Talk:Band (algebra)

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)[reply]

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

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)[reply]

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)[reply]

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/abs/math/0503242v3 . --Vaughan Pratt (talk) 22:36, 12 May 2008 (UTC)[reply]

Normal Bands

It is obvious why the equation for a medial band implies that for a normal band.

But I do not understand how normal implies medial.

Is this a misreading of footnote 4 in the reference 3 to Yamada 1971?

See pdf p7 of 8, paper p166.

I do not understand the paper or the theory well enough to see whether there is some proof that normal implies medial.

My impression is that the footnote is intended to merely state the obvious that medial implies normal.

Unless a clear proof is referenced I propose the claim should be rephrased:

"We can also say a band S satisfying the next equation below is normal"

"This is the same equation used to define medial magmas, and so a medial band may also be called a normal band, and medial bands are examples of normal bands.[3]"

I already did this diff below to rearrange in preparation for this change but am proposing the substantive change here for others to implement:

https://wiki.riteme.site/w/index.php?title=Band_(algebra)&diff=prev&oldid=1056377984

Including an example normal band that is not medial might also be useful. ArthurD8 (talk) 13:35, 23 November 2021 (UTC)[reply]

Yamada p166 gives the correct equation for medial at the text referencing footnote 4 on p166 - xyzw = xzyw

This was incorrectly copied as xyzx = xzyx when Normal Bands were first added to the page, together with ref to Yamada 1971.

https://wiki.riteme.site/w/index.php?title=Band_(algebra)&diff=410122041&oldid=410121163

Error not noticed when letters changed to correspond to the lattice diagram:

https://wiki.riteme.site/w/index.php?title=Band_(algebra)&diff=666432206&oldid=666431650

Elaborated into two separate equations so that the second actually is the medial identity: axyb=ayxb

https://wiki.riteme.site/w/index.php?title=Band_(algebra)&diff=889418934&oldid=888814875

This was the version I rearranged. It should also be respelled as zxyw=zyxw to correspond to the Yamada reference using the letters on the lattice diagram.

So I am now reasonably certain the claim is an error.

But I still don't have a counter example. Clearly any rectangular band is both normal and medial since both identities are satisfied by the fact that any word longer than 2 letters is identical to the product of the first and last letter in a Nowhere commutative semigroup eg zxyw=zw=zyxw.

Any counter example that satisfies the normal identity with the same letter at both ends must also fail to satisfy medial entity with different letters at the start and end but it cannot be rectangular. ArthurD8 (talk) 00:36, 27 November 2021 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

I replaced the longer equation with the shorter and noted the longer parenthetically. --Vaughan Pratt (talk) 23:37, 12 May 2008 (UTC)[reply]

I added a fair bit. See my comment at Talk:Nowhere_commutative_semigroup which ends with:

"Hopefully somebody else could fix my clumsiness and should consider whether to move stuff from that section on Rectangular Bands to Nowhere commutative semigroups (or bands) or merge the two or move stuff from both to a page on Rectangular Bands." ArthurD8 (talk) 18:23, 25 November 2021 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

"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)[reply]

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

Left-zero and right-zero

The null semigroup and special classes of semigroups articles use left-zero for a semigroup in which xy = x (Cayley table has constant rows) and right-zero for a semigroup in which xy = y (Cayley table has constant columns). The current article seems to contradict this convention. Please verify. —04:49, 25 November 2009 Classicalecon

Yes, it's backwards from the usual convention in semigroup theory. I fixed it. Michael Kinyon (talk) 18:13, 12 September 2010 (UTC)[reply]

Include some examples

I have a decent grasp of abstract algebra, including semilattices, but the concept of a non-commutative band (and its many varieties) is so abstract that I can't relate it to anything familiar. Some examples of real-world applications would help a lot. Thank you! 50.123.77.139 (talk) 17:14, 20 December 2021 (UTC)[reply]

There's an example of a rectangular band in the lead image of Reversible cellular automaton, detailed later in that article. Maybe something like that could be adapted here? Rectangular bands are easy: each object has two attributes (like in the example, color and shape) and the operation combines the first attribute of its first argument and the second attribute of the second argument. —David Eppstein (talk) 17:50, 20 December 2021 (UTC)[reply]