an existing article List of algebraic structures. The first paragraph is also strange as well because its tone is completely different from the intro paragraph. Angry bee ( talk) 22:38, 14 April 2011 (UTC)
This article seems to contradict the article Algebra over a field, on the number of binary operations. The current article places Algebra over a field under the subsection "Four binary operations." But the referenced article says, "An algebra over a field is a set together with operations of multiplication, addition, and scalar multiplication by elements of the field." The way I count, multiplication, addition, and scalar multiplication are only three binary operations. What is the fourth binary operation? —Preceding unsigned comment added by 134.79.192.112 ( talk) 22:09, 13 October 2010 (UTC)
Several editors have complained about the confusing notation that uses the word "structure" in two senses:
I have now boldly reworded (hopefully) all references to this second concept. A structure in the sense of this article is now a "single" structure -- a set (possibly many-sorted) together with operations.
-- Aleph4 ( talk) 15:20, 31 October 2011 (UTC)
Hello: it recently came to the attention of the math wikiproject how much work was needed on this page. There will probably be huge changes to deal with the organization of the page, which seems to be entirely from a universal algebra standpoint, and is not accessable to laypeople. Some of the content is fine, but the major task is to avoid the "identities" "not by identities" dichotomy, and to limit repetition. There are also some really questionable organizational issues like discussing domains and fields and ringlike structures separately... that could just be me though.
I've recommended the following tasks:
I'll begin work on #2, but #1 and #3 are best explained by someone other than me :) Rschwieb ( talk) 14:18, 1 February 2012 (UTC)
Should we mention that constants such as 0 and 1 are " nullary operations", by the analogy to situation with logical connectives? Incnis Mrsi ( talk) 10:16, 12 March 2012 (UTC)
I would like to call the "no binary operation" section into question. Brief consultation with my common sense says that a set alone is not an algebraic object. I feel the same way about a pointed set. A case might be made for keeping a set with a unary operation, but we would need more evidence. Overall this little subsection seems a little shaky, so it should be talked about. Anyone have references which would lend credibility to a set with unary operation (without binary operations) as being an algebraic structure? Rschwieb ( talk) 13:17, 12 March 2012 (UTC)
From what I can tell, a lot of the confusion here comes from the fact that some editors are coming from a model theory background, where a variety is just the olde-fashioned name for an equational theory, while other editors are coming from a category theory background, and want to see everything as a class. Most readers probably don't have a background in either, and so get stumped by the statement: "an algebraic structure is a set with binary operators". But what is that set? So, for example: consider the set V={x,y,z,...} whose elements are interpreted as variables. Then (V, 0, +) would seem to be an "algebraic structure" according to this article. But other readers imagine a set Z={0,1,2,...} and so imagine that (Z,+) is an "algebraic structure". I think this confusion needs to be clarified first. It seems that Structure (mathematical logic) provides the correct definition of an algebraic structure, but this article does not seem to link to that one... or, at least, doesn't underscore that an algebraic structure is a special case of a structure.
This article also evokes other, related ideas that serve only to confuse, so for example, the relationship between a representation (say, of a group) and the abstract group itself (both have the same "algebraic structure"(?) but are not the same). Similarly, the relationship between the interpretation (model theory) and the model itself. That is, since this article implies that "algebraic structures" are like universal algebra, but with binary operators only, this really just implies that algebraic structures are just a special case of an equational theory. So there should be at least a tiny hint, in this article, that there is a model, and that there is a theory of all of the valid interpretations of the model. Otherwise, one risks, potentially, of having "algebraic structures" which have interpretations that don't fit the theory, yeah? (I'm hand-waving here, but see the point?). And... on a completely different vein, the relationship between this and standard category theory should be clarified. Anyway, all these should be given at least a nod, so as to draw a sharper distinction. linas ( talk) 15:20, 13 April 2012 (UTC)
Reply, part 2: I have three books that define algebraic structures: Paul Cohen, "Universal algebra", page 48ff Baader & Nipkow, "Term rewriting" page 44ff and Wilifred Hodges, "A shorter model theory" starting on page 2. The 2011 version of this article Some earlier version of this article, that I can no longer find, defined something that, at least vaguely, if incompletely, resembled what these books say. The current article does not do even that.
This is not an appropriate place to perform "original research" as to the definition of an "algebraic structure". Many authors have already defined this for us. We need merely reproduce what they have written. An "algebraic structure" is not supposed to be a vague, intuitive thing: its supposed to be a crisp definition; one is supposed to be able to go off and state theorems about them, etc. linas ( talk) 00:09, 14 April 2012 (UTC)
An algebraic structure on a set A is essentially a collection of finitary operations on A...
Just some suggestions:
-- J58660 ( talk) 04:06, 1 November 2012 (UTC)
A. G . Kurosch and Birkhoff use the expression Algebraic system . Bourbaki use Algebraic structure.-- Tarpuq ( talk) 17:58, 31 December 2013 (UTC)