Talk:Band (algebra)

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Mathematics rating: Start Class Low Priority  Field: Algebra
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[edit] Proposed merger

Since this article and Rectangular band are stubs, and since Left-zero band and Right-zero band don't even exist yet, I propose merging them all into here. A brief section would describe the two types of zero bands, and another section would incorporate the current contents of Rectangular band, as well as describe in a bit more detail why rectangular bands are direct products of left-zero and right-zero bands. The usual redirects will be set up. Also, the examples section of Semigroup, which mentions bands in two separate spots, would be appropriately adjusted.

I suspect this proposal isn't controversial, and I would have simply been been bold, but better safe than sorry. --Michael Kinyon 16:14, 31 July 2006 (UTC)

I'll go ahead and do it. Pascal.Tesson 16:17, 28 August 2006 (UTC)

Oh, OK. Thanks! Seeing that a month is plenty of time for comments, I was going to get to it next week. You just saved me some work! :-) Michael Kinyon 20:55, 28 August 2006 (UTC)

Nice work so far, Pascal. Where do normal bands, i.e., those satisfying xyzx = xzyx, fit into the lattice? (I am not a semigroup theorist, and apologize if this is basic stuff.) Michael Kinyon 07:18, 29 August 2006 (UTC)

The lattice of varieties of normal bands fits within that of regular bands. It has eight varieties forming a cube at the bottom, namely 0: x = y, 1: xy = x, 2: xy = yx, 4: xy = y, 3: zxy = zyx, 5: x = xyx, 6: xy = yxy, 7: xyzx = xzyx, with inclusions given by the bitwise ordering of 0-7 written in binary (e.g. 3 = 011 is the join of 1 = 001 and 2 = 010). The variety of normal bands itself is 7 = 111, at the top of this cube. Further up are 11: xy = xyx, 22: xy = yxy, 15: xzy = zxzy, 23: yxz = yzxz, 31: zxyz = zxzyz, which together with the upper face 2,3,6,7 of the cube form the 3x3 lattice of the nonrectangular varieties of regular bands. See e.g. the diagram at the end of http://arxiv.org/PS_cache/math/pdf/0503/0503242v3.pdf . --Vaughan Pratt (talk) 22:36, 12 May 2008 (UTC)

[edit] Rectangular band

It would be good to include some references. I have consulted briefly my (Russian translation of Clifford-Preston) where rectangular band is defined as a semigroup on I\times J with the operation given by (i,j)(k,l)=(i,l) (Clifford, Preston. Algebraicheskaja teorija polugrupp, Mir, Moskva, 1972, s. 45-46). (Maybe somewhere later they mention that this is equivalent to the definition from the article. They also mention the paper of Klein-Barmen where this notion was defined - I do not have the book here with me right now, but I remember the title was something like Verallgemeinerung des Verbandsbegriffs.) The book A. Nagy: Special Classes of Semigroups has a slightly diferrent definition from this article (xyx=x) and it mentions that this is equivalent to (i,j)(k,l)=(i,l). The author refers to Clifford-Preston for this result. --Kompik 22:38, 21 April 2007 (UTC)

You could pretty much define rectangular bands in any of these three ways (xyx =x or xyz=xz or the I times J thing). Going from one of these to the other is transparent enough. For instance, note that if xyz = xz then clearly xyx = x. Conversely, if the band satisfies xyx = x then xyz = (xzx)y(zxz) = xz(xyz)xz = xz. Pascal.Tesson 23:04, 21 April 2007 (UTC)
Yes, this one was clear. Nevetherless, it would be good to include some references WP:CITE. I have included the books I have used. --Kompik 08:37, 22 April 2007 (UTC)
I replaced the longer equation with the shorter and noted the longer parenthetically. --Vaughan Pratt (talk) 23:37, 12 May 2008 (UTC)

[edit] Stub class?

Let me preface this by saying that I know little about math. The upper reaches of my mathematical knowledge is basic differential and integral calculus. (this is not to say I don't like math, quite the opposite in fact)

That being said, I don't think this article is a stub. I have (or will have actually) taken the liberty of moving it to start class, as it seems to contain a good amount of information. I have also added the technical template, as while I respect that it is a very specific subject, but this article seems that (bolded for emphasis) if you can understand what it is saying, you already know what it is. Discuss. one/zero 17:15, 28 July 2007 (UTC)

That's a bit of an overstatement. I would expect any math undergraduate to make perfect sense of this article. It is sketchy and certainly could do a better job of discussing the contexts in which bands occur naturally but other than that it's unfair to label the article as technical. Sure, if you know nothing about abstract algebra the article is completely out of reach but then again nobody without basic mathematical training will ever read it. This happens with all advanced math articles (see things like E8 lattice or Tarski-Vaught test) and is not only acceptable but desirable to avoid unwieldy articles. Pascal.Tesson 21:25, 28 July 2007 (UTC)

[edit] Idempotent semigroup

I think that some authors use the name idempotent semigroup for the same notion as described in the article. Try google [1] [2] [3] --Kompik 11:51, 15 September 2007 (UTC)

[edit] "Zero band"?

Is this alluding to some sort of "operation written as multiplication, identity written as zero" notation, or should that be a "one"? --Tropylium (talk) 21:16, 7 December 2007 (UTC)

I added a bit about the Cayley table having constant columns, did that help? --Vaughan Pratt (talk) 23:38, 12 May 2008 (UTC)