Talk:Boolean algebra (structure)/Archive 4

717:.) I wish that were the source of that problem, but it's not; the same thing happens with any font. I spent considerable time with a number of SVG implementations trying to understand exactly what caused each to fail. The one used by Knowledge (XXG) was, by far, the worst of them all. (That's well known. The rationale is that it's fast; but it's easy to be fast if you don't have to be correct!) The thing is, it's not a matter of understanding the SVG standard, but of teasing out exactly what combination of 454:

concept that does not depend on any explicitly given list of either operations or equations. A Boolean algebra is just any model of the equational theory of the algebra of finitary operations on {0,1}. The existing definitions create the incorrect impression that "Boolean algebra" can't be defined without first choosing suitable operations like and-or-not or plus-minus-times or NOR and equations like those for a complemented distributive lattice or a Boolean ring.

2413:" shows that Absorption implies Idempotency and states that Distributivity implies Modularity, using the lattice properties. Birkhoff proves that one Distributivity law implies the other, using Commmutativity, Associativity, and Absorption. However, some confusion seems to have happened in his book, as L6 is presented before L5 (which in turn would not be needed), while L7 is not defined at all. Absorption + Complement implies Identity (x∨0 = x∨(x∧¬x) = x). 31: 497:

restrictions to a nonzero element, existence of nontrivial automorphisms, model-theoretic saturation, stuff like that. Getting everything in while keeping a tight narrative flow is going to be an extreme challenge. But as I say, let's first get the header files right. If there are no objections, I'll go ahead and do the move and dab page as outlined above, leaving "canonically defined" and "introduction" out of the picture for the moment. --

658:

diagram of this lattice, and the style of nodes with labels in ovals is I think the graphviz default. So really the only "artistic" choice made by KSmrq and copied by Fibonacci is to make the arrowheads hollow instead of solid. Is that all we're arguing about? All this is convincing me that I'm making the right choice by uploading my own diagrams PD and not having to worry about whether I'm being properly credited. —

944:

the topics discussed here. There is a history section which names names of contributors to the topic, none of which match up to the actual references. I'm tempted to rip out the whole references section, except for the Halmos book, and add a note at the top of the article stating that the material here can be found discussed in greater depth in that book. Can someone dissuade me from this course of action? —

128:

of the subject, which reflect the cultural biases of the authors. Some of the Boolean algebra expositors in the literature, e.g. Halmos, come from quite a different background than those who grew up convinced a Boolean algebra could only be defined formally as a kind of lattice. The present write-up tries to express these different cultural backgrounds as neutrally as possible.

89:(I moved the following paragraph down from the end of the (previously) second last section, where it might not have been so visible, and started a new section since the "French" section was getting long. Also probably time to archive some of the earlier sections if someone would like to volunteer to do that, this page is now up to 155 kilobytes. 203:, and if you want to learn from it (just as if you want to learn from a print encyclopedia) you have to expect that things will be presented more synthetically than they would be in a textbook, in a more "just the facts" style, and that you have to find your own approach into the material rather than having one spelled out for you in easy lessons. 413:

and OR, and that it forms a ring structure, and that whenever it's convenient to use this terminology, you just do it and don't need to say much about it. And that there are other possible choices. But is it worth a whole article? I'd just add an "alternative characterizations" section to the "structure" article, and write a paragraph or two. --

389:

exemplifies two very distinct meanings of the phrase "Boolean algebra", so we pretty much have to have two articles. But the problem remains that neither of those articles, as written by mathematicians, is going to serve the needs of people wanting to understand how to do a Boolean search on an ORION database, or things like that.

1993:. This article is about something that is almost exclusively of interest to mathematicians, which is of course not true for Boolean algebra. The similar names (Boolean algebra the theory vs. Boolean algebras the structures) is no accident. You can think of the structures as the "advanced" concept behind the "simple" algebra. 1561:) that is both accessible by a general audience and reasonable for mathematicians. Therefore it seems reasonable to cut off some of the more elementary material here that is also presented in the introductional article. I am also going to shift the focus slightly towards lattice theory, an aspect that is 923:; it looks much cleaner. Could possibly use a little more text on motivations -- that's always a delicate balance as it's tricky to avoid sliding into a textbook style, but as it stands I think even a lot of sophisticated readers are going to be scratching their heads about the why of the whole thing. -- 2255:

Do we really want to define a Boolean algebra as a "six-tuple" anyway? For my taste this is too "implementation-specific" and not very informative. Can't we say that a Boolean algebra is a set, equipped with two binary operations and one unary operation, and having two distinguished elements? It's

2134:

is all we can say given the proto-mathematical way (relative to today's standards) in which Boole wrote: "Boole's algebra was connected with the origins of both abstract algebra and symbolic logic." "Boolean algebra was perfected by Jevons, Schröder, Huntington, and others until it reached the modern

981:

Really my main objection was to the template on the article page, not to adding inline refs. I think such templates should go on article pages only when it's necessary to warn the reader that either (1) there are statements in the article that may not be accurate, or (2) there are problems that would

792:

to support SVG (which the above comments indicate it does badly) while offering no support at all for PostScript. I can readily crank out PostScript figures that don't seem to have SVG counterparts, and end up having to convert my nice resolution-independent PostScript files into bitmaps in order to

512:

OK, couldn't anyone have dropped me a note on my talk page to tell me you were planning on undoing all my careful work (to make an accessible article on Boolean logic aimed at a general audience and with computer science and electronics applications) ? This article remains readable only by PhDs, and

127:

and worked it up into an article (a) focusing just on Boolean algebras, little or no syntax, and (b) replacing the idiosyncratic bits by material reflecting the status quo as neutrally as possible. The original reason I wrote that article is that it really bugged me to see such basis-biased accounts

2966:

Alternative axiomatizations for Boolean algebras, and their history, are an interesting topic. However, the current form of the section is confusing: it doesn't say what Whitehead's axioms were, it doesn't say where the current axiomatization (used in the Definition section) fits in (is it the same

1932:

Perhaps not, since you talk about multiplication and addition, which are more appropriate to algebraic structures. I'm afraid the usual operations in a Boolean algebra do not directly translate into multiplication and addition, although you can recombine them into operations that do so correspond —

1502:

But after all, is the intended meaning of the equation not to have \emptyset rather than \varnothing? If one would change \varnothing to \emptyset, then, for LaTeX typesetting, the difference would be to have an oval with a slash (instead of a circle), and, for HTML rendering, the gain would be that

943:

for situations in which no inline citations are necessary; I don't think this article is short enough to qualify. There are ten publications listed in the references, some of which I know to be relevant but some of which look dodgy, and I have no idea which, if any, of them, are relevant to which of

745:

I believe I abandoned XML entities early on, sadly. I had wanted to be able to name a color, for example, and an entity seemed a natural approach. I switched to CSS classes, which proved more robust, though still a little delicate. The automatically generated stuff (including output of dot, gnuplot,

3009:

The "Proven properties" box is intended for readers (like me) who wish to see how certain properties are proved (click on "show" to look at a proof); axioms and properties are referred to by their labels ("Idn" etc.) in the proofs; this is the purpose of these labels. I think such proof-boxes could

2324:

Finally, I note that this decription has variables (here a, b and c). These are not mentioned explicitly as a feature of Boolean algebra (ie: as part of the six-tuple), unless they are implied by "set A". Clearly it's intended that set A could contain 0 and 1, but what is it that lets us know that

1966:

Indeed. I would be in favour of a "didactic" approach, starting with the simplistic notion of Boolean algebra as an algebra of elements that can only be TRUE or FALSE and involve operators like AND and OR, and then perhaps embark on the generalisations mathematicians are so fond of. I am aware that

1947:

Constructive criticism as to how to make it more useful for self-teaching, while not detracting from its quality as a reference work, is always welcome, and I take the remarks about multiplication and addition in that spirit. I hope I've explained why I don't think talking about multiplication and

1761:

are standard freshman-level computer science topics while the subject of this article is more something one would see in an upper-division modern algebra class for mathematics majors only. They have different audiences, neither of which would find what they are looking for in an article written for

954:

Most of the citations seem to be for the section "Axiomatics for Boolean algebras" which does have inline citations (to Huntington (1933) and Dahn (1998)). Some of the references do seem to need further investigation, such as Brown and Vranesic (2002), Cori and Lascar (2000), Mendelson (1970), and

678:

The SVG I replaced was nothing more than a copy of my PNG, and yet it did not comply with the generous license under which I contributed the PNG, which requires both attribution and use of the same license. Incidentally, I uploaded the PNG at very high resolution so it was not visibly inferior, and

2218:

Ah, the "one element" statement is really talking about the case where there are still two elements in the six-tuple, it's just that the two elements symbolized as 0 and 1 (⊥ and ⊤) are not distinct values. The point of confusion was that if there was literally "only one element" then the algebra

1706:

article. Boolean algebras are a widely studied topic with a large diffusion in the literature; modular lattices I think I just heard of today. While it's certainly plausible that a certain number of readers aren't just sure of the distinction between "logic" and a "structure", and wind up at the

412:

Oh, I haven't addressed the "canonically defined" article. Vaughan, to be honest, I can't really understand why you think this issue is so important; to me it appears to be a detail of presentation. It's nice to know that you can characterize Boolean algebras in terms of AND and XOR instead of AND

741:

Most of my SVG images are created using a text editor. I use the standard in ways that should be unremarkable; yet CSS, nested transforms, length units, markers, and various other features cause renderers to misbehave. What works and what does not is bizarre. Since each renderer breaks in its own

496:

Your stuff is good and I don't want to see it lost either. I'd like to see it merged into the structure article (hopefully it'll pack down a bit in transition, though; there's lots of other stuff I'd like to see go into the structure article too, like atomicity, atomlessness, quotients by ideals,

134:

is concerned that I'm writing too much about the subject. If that's a universally shared concern, I guess the message would have to be that Knowledge (XXG) is aiming to avoid encyclopedic articles (!). Is that in fact the case, or is the brevity of many articles more a reflection of the limited

2320:

Then instead of the obtuse statement "A Boolean algebra with only one element is called a trivial Boolean algebra", we could have something more concrete like "The most familiar Boolean algebra allows two distinct values for elements of A, and results. If instead the two values are the same, the

657:

Superficially, the resemblance is much closer than one would expect from an independent re-drawing, but if you pay attention to KSMrq's drawing it says it was created by graphviz, and it's reasonable to believe that the same is true for Fibonacci. The node layout is an obvious choice for a Hasse

388:

Actually, having allowed that possibility into my internal discussion, I find myself more and more drawn to the idea that it may be the only practical solution here, much as I would dislike to see it become commonplace. The structure-v-equational-logic distinction is, to me, extremely clear, and

453:

On the "canonically defined" article, the point was not that rings are an alternative to lattices in the definition of "Boolean algebra", nor that there are yet other choices of operations on which to base a definition such as NOR, but rather that there's a simple yet rigorous definition of the

1584:

I've replaced a lot of the formulas with math notation, primarily because Firefox 3 (unfortunately) does not render the ∨ (logical and) character (I said IE in my edit summary by mistake; I was actually using Firefox at the time). It may also not be supported by IE6 but I haven't checked. All

206:

As regards the "canonically defined" article -- I would add a section to the algebraic structure article, called "alternative characterizations" or some such, and mention there the definition with the XOR operator. I never completely digested the article (just got the general "too essay-like"

731:

I've been using SVG as generated by a recent version of Adobe Illustrator, with only a few problems: (1) it defines and uses xml entities, which must be expanded before Knowledge (XXG)'s renderer can handle it, and (2) it uses idiosyncratic font names which must be replaced by something more

2344:

I noticed that the text claims that the finite or cofinite subsets of an arbitary set S is a Boolean algebra, but that it didn't define the operations. Obviously meet and join are union and intersection, but what is the negation operation? It certainly can't be set-theoretic complement.

536:

I've done the move and started the very tedious process of fixing all the links. I'll come back and do some more later; sooner or later we'll get them done. Anyone who wants to help out should note that some of the links should not be to either of the "Boolean algebra" articles, but to

513:

now that article is similarly readable only by PhDs. (The simple illustrations with Venn diagrams were stripped out, and mathematician-speak was put in, for example.) I have now put the general audience article back at "Boolean logic" and left the PhD level article at the new name of

402:

itself becomes a strict dab page, not "dab plus". Anything that might have gone in "dab plus" should be worked into one of the three real articles. The naming of the "Introduction" article addresses my concern about linking (it won't be accidentally linked to by authors intending the

2967:

as Whitehead's?); it's not clear why it says "even proving the associativity laws"; etc. The "Proven properties" box seems unnecessary; showing the dual forms of every equation seems unnecessary; labelling the axioms with Idn/Cmm/etc. seems unnecessary; repeating the content of the

1892:

As usual, Knowledge (XXG) pages on mathematical topics are virtually useless to anyone who actually needs to use them. If you can understand this technical gobbledygook, then you already know mathematics well enough, probably, to have little need for a wikipedia page on this topic.

259:

Even though Knowledge (XXG) is not a textbook, there are too many maths articles that are neither well written nor readily accessible even to mathematicians. I'd say, it is generally better to err on the side of too much well written and accessible material, than too little.

151:(the latter name definitely needs to disappear one way or another) and comment on suitability for respectively the symbolic and abstract articles -- too fast, too slow, too long, too short, too many figures, too few figures, not enough references to Chad Boole, etc. etc. 2360:

the article (which could be clearer on this point) asserts that the collection of subsets that are finite or cofinite forms a Boolean algebra. That is, all of the finite subsets together with all of the cofinite subsets. So complementation is indeed the negation.

3010:

be an(other) advantage web-base encyclopedias have over paper based ones: they do not take considerable space when collapsed (by default), nor do they distract a reader who is not interested in them; but then can provide valuable information to an interested reader.

703:

Your comparison example uses a nonstandard font which is not linked to nor defined within the svg (Nimbus-Roman). Perhaps that is related to some of the rendering difficulties. Though I don't understand some other issues (the huge curve width and missing axes).

142:

into an article on the elementary or symbolic aspects and a separate one on Boolean algebras as abstract structures, I'm happy to do what it takes to move that forwards. Meanwhile let me encourage those who'd like to see such a split take a look at each of

3005:

The text says "even proving the associativity laws", because this is unusual, they are normally required as axioms. I was really impressed when I read Huntington's 1904 article. Maybe rephrasing as "which doesn't even require associativity axioms" is more

3079:

Although of course Huntington is an important figure in the axiomatization of Boolean algebra, I don't get the impression that the Huntington axiomatization is considered as a core part of the subject today. I am not an expert -- but for example,

2309:

Thanks for your responses! At this point, I understand what this is trying to say. But the more I look at it, the more I think that the "is a six-tuple" is a red herring -- why not just say "A Boolean algebra has six features: a set A..." etc.

2939:

The definition we should be using here is the "standard" one (whatever that is). I suspect that such a definition probably includes Absorption. In any case whatever definition we use should cite a standard reference work not a research paper.

938:

I put a citation cleanup tag on this article and was immediately reverted. This is not an instance of inline-citation purity: I don't want a citation on every sentence. But this article, as it stands now, has zero inline citations. None. See

329:

respectively? The above discussion suggests that this is the preferred choice, and it does have the advantage of being more or less self-explanatory. My own preference would be to track the standard distinction drawn in sections 2 and 3 of

2031:

No, it's not about "advanced topics in Boolean algebra" in general; that's a misunderstanding. (I don't think it's really what Hans meant either.) This article is about a particular sort of structure, in the mathematical sense of the word

721:

input features causes the renderer to misbehave. Not only is it tedious, it's really annoying! Eventually I decided it was not a good use of my time, so I uploaded a PNG. I shall continue to do so until the rendering software is fixed.

292:(or whatever final name is agreed on) in order to separate the question of article name from the details of what to merge and what to displace, dealing with the latter first. Hopefully people running across the merge tags will notice 2325:

the set A can contain variables? If variables are a sort of metafeature of set description, and actually the set can contain only values, then how does a set with more than two members (0 and 1) arise to correspond to a, b and c?

1707:

wrong article, these data themselves don't appear to indicate that it's a huge problem. If there were no misnavigation at all, I would frankly expect the discrepancy between the two articles to be greater than this. --

793:

turn them into figures for Knowledge (XXG) articles. There is no translator from PostScript to SVG. Why can't Knowledge (XXG) at least look into rendering PostScript? Ghostscript is readily available. SVG sucks. --

2196:

The usual convention in mathematics is that, unless the text says that things have to be distinct, they might be identical. Thus "two elements" can be the same element twice; "two distinct elements" could not. — Carl

982:

bring Knowledge (XXG) into disrepute, such as bad grammar/spelling errors or unclear writing. When it's only editors that need to be informed, the tag ought to go on the talk page, so as not to uglify the article. --

761:

renderer seems far superior to everything except ASV6β. It's built on Java, which makes it cross-platform (good) and a bit slow (not so good). Rapidly improving for rendering, and already nicest for GUI editing, is

643:

It's a near-identical re-drawing of an image made by KSmrq, who was not credited (and the license is different from KSmrq's license). KSmrq objected. I don't really know whether this is the sort of thing actually

475:(structure) article is fine as it stands. I do feel however that as a self-contained article it has a lot of useful material and insights about Boolean algebra that is hard to find elsewhere. Merging it in with 2381:

I'm wondering about the different axiomatizations of Boolean algebras found in the literature and started the synopsis shown below (additional columns are welcome!). A "." in a cell means that the law of the row

2985:

I'll try to find out what Whitehead's axioms were, and then supplement them, and relate them to the current axiomatization (by Davey+Priestley). Do you suggest to have a different axiomatization in the article?

1177: 628:

has been replaced by an inferior png substitute image? KSmrq says "illegal image" and Cronholm144 says "not attributed" but to me it appears as properly licenced under GFDL and CC, attributed as "own work" by

3013:

I included all dual formulas as I found this in a similar way in section "Definition" of the article; they could be omitted in the boxes, adding an appropriate remark on dual forms (and their labels, as e.g.

1004:

The "investigation" I mentioned is just to make sure that the books discuss boolean algebras (qua lattices); the title of Stoll's book isn't a clear indication. I'll strike it out above since it does. — Carl

1222: 787:

Support for SVG seems minimal everywhere. Given this, and given the huge support worldwide for the past 25 years for PostScript, both embedded and not, I find it extremely frustrating that Knowledge (XXG)

1120: 1654:

Boolean algebras are basically the same objects as boolean lattices, and the homomorphisms are also the same. By moving to the unique name we can avoid the unfortunate name clash. (This is recommended by

1493: 1533:\emptyset is rather ugly. The best solution would be for the developers to fix the code so that \varnothing is converted to HTML in the same way as \emptyset. That way it would work for all articles. -- 675:

I decided rather than waiting to see if the attribution and licensing problems would be fixed, I would simply upload my own original SVG. So feel free now to use my SVG, which has no copyright problems.

1896:

those of us who are just learning this stuff can't even decipher the page. I hope contributors to wikipedia pages do not actually teach students using the same way they write Knowledge (XXG) pages.

3203:

It would be nice to simply state that Boolean algebras and Boolean rings are isomorphic in the lead section, if such an isomorphism exists. Does anyone have such a source for the footnote citation?

2051:

I don't exactly recall the arguments by which this one won out over those. I'm not enthusiastic about relitigating the issue, but I could live with any of those three titles if others preferred. --

854:

However there are still about 140 links to the disambig page (from talk, wikipedia, and user pages). I don't think it's really policy to change most of those. This is a little unfortunate because

517:. I suggest you merge the two "Boolean algebra" articles together (or leave them separate, if you prefer) and leave "Boolean logic" alone. If you wish, I am willing to rename "Boolean logic" to 3178: 1899:

How about making a list that shows how to translate each of the operations into standard mathematical operations-- like multiplication or addition, without using technical mathematical jargon?

1503:\emptyset has a rendering which is ∅. So would somebody oppose to a change of \varnothing to \emptyset so that the readers that use HTML as much as possible in rendering see the HTML formula 1447: 1356: 207:

impression), so I'm not sure what else from it ought to be merged. You do write well and include a lot of good information, so probably there is more material that should come over. --

2017:

Right. I was mistaken. Would you agree that the title of the lemma is confusing? Adding the word "structure" in brackets is not very clarifying. Perhaps "advanced topics" is clearer.

364:

article is about algebra treated abstractly, not about algebras in the sense of modules plus multiplication. The "structure" article should be analogous to the latter, not the former.

732:

standard. Neither issue is hard to handle; I use a small script to do the replacements. But it was, I agree, very frustrating before I figured out what was causing the problems. —

1257: 1810:, then revived it in its original problematic form without bothering to address any of the many criticisms made of it (and apparently not understanding them either). Ideally 107:

into a single suitably named article about the purely syntactic side of Boolean algebra I won't object. I don't mind doing it myself if no one else minds, just let me know.

1075: 1850:. If you have something new to add, would be better to comment there. If possible, contribute things that tend to reduce rather than increase the number of open issues. -- 1324: 991:

I'm familiar with the Stoll book. It has a nice chapter on Boolean algebras. It seems to be a perfectly reasonable reference to me. What seems to be the problem with it?

1826:, but until the debate over the appropriate organization of the main article(s) is settled there isn't a good case for implementing the redirect, so in the meantime 1291: 196:

exactly. It's that it's too essay-like, too pedagogical, and also I was concerned that implementation code was being written before we'd agreed on the header files.

778:

have some SVG support, but leave much to be desired. Signs are that Adobe itself is moving away from SVG, so we can only hope everyone else picks up the slack. --

1940:

Regarding your remarks about teaching: Knowledge (XXG)'s purpose is not in fact to teach. It's a reference work, not a textbook. It's a valuable resource for

228:

or the other way around. Basically, create anywhere by any means a text that is the desired merged text (or a reasonable approximation). Then, if not already at

2241:

No, in that case it would be a six-tuple, two of whose entries are equal. The axioms would not change, but the symbols 0 and 1 would denote the same value. --

1518:

PS: I already changed \lnot \to neg for the same purpose (to the exception that lnot and neg are LaTeX synonymous while \varnothing and \emptyset are not).

742:

way, dodging the bugs is a very unpleasant exercise. (In sympathy for the implementors, I should say that the standard itself does not make their job easy.)

398:

So I'm now leaning to a three-article-plus-dab-page model: The two articles based on the structure/logic distinction, the "Introduction to..." article, and

199:

The thing to keep in mind here is that Knowledge (XXG) is a reference work. It's certainly a learning tool, but only secondarily; primarily it's a place to

940: 682:

At one time it was not possible to upload SVG files. Even today, perfectly correct SVG files may be improperly rendered by the defective SVG rasterizer (

69: 64: 59: 559:

I have this nagging suspicion that the typical innocent reader wanting to read up on "Boolean algebra", having read the disambiguating definitions at

2416:

I'm unable to see whether all axiomatizations are equivalent, but I'd like to know that. Maybe, at least the main variants could be mentioned in the

367:

A complicating factor: There's a current active discussion about "introduction to..."-type articles, triggered by the featured-article candidacy of

625: 1456:

To respect the editor choice of having \varnothing in the equation rather than, say \emptyset, I tried to replace \varnothing by \O (which is in

463:

But I agree in retrospect that that point alone doesn't justify a separate article. However most of what's in my article isn't in the existing

648:

by copyright, but given that KSmrq objects and the costs of meeting his conditions are small, I'd encourage Fibonacci to go ahead and do it. --

360:

I definitely prefer the structure/logic distinction. The distinction should be based on the subject matter, not on the approach. Note that the

428:

I don't have any strong disagreement with any of the above. So it sounds like whoever's offering to do the dab page should at that time move

858:

is hard to interpret. Does anyone know whether there's any way to filter the "what links here" tool, so that only mainspace links show up? --

3177:

I wonder how that is. Even if it is indeed so, wouldn't it be simpler to attribute this corollary to commutativity and absorption instead?

1648: 468: 148: 124: 103:

so as to make it purely about syntax, and added some figures. At this point I'll stop working on it. If someone sees fit to merge it and

3095:. R. Padmanabhan and S. Rudeanu certainly discuss it, but that's a book-length treatment -- which also mentions that Huntington proposed 1967:

for a genuine mathematician, the only reposible route is the other way round, but, indeed, Knowledge (XXG) primarily targets non-experts.

1557:, I am going to remove some redundancy from this one. Vaughan Pratt has done an excellent job of writing an introduction (under the title 749:

Only one renderer supports declarative animation and most static features well, and that's the Adobe 6 beta, only available for Windows. (

588:

Should we, perhaps, expand the first definition by adding "used for describing logic circuits and formulating search engine queries"? --

2346: 467:

article. However as an attempt to improve overall on the status quo it didn't succeed the way I'd hoped it might. I'm happy to remove

2889: 1125: 3182: 1906: 368: 2090: 746:

and so on) seems to mostly avoid all the features that lend convenience and economy, perhaps as a response to the broken renderers.

1182: 1080: 518: 47: 17: 1468: 1823: 372: 2036:, which is itself not really understood by the general public. When we were originally debating it, other possibilities were 1365: 1558: 563:, will have no clue as to which of the two to navigate to. For reference, here is the current wording of the disambiguation: 237: 691: 687: 3034: 3030: 2429: 2417: 1799: 1794:

had too many problems to fix incrementally, and I rewrote it from scratch. The result was then subsequently moved to

579: 433: 322: 2093:) that has oodles of axiomatizations, but it's not organized historically, so it's hard to find the answer in there. 1738:

has the sensible approach of not forking the article based on the approach (order vs algebra). That is lacking here.

375:

article, while maintaining the content-based logic/structure distinction in the articles aimed at mathematicians. --

3042: 2994: 2957: 2930: 2897: 2858: 2219:

would consist of only a five-tuple, and the identity and complements axioms would have to be written differently.

894: 479:

might make sense if it were clear how to do the merging without breaking up the continuity of the two articles. --

281: 221: 182: 174: 144: 120: 100: 38: 1625:, but the redirect has a history and Carl (whom I asked) is a bit more cautious than I am. Here is the rationale: 343: 1644:; renaming makes it even clearer (especially to Google) that the article is not about elementary school material. 906: 347: 1779: 1407: 1329: 2274:

No objection from me. Though it doesn't solve Gwideman's confusion, it seems like a completely sound idea. --

1465:

as a circle with a slash (i.e. ∅, i.e. ∅) but it interprets it for LaTeX rendering as an oval (i.e.

2952:

Ok, I adapted the axiomatization to Davey.Priestley.1990, but added a remark (and 2 boxes) about Huntington.

1495:, as if it actually were a \emptyset), so that \O as an alternative for \varnothing does not work as far as 2366: 2350: 2279: 2246: 2117: 1926: 1910: 1873: 1835: 1819: 1795: 1763: 1754: 1633: 1523: 798: 573: 514: 441: 371:. I personally don't like them. However, if they do become standard, then it would be reasonable to have an 326: 289: 3025:

The text on the Robbins algebra has not been written by me; it is in fact a lot of repetition. Maybe, the "

1902: 138:

If there's general support for incorporating some part of each of my two articles in the proposed split of

2925:", where Huntington presents his "First axiom set"; the latter seems to be identical with the WP axioms. 2079:

Ok, so who gave that? There are ton of engineering books saying that Boole discovered the Boolean algebra

1400:

My user preferences are set so to render math formulas as HTML as much as possible. This provides a nice

3214: 3038: 2990: 2953: 2944: 2926: 2893: 2876: 2854: 2292: 2186: 1725: 1362: 1026: 995: 2317:

which the elements of A are allowed to take on, and which are allowed as the results of expressions?

1647:

We currently have no less than 3 articles named "boolean algebra (something or other)" and one article

3220: 3186: 3108: 3046: 2998: 2980: 2961: 2947: 2934: 2901: 2879: 2862: 2370: 2354: 2334: 2295: 2283: 2269: 2250: 2228: 2209: 2189: 2172: 2144: 2121: 2102: 2060: 2026: 2000: 1976: 1957: 1914: 1877: 1859: 1839: 1783: 1747: 1728: 1716: 1696: 1673: 1615: 1589: 1574: 1542: 1527: 1453:

image. The reason is that the math environment does not recognize \varnothing as HTML interpretable.

1390: 1029: 1017: 998: 986: 975: 948: 927: 913: 881: 872: 862: 802: 782: 736: 726: 708: 698: 662: 652: 637: 611: 602: 592: 549: 525: 501: 483: 417: 379: 354: 304: 268: 248: 211: 155: 111: 93: 3139:

It follows from the last three pairs of axioms above (identity, distributivity and complements) that

2109: 1997: 1994: 1847: 1669: 1665: 1570: 1566: 2890:

http://www.ams.org/journals/tran/1933-035-01/S0002-9947-1933-1501684-X/S0002-9947-1933-1501684-X.pdf

1230: 3204: 2326: 2265: 2220: 2164: 2056: 1953: 1855: 1775: 1712: 1692: 945: 869: 733: 705: 659: 634: 350:

as being more in line with standard usage. Are there strong feelings either way at this point? --

165:-- it was split quite some time ago, and now does not treat equational logic. The article that was 1048: 3104: 2976: 2362: 2330: 2275: 2242: 2224: 2168: 2140: 2113: 2098: 1869: 1831: 1743: 1519: 1385: 1297: 910: 898: 794: 480: 351: 335: 301: 297: 236:

page is not deleted (something only admins can do), its edit history will be preserved. See also

152: 108: 90: 1656: 1585:

formulas should still render as text if you have the right option selected in your preferences.

1753:

The problem is not so much order vs algebra, it's more that (at least in US college curricula)

855: 630: 232:, paste and copy the merged text there wholesale, superseding the old contents. As long as the 3093: 2087: 2022: 1972: 1538: 753:

while you can; it will soon disappear permanently.) Except for lacking animation (which is in

608: 589: 264: 244: 3085: 3209: 2941: 2873: 2289: 2183: 2131: 1722: 1023: 992: 920: 890: 848: 840: 836: 832: 542: 538: 521:, in order to make it perfectly clear that this is not an article for or by mathematicians. 361: 339: 1622: 3193: 3026: 2968: 2410: 2313:

Also, would there anything wrong with specifically saying that the "two elements" are the

1990: 1815: 1703: 1641: 1629: 1600: 1554: 1262: 828: 775: 560: 476: 472: 464: 429: 399: 314: 162: 139: 2157:

In the Definition section: "A Boolean algebra is a six-tuple and two elements 0 and 1 "

2127: 847:

is currently protected so I couldn't get to that one, and I didn't know how to deal with

220:

Pragmatically there is no difference in the editing needed in either direction, whether

2261: 2052: 1949: 1851: 1708: 1702:

I would also add that I'm less than convinced by the data from the comparison with the

1688: 983: 924: 902: 878: 859: 844: 649: 599: 546: 498: 414: 376: 293: 208: 131: 3100: 2972: 2204: 2136: 2094: 1922: 1827: 1811: 1803: 1791: 1771: 1767: 1758: 1739: 1637: 1610: 1586: 1460:

a slightly larger form of \varnothing). But it did not work because the <math: -->

1380: 1012: 970: 437: 318: 310: 285: 233: 229: 225: 186: 178: 170: 104: 2321:

axioms still hold in a trivial way, and this is termed a "trivial Boolean Algebra".

1683:

is the standard name. It would be more convenient for us if the standard name were

750: 3197: 2018: 1968: 1934: 1534: 754: 277: 261: 241: 2909: 1375:

It would be nice if possible (interesting) properties of boolean algebras such as

1640:. Until I fixed this today, the article was the first on the disambiguation page 1179:

is not tautological (it is in fact a contradiction). The correct dual formula is

313:

is now completed. For the purposes of a dab page, is everyone in agreement that

181:

has a long history with many authors, so for the sake of preserving the history,

3081: 1807: 1735: 963:

seems like a fine reference to include even if it is not directly cited. — Carl

522: 46:

If you wish to start a new discussion or revive an old one, please do so on the

1376: 779: 723: 695: 1515:

images it would be rendered with a slashed oval instead of a slashed circle.

1404:

rendering for all equations of the article to the exception of the equation

714: 582:, a type of algebraic structure modeling the aforementioned logical calculus 1632:. To me this suggests that most of these hits are from readers looking for 831:

to one of the articles (or in a few cases to some different article, like

2200: 1606: 1008: 966: 767: 99:

Meanwhile I've removed the material about Boolean algebras (plural) from

3022:). On the other hand, they use space only if the box has been expanded. 771: 683: 331: 897:

based on work of Jónsson and Tsinakis, and also a complete rewrite of

827:

I have completed, for now, the task of sorting the mainspace links to

770:

editor is based on Batik, but is somewhat disappointing. The browsers

1553:

As a first step in fighting the current proliferation of articles on

1511:) = ∅ instead of a PNG image, knowing that for readers that rely on 2386:

used by the axiomatiation of the column, all other entries mean it

2084: 1457: 1172:{\displaystyle (A\geq B)\leftrightarrow ({\overline {A}}\wedge B)} 694:.) I'm a big proponent of SVG, and use it as much as possible. -- 471:

from Knowledge (XXG) altogether if people feel that the existing

135:

resources of their authors than of a deliberately imposed style?

1462: 1401: 758: 2837:. Cambridge Mathematical Textbooks. Cambridge University Press. 2260:

much more intuitive, but it's at least a small improvement. --

1770:

could be merged, though, as they're on similar topics and the

1512: 1496: 1450: 25: 1217:{\displaystyle A<B\Leftrightarrow {\overline {A}}\wedge B} 856:

http://en.wikipedia.org/Special:Whatlinkshere/Boolean_algebra

169:

to treat the equational logic, or at least so I thought, was

3029:" article should be merged and redirected into the section " 2160:

Later: "A Boolean algebra with only one element is called "

1379:

are mentioned in the article, and what they mean of course.

1115:{\displaystyle A\leq B\Leftrightarrow {\overline {A}}\vee B} 192:

As for the "too much" thing: It's not that it's too much in

1488:{\displaystyle \emptyset {\mbox{ instead of }}\varnothing } 161:

So first of all, there's no proposal to split the existing

3092:

for an example of how various axiomatizations are covered

1687:, but it isn't, and there's nothing we can do about it. -- 893:

article is now revised. It makes use of a new article on

288:. I did that in preference to proposing to merge both to 2910:"Sets of Independent Postulates for the Algebra of Logic" 1628:

This article had 7686 hits in March, compared to 595 for

3133:

Currently, the definition section contains these lines:

2868:

Hmm ... absorption was included in the definition until

2178:

As the article says, some authors require 0 and 1 to be

713:

The font is a standard font — for Knowledge (XXG). (See

280:. I've started the merge process with a mergeto tag in 2869: 2112:

of the Boolean algebra talk page gives some answers. --

763: 2390:

used, and give some reference (rank of appearance for

1944:-teaching; this is a subtle but important distinction. 1476: 1045:

I have my doubts as to the duality of the connectives

3192:

Citation for the existence of an isomorphism between

1599:

Does anyone object to moving (renaming) this page to

1471: 1410: 1396:

About the math rendering of boolean algebra equations

1332: 1300: 1266: 1233: 1185: 1128: 1122:, i.e., is a tautology, the supposedly dual formula 1083: 1051: 576:, a logical calculus applying to truth values or sets 2420:. In particular: isn't Absorption missing for sure? 1948:

addition would actually be all that helpful here. --

1461:

environment is bugged: it correctly interpret \O in

2126:Thanks, that is quite in-depth. However, given the 1227:It is my impression that the dual connectives are: 342:, which for Boolean algebra would seem to indicate 238:

Help:Merging and moving pages#Performing the merger

2849:. Colloquium Publications. Vol. 25. Am. Math. Soc. 1830:continues to exist as a blot on the landscape. -- 1487: 1441: 1350: 1318: 1284: 1251: 1216: 1171: 1114: 1069: 626:commons:Image:Hasse diagram of powerset of 3.svg 901:, which connects up also with a new article on 1888:The lack of usefulness of this page in general 941:Knowledge (XXG):Scientific citation guidelines 686:) Knowledge (XXG) has chosen to use. (Compare 624:Can someone please explain to me clearly why 8: 3035:Boolean_algebra_(structure)#Robbins algebra 851:-- I think Vaughan may be looking into it. 1921:Hi 72. Is it possible you're looking for 1442:{\displaystyle A\cap (A^{C})=\varnothing } 1351:{\displaystyle (\oplus ,\leftrightarrow )} 2108:First comment from Vaughan Pratt in this 1603:, which currently redirects here? — Carl 1565:a bit underdevelopped in all articles. -- 1475: 1470: 1424: 1409: 1331: 1299: 1265: 1232: 1198: 1184: 1150: 1127: 1096: 1082: 1050: 2422: 2042:Boolean algebra (mathematical structure) 1989:I guess you are looking for the article 620:Image:Hasse diagram of powerset of 3.svg 1482: 1436: 3179:2A02:8109:9340:42C:8410:961C:6FA1:7170 3031:Boolean_algebra_(structure)#Axiomatics 679:the unlicensed SVG was the substitute. 44:Do not edit the contents of this page. 2038:Boolean algebra (algebraic structure) 1864:Excellent advice, Trovatore. I hope 1814:would be merely a redirect to one of 1798:so as to pair it more naturally with 1041:A comment on 'Principles of duality': 1040: 7: 1649:boolean algebras canonically defined 469:Boolean algebras canonically defined 149:Boolean algebras canonically defined 125:Boolean algebras canonically defined 2888:Thanks! I found the cited paper at 2833:B.A. Davey, H.A. Priestley (1990). 919:Vaughan, thanks for the rewrite at 2835:Introduction to Lattices and Order 1472: 369:Introduction to general relativity 24: 2769:x ≤ z ⇒ x ∨ (y ∧ z) = (x ∨ y) ∧ z 532:Move completed; links need fixing 177:does a better job of it. However 1846:There's a massive discussion on 1595:Possible move to Boolean lattice 519:Boolean logic (computer science) 29: 18:Talk:Boolean algebra (structure) 2557:x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) 2540:x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) 1824:Introduction to Boolean algebra 1252:{\displaystyle (\vee ,\wedge )} 373:Introduction to Boolean algebra 3033:", or even an own subsection " 2086:. There is also an monograph ( 2083:, but that is contradicted by 1802:. The author of the original 1559:Boolean algebra (introduction) 1430: 1417: 1345: 1342: 1333: 1313: 1301: 1279: 1267: 1246: 1234: 1195: 1166: 1147: 1144: 1141: 1129: 1093: 1064: 1052: 526:03:23, 25 September 2007 (UTC) 1: 3187:01:05, 11 February 2014 (UTC) 3109:20:40, 7 September 2013 (UTC) 3047:19:21, 7 September 2013 (UTC) 2999:20:49, 7 September 2013 (UTC) 2981:17:50, 7 September 2013 (UTC) 2962:14:38, 7 September 2013 (UTC) 2948:17:25, 6 September 2013 (UTC) 2935:08:15, 6 September 2013 (UTC) 2908:Edward V. Huntington (1904). 2902:07:11, 6 September 2013 (UTC) 2880:21:10, 5 September 2013 (UTC) 2863:20:41, 5 September 2013 (UTC) 2371:01:21, 28 December 2012 (UTC) 2355:23:25, 27 December 2012 (UTC) 1762:the other audience. Possibly 1543:12:45, 31 December 2007 (UTC) 1528:12:14, 31 December 2007 (UTC) 1391:12:36, 6 September 2007 (UTC) 1070:{\displaystyle (\leq ,\geq )} 3084:'s textbook relegates it to 1590:20:34, 7 December 2008 (UTC) 1575:22:04, 27 January 2008 (UTC) 1319:{\displaystyle (\leq ,<)} 1203: 1155: 1101: 961:Handbook of Boolean Algebras 868:I handled the Peirce edit. — 803:09:03, 28 January 2008 (UTC) 344:Boolean algebra (elementary) 296:'s pointer at the bottom of 2398:, page number:law name for 2377:Synopsis of axiomatizations 2335:23:24, 31 August 2012 (UTC) 2296:21:36, 28 August 2012 (UTC) 2284:13:41, 28 August 2012 (UTC) 2270:04:50, 28 August 2012 (UTC) 2251:03:30, 28 August 2012 (UTC) 2229:23:35, 27 August 2012 (UTC) 2210:01:44, 27 August 2012 (UTC) 2190:00:38, 27 August 2012 (UTC) 2173:00:16, 27 August 2012 (UTC) 2153:Contradiction in definition 1958:21:35, 3 October 2009 (UTC) 1915:20:50, 3 October 2009 (UTC) 1800:Boolean algebra (structure) 1679:I do object to this idea. 1366:07:52, 28 August 2007 (UTC) 1285:{\displaystyle (\geq ,: --> 928:21:37, 17 August 2007 (UTC) 914:23:58, 16 August 2007 (UTC) 895:residuated Boolean algebras 580:Boolean algebra (structure) 434:Boolean algebra (structure) 323:Boolean algebra (structure) 284:and a mating mergefrom in 3236: 2906:It seems I'd better read " 2145:13:26, 27 March 2011 (UTC) 2122:12:42, 27 March 2011 (UTC) 2103:12:15, 27 March 2011 (UTC) 1878:19:53, 25 March 2011 (UTC) 1860:01:21, 25 March 2011 (UTC) 1840:19:53, 25 March 2011 (UTC) 1790:It was clear in 2006 that 1784:01:13, 25 March 2011 (UTC) 1748:23:06, 24 March 2011 (UTC) 1729:04:39, 26 April 2009 (UTC) 1717:01:33, 22 April 2009 (UTC) 1697:20:41, 21 April 2009 (UTC) 1674:19:40, 21 April 2009 (UTC) 1616:19:08, 21 April 2009 (UTC) 1030:04:46, 6 August 2007 (UTC) 1018:00:15, 6 August 2007 (UTC) 999:22:00, 5 August 2007 (UTC) 987:20:30, 5 August 2007 (UTC) 976:20:27, 5 August 2007 (UTC) 949:20:16, 5 August 2007 (UTC) 907:residuated Kleene algebras 348:Boolean algebra (abstract) 282:Elementary Boolean algebra 222:Elementary Boolean algebra 183:elementary Boolean algebra 175:elementary Boolean algebra 145:Elementary Boolean algebra 121:Elementary Boolean algebra 101:Elementary Boolean algebra 3018:is used in the proof of H 2989:((done: Whitehead.1898)) 2845:Garrett Birkhoff (1967). 2785: 2761: 2737: 2696: 2655: 2614: 2573: 2532: 2516:x ∧ (y ∧ z) = (x ∧ y) ∧ z 2499:x ∨ (y ∨ z) = (x ∨ y) ∨ z 2491: 2450: 882:19:07, 25 July 2007 (UTC) 873:19:06, 25 July 2007 (UTC) 863:18:59, 25 July 2007 (UTC) 783:21:43, 25 July 2007 (UTC) 737:19:15, 25 July 2007 (UTC) 727:19:56, 24 July 2007 (UTC) 709:18:06, 24 July 2007 (UTC) 699:17:59, 24 July 2007 (UTC) 663:17:48, 24 July 2007 (UTC) 653:17:27, 24 July 2007 (UTC) 638:17:22, 24 July 2007 (UTC) 612:15:32, 23 July 2007 (UTC) 603:06:05, 23 July 2007 (UTC) 598:Sure, that seems fine. -- 593:03:23, 23 July 2007 (UTC) 550:20:04, 20 July 2007 (UTC) 502:20:01, 19 July 2007 (UTC) 484:19:40, 19 July 2007 (UTC) 418:22:40, 17 July 2007 (UTC) 380:21:47, 17 July 2007 (UTC) 355:21:36, 17 July 2007 (UTC) 305:19:36, 11 July 2007 (UTC) 116:Now back to the present. 2340:Finite/Cofinite subsets? 2061:21:09, 8 June 2011 (UTC) 2046:Boolean algebra (object) 2027:20:29, 8 June 2011 (UTC) 2001:15:29, 8 June 2011 (UTC) 1977:15:14, 8 June 2011 (UTC) 1868:involved follows it. -- 1721:I agree with Trovatore. 934:Lack of inline citations 269:20:33, 9 July 2007 (UTC) 249:20:26, 9 July 2007 (UTC) 212:16:45, 9 July 2007 (UTC) 156:09:25, 9 July 2007 (UTC) 112:09:10, 8 July 2007 (UTC) 94:09:25, 9 July 2007 (UTC) 3221:01:45, 3 May 2015 (UTC) 3207:, do you know of one? — 3129:Derivation of the order 1927:Boolean algebra (logic) 1820:Boolean algebra (logic) 1796:Boolean algebra (logic) 1764:boolean algebra (logic) 1755:boolean algebra (logic) 1634:boolean algebra (logic) 1449:which is rendered as a 574:Boolean algebra (logic) 515:Boolean algebra (logic) 442:Boolean algebra (logic) 327:Boolean algebra (logic) 290:Boolean algebra (logic) 119:Having stopped work on 2892:and start reading it. 1489: 1443: 1352: 1320: 1287: 1253: 1218: 1173: 1116: 1071: 631:commons:User:Fibonacci 185:should be merged into 3037:" (to be created)? - 2971:seems unnecessary. -- 2406:) to ease verifying. 1937:for more information. 1621:I would have done it 1549:Reducing this article 1490: 1444: 1353: 1321: 1288: 1254: 1219: 1174: 1117: 1072: 823:Links (mostly) sorted 42:of past discussions. 2130:rules here, perhaps 1848:talk:Boolean algebra 1774:article is a mess. — 1469: 1408: 1330: 1298: 1264: 1231: 1183: 1126: 1081: 1049: 3205:User:David Eppstein 3155:if and only if 3099:axiomatizations. -- 2075:1st axiomatization? 1358:, and (NOR, NAND). 899:residuated lattices 321:should be moved to 2917:These Transactions 2721:¬(x ∨ y) = ¬x ∧ ¬y 2704:¬(x ∧ y) = ¬x ∨ ¬y 1485: 1480: 1439: 1361:Shimon P. Vingron 1348: 1316: 1282: 1249: 1214: 1169: 1112: 1067: 336:Elementary algebra 298:Talk:Boolean logic 2826: 2825: 2208: 2182:, others do not. 1905:comment added by 1614: 1479: 1389: 1206: 1158: 1104: 1016: 974: 607:Thus amended. -- 300:to this page. -- 266: 246: 224:gets merged into 123:, I went back to 82: 81: 54: 53: 48:current talk page 3227: 3219: 3217: 3212: 3194:Boolean algebras 3088:; see Gratzer's 3039:Jochen Burghardt 2991:Jochen Burghardt 2954:Jochen Burghardt 2927:Jochen Burghardt 2924: 2914: 2894:Jochen Burghardt 2855:Jochen Burghardt 2850: 2838: 2423: 2198: 1917: 1604: 1494: 1492: 1491: 1486: 1481: 1477: 1448: 1446: 1445: 1440: 1429: 1428: 1383: 1357: 1355: 1354: 1349: 1325: 1323: 1322: 1317: 1293: 1290: 1289: 1283: 1258: 1256: 1255: 1250: 1223: 1221: 1220: 1215: 1207: 1199: 1178: 1176: 1175: 1170: 1159: 1151: 1121: 1119: 1118: 1113: 1105: 1097: 1076: 1074: 1073: 1068: 1006: 964: 921:relation algebra 891:relation algebra 849:relation algebra 841:Boolean operator 837:Boolean datatype 833:Boolean function 543:Boolean datatype 539:Boolean function 362:abstract algebra 340:Abstract algebra 265: 245: 78: 56: 55: 33: 32: 26: 3235: 3234: 3230: 3229: 3228: 3226: 3225: 3224: 3215: 3210: 3208: 3201: 3131: 3027:Robbins algebra 3021: 3017: 2969:Robbins algebra 2912: 2907: 2844: 2832: 2739:Double Negation 2598:x ∧ (x ∨ y) = x 2581:x ∨ (x ∧ y) = x 2411:Lattice_(order) 2379: 2342: 2155: 2077: 1991:Boolean algebra 1900: 1890: 1816:Boolean algebra 1704:modular lattice 1685:Boolean lattice 1681:Boolean algebra 1642:boolean algebra 1630:modular lattice 1601:Boolean lattice 1597: 1582: 1555:Boolean algebra 1551: 1467: 1466: 1420: 1406: 1405: 1398: 1373: 1328: 1327: 1296: 1295: 1261: 1260: 1229: 1228: 1181: 1180: 1124: 1123: 1079: 1078: 1047: 1046: 1043: 936: 903:action algebras 829:Boolean algebra 825: 622: 568:Boolean algebra 561:Boolean algebra 557: 534: 477:Boolean algebra 473:Boolean algebra 465:Boolean algebra 430:Boolean algebra 400:Boolean algebra 315:Boolean algebra 294:User: Trovatore 163:Boolean algebra 140:Boolean algebra 132:User: Trovatore 87: 74: 30: 22: 21: 20: 12: 11: 5: 3233: 3231: 3200: 3190: 3175: 3174: 3173: 3172: 3171: 3170: 3169: 3168: 3130: 3127: 3126: 3125: 3124: 3123: 3122: 3121: 3120: 3119: 3118: 3117: 3116: 3115: 3114: 3113: 3112: 3111: 3090:Lattice Theory 3062: 3061: 3060: 3059: 3058: 3057: 3056: 3055: 3054: 3053: 3052: 3051: 3050: 3049: 3023: 3019: 3015: 3011: 3007: 3003: 3002: 3001: 2883: 2882: 2852: 2851: 2847:Lattice Theory 2839: 2824: 2823: 2820: 2817: 2814: 2811: 2807: 2806: 2803: 2800: 2797: 2794: 2790: 2789: 2783: 2782: 2779: 2776: 2773: 2770: 2766: 2765: 2759: 2758: 2755: 2752: 2749: 2746: 2742: 2741: 2735: 2734: 2731: 2728: 2725: 2722: 2718: 2717: 2714: 2711: 2708: 2705: 2701: 2700: 2694: 2693: 2690: 2687: 2684: 2681: 2677: 2676: 2673: 2670: 2667: 2664: 2660: 2659: 2653: 2652: 2649: 2646: 2643: 2640: 2636: 2635: 2632: 2629: 2626: 2623: 2619: 2618: 2612: 2611: 2608: 2605: 2602: 2599: 2595: 2594: 2591: 2588: 2585: 2582: 2578: 2577: 2571: 2570: 2567: 2564: 2561: 2558: 2554: 2553: 2550: 2547: 2544: 2541: 2537: 2536: 2534:Distributivity 2530: 2529: 2526: 2523: 2520: 2517: 2513: 2512: 2509: 2506: 2503: 2500: 2496: 2495: 2489: 2488: 2485: 2482: 2479: 2476: 2472: 2471: 2468: 2465: 2462: 2459: 2455: 2454: 2448: 2447: 2442: 2437: 2433: 2426: 2378: 2375: 2374: 2373: 2341: 2338: 2307: 2306: 2305: 2304: 2303: 2302: 2301: 2300: 2299: 2298: 2288:Fine with me. 2286: 2234: 2233: 2232: 2231: 2213: 2212: 2193: 2192: 2154: 2151: 2150: 2149: 2148: 2147: 2076: 2073: 2072: 2071: 2070: 2069: 2068: 2067: 2066: 2065: 2064: 2063: 2049: 2008: 2007: 2006: 2005: 2004: 2003: 1982: 1981: 1980: 1979: 1961: 1960: 1945: 1938: 1930: 1889: 1886: 1885: 1884: 1883: 1882: 1881: 1880: 1844: 1843: 1842: 1776:David Eppstein 1732: 1731: 1719: 1677: 1676: 1662: 1661: 1660: 1652: 1645: 1596: 1593: 1581: 1578: 1550: 1547: 1546: 1545: 1499:is concerned. 1484: 1474: 1438: 1435: 1432: 1427: 1423: 1419: 1416: 1413: 1397: 1394: 1372: 1369: 1347: 1344: 1341: 1338: 1335: 1315: 1312: 1309: 1306: 1303: 1281: 1278: 1275: 1272: 1269: 1248: 1245: 1242: 1239: 1236: 1213: 1210: 1205: 1202: 1197: 1194: 1191: 1188: 1168: 1165: 1162: 1157: 1154: 1149: 1146: 1143: 1140: 1137: 1134: 1131: 1111: 1108: 1103: 1100: 1095: 1092: 1089: 1086: 1066: 1063: 1060: 1057: 1054: 1042: 1039: 1037: 1035: 1034: 1033: 1032: 979: 978: 946:David Eppstein 935: 932: 931: 930: 887: 886: 885: 884: 870:David Eppstein 845:Charles Peirce 824: 821: 820: 819: 818: 817: 816: 815: 814: 813: 812: 811: 810: 809: 808: 807: 806: 805: 768:GLIPS Graffiti 747: 743: 734:David Eppstein 706:David Eppstein 680: 676: 668: 667: 666: 665: 660:David Eppstein 635:David Eppstein 621: 618: 617: 616: 615: 614: 586: 585: 584: 583: 577: 556: 555:Disambiguation 553: 533: 530: 529: 528: 509: 508: 507: 506: 505: 504: 489: 488: 487: 486: 458: 457: 456: 455: 448: 447: 446: 445: 423: 422: 421: 420: 407: 406: 405: 404: 393: 392: 391: 390: 383: 382: 365: 274: 273: 272: 271: 254: 253: 252: 251: 215: 214: 204: 201:look things up 197: 190: 86: 85:Execution time 83: 80: 79: 72: 67: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 3232: 3223: 3222: 3218: 3213: 3206: 3199: 3198:Boolean rings 3195: 3191: 3189: 3188: 3184: 3180: 3166: 3162: 3158: 3154: 3150: 3146: 3143: 3142: 3141: 3140: 3138: 3137: 3136: 3135: 3134: 3128: 3110: 3106: 3102: 3098: 3094: 3091: 3087: 3083: 3078: 3077: 3076: 3075: 3074: 3073: 3072: 3071: 3070: 3069: 3068: 3067: 3066: 3065: 3064: 3063: 3048: 3044: 3040: 3036: 3032: 3028: 3024: 3012: 3008: 3004: 3000: 2996: 2992: 2988: 2987: 2984: 2983: 2982: 2978: 2974: 2970: 2965: 2964: 2963: 2959: 2955: 2951: 2950: 2949: 2946: 2943: 2938: 2937: 2936: 2932: 2928: 2922: 2918: 2911: 2905: 2904: 2903: 2899: 2895: 2891: 2887: 2886: 2885: 2884: 2881: 2878: 2875: 2871: 2867: 2866: 2865: 2864: 2860: 2856: 2848: 2843: 2840: 2836: 2831: 2828: 2827: 2821: 2818: 2815: 2812: 2809: 2808: 2804: 2801: 2798: 2795: 2792: 2791: 2788: 2784: 2780: 2777: 2774: 2771: 2768: 2767: 2764: 2760: 2756: 2753: 2750: 2747: 2744: 2743: 2740: 2736: 2732: 2729: 2726: 2723: 2720: 2719: 2715: 2712: 2709: 2706: 2703: 2702: 2699: 2695: 2691: 2688: 2685: 2682: 2679: 2678: 2674: 2671: 2668: 2665: 2662: 2661: 2658: 2654: 2650: 2647: 2644: 2641: 2638: 2637: 2633: 2630: 2627: 2624: 2621: 2620: 2617: 2613: 2609: 2606: 2603: 2600: 2597: 2596: 2592: 2589: 2586: 2583: 2580: 2579: 2576: 2572: 2568: 2565: 2562: 2559: 2556: 2555: 2551: 2548: 2545: 2542: 2539: 2538: 2535: 2531: 2527: 2524: 2521: 2518: 2515: 2514: 2510: 2507: 2504: 2501: 2498: 2497: 2494: 2493:Associativity 2490: 2486: 2483: 2480: 2477: 2475:x ∧ y = y ∧ x 2474: 2473: 2469: 2466: 2463: 2460: 2458:x ∨ y = y ∨ x 2457: 2456: 2453: 2452:Commutativity 2449: 2446: 2443: 2441: 2438: 2436: 2434: 2432: 2431: 2427: 2425: 2424: 2421: 2419: 2414: 2412: 2409:The article " 2407: 2405: 2401: 2397: 2393: 2389: 2385: 2376: 2372: 2368: 2364: 2359: 2358: 2357: 2356: 2352: 2348: 2347:67.79.154.194 2339: 2337: 2336: 2332: 2328: 2322: 2318: 2316: 2311: 2297: 2294: 2291: 2287: 2285: 2281: 2277: 2273: 2272: 2271: 2267: 2263: 2259: 2254: 2253: 2252: 2248: 2244: 2240: 2239: 2238: 2237: 2236: 2235: 2230: 2226: 2222: 2217: 2216: 2215: 2214: 2211: 2206: 2202: 2195: 2194: 2191: 2188: 2185: 2181: 2177: 2176: 2175: 2174: 2170: 2166: 2161: 2158: 2152: 2146: 2142: 2138: 2135:conception." 2133: 2129: 2125: 2124: 2123: 2119: 2115: 2114:Hugo Herbelin 2111: 2107: 2106: 2105: 2104: 2100: 2096: 2092: 2091:981-283-454-0 2089: 2085: 2082: 2074: 2062: 2058: 2054: 2050: 2047: 2043: 2039: 2035: 2030: 2029: 2028: 2024: 2020: 2016: 2015: 2014: 2013: 2012: 2011: 2010: 2009: 2002: 1999: 1996: 1992: 1988: 1987: 1986: 1985: 1984: 1983: 1978: 1974: 1970: 1965: 1964: 1963: 1962: 1959: 1955: 1951: 1946: 1943: 1939: 1936: 1931: 1928: 1924: 1923:Boolean logic 1920: 1919: 1918: 1916: 1912: 1908: 1904: 1897: 1894: 1887: 1879: 1875: 1871: 1870:Vaughan Pratt 1867: 1863: 1862: 1861: 1857: 1853: 1849: 1845: 1841: 1837: 1833: 1832:Vaughan Pratt 1829: 1828:Boolean logic 1825: 1821: 1817: 1813: 1812:Boolean logic 1809: 1805: 1804:Boolean logic 1801: 1797: 1793: 1792:Boolean logic 1789: 1788: 1787: 1786: 1785: 1781: 1777: 1773: 1772:boolean logic 1769: 1768:boolean logic 1765: 1760: 1759:boolean logic 1756: 1752: 1751: 1750: 1749: 1745: 1741: 1737: 1730: 1727: 1724: 1720: 1718: 1714: 1710: 1705: 1701: 1700: 1699: 1698: 1694: 1690: 1686: 1682: 1675: 1671: 1667: 1663: 1658: 1653: 1650: 1646: 1643: 1639: 1638:boolean logic 1635: 1631: 1627: 1626: 1624: 1620: 1619: 1618: 1617: 1612: 1608: 1602: 1594: 1592: 1591: 1588: 1580:Math notation 1579: 1577: 1576: 1572: 1568: 1564: 1560: 1556: 1548: 1544: 1540: 1536: 1532: 1531: 1530: 1529: 1525: 1521: 1520:Hugo Herbelin 1516: 1514: 1510: 1506: 1500: 1498: 1464: 1459: 1454: 1452: 1433: 1425: 1421: 1414: 1411: 1403: 1395: 1393: 1392: 1387: 1382: 1378: 1370: 1368: 1367: 1364: 1363:81.217.16.172 1359: 1339: 1336: 1310: 1307: 1304: 1276: 1273: 1270: 1243: 1240: 1237: 1225: 1211: 1208: 1200: 1192: 1189: 1186: 1163: 1160: 1152: 1138: 1135: 1132: 1109: 1106: 1098: 1090: 1087: 1084: 1077:. For, while 1061: 1058: 1055: 1038: 1031: 1028: 1025: 1021: 1020: 1019: 1014: 1010: 1003: 1002: 1001: 1000: 997: 994: 989: 988: 985: 977: 972: 968: 962: 958: 953: 952: 951: 950: 947: 942: 933: 929: 926: 922: 918: 917: 916: 915: 912: 911:Vaughan Pratt 908: 904: 900: 896: 892: 883: 880: 876: 875: 874: 871: 867: 866: 865: 864: 861: 857: 852: 850: 846: 842: 838: 834: 830: 822: 804: 800: 796: 795:Vaughan Pratt 791: 786: 785: 784: 781: 777: 773: 769: 765: 760: 756: 752: 748: 744: 740: 739: 738: 735: 730: 729: 728: 725: 720: 716: 712: 711: 710: 707: 702: 701: 700: 697: 693: 689: 685: 681: 677: 674: 673: 672: 671: 670: 669: 664: 661: 656: 655: 654: 651: 647: 642: 641: 640: 639: 636: 632: 627: 619: 613: 610: 606: 605: 604: 601: 597: 596: 595: 594: 591: 581: 578: 575: 572: 571: 569: 566: 565: 564: 562: 554: 552: 551: 548: 544: 540: 531: 527: 524: 520: 516: 511: 510: 503: 500: 495: 494: 493: 492: 491: 490: 485: 482: 481:Vaughan Pratt 478: 474: 470: 466: 462: 461: 460: 459: 452: 451: 450: 449: 443: 439: 438:Boolean logic 435: 431: 427: 426: 425: 424: 419: 416: 411: 410: 409: 408: 401: 397: 396: 395: 394: 387: 386: 385: 384: 381: 378: 374: 370: 366: 363: 359: 358: 357: 356: 353: 352:Vaughan Pratt 349: 345: 341: 337: 333: 328: 324: 320: 319:Boolean logic 316: 312: 311:Boolean logic 309:The merge at 307: 306: 303: 302:Vaughan Pratt 299: 295: 291: 287: 286:Boolean logic 283: 279: 278:User: Lambiam 270: 267: 263: 258: 257: 256: 255: 250: 247: 243: 239: 235: 234:Boolean logic 231: 230:Boolean logic 227: 226:Boolean logic 223: 219: 218: 217: 216: 213: 210: 205: 202: 198: 195: 191: 188: 187:Boolean logic 184: 180: 179:Boolean logic 176: 172: 171:Boolean logic 168: 164: 160: 159: 158: 157: 154: 153:Vaughan Pratt 150: 146: 141: 136: 133: 129: 126: 122: 117: 114: 113: 110: 109:Vaughan Pratt 106: 105:Boolean logic 102: 97: 95: 92: 91:Vaughan Pratt 84: 77: 73: 71: 68: 66: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 3202: 3176: 3164: 3160: 3156: 3152: 3148: 3144: 3132: 3096: 3089: 2920: 2916: 2853: 2846: 2841: 2834: 2829: 2786: 2762: 2738: 2697: 2656: 2615: 2574: 2533: 2492: 2451: 2444: 2439: 2435: 2428: 2415: 2408: 2403: 2399: 2395: 2391: 2387: 2383: 2380: 2343: 2323: 2319: 2314: 2312: 2308: 2257: 2179: 2162: 2159: 2156: 2080: 2078: 2045: 2041: 2037: 2033: 1941: 1935:Boolean ring 1907:72.234.37.70 1898: 1895: 1891: 1865: 1733: 1684: 1680: 1678: 1598: 1583: 1562: 1552: 1517: 1508: 1504: 1501: 1455: 1399: 1374: 1360: 1226: 1044: 1036: 990: 980: 960: 957:Stoll (1963) 956: 937: 888: 853: 826: 789: 718: 645: 623: 587: 567: 558: 535: 308: 275: 200: 193: 166: 137: 130: 118: 115: 98: 88: 75: 43: 37: 3086:an exercise 3082:Paul Halmos 2942:Paul August 2874:Paul August 2787:Idempotency 2616:Complements 2290:Paul August 2184:Paul August 1901:—Preceding 1808:User:StuRat 1736:Semilattice 1723:Paul August 1478:instead of 1024:Paul August 993:Paul August 751:Grab it now 403:structure). 173:. But your 36:This is an 2923:: 288–309. 2763:Modularity 2639:x ∧ ¬x = 0 2622:x ∨ ¬x = 1 2575:Absorption 1666:Hans Adler 1567:Hans Adler 1292:)}" /: --> 877:Thanks. -- 570:may mean: 334:, between 2870:this edit 2810:x ∨ x = x 2793:x ∧ x = x 2698:De Morgan 2680:x ∧ 1 = x 2663:x ∨ 0 = x 2262:Trovatore 2053:Trovatore 2034:structure 1950:Trovatore 1852:Trovatore 1806:article, 1734:At least 1709:Trovatore 1689:Trovatore 1377:atomicity 1371:Atomicity 984:Trovatore 925:Trovatore 879:Trovatore 860:Trovatore 650:Trovatore 600:Trovatore 547:Trovatore 499:Trovatore 415:Trovatore 377:Trovatore 209:Trovatore 76:Archive 4 70:Archive 3 65:Archive 2 60:Archive 1 3101:Macrakis 3097:multiple 2973:Macrakis 2657:Identity 2327:Gwideman 2221:Gwideman 2180:distinct 2165:Gwideman 2137:Tijfo098 2095:Tijfo098 1903:unsigned 1866:everyone 1740:Tijfo098 1657:WP:NCDAB 1587:Dcoetzee 1263:)}": --> 764:Inkscape 692:this PNG 688:this SVG 276:Thanks, 194:quantity 167:supposed 2745:¬¬x = x 2733:17:L10 2716:17:L10 2552:11:L6' 2418:article 2110:section 2019:Rbakels 1969:Rbakels 1535:Zundark 1022:OK ;-) 772:Firefox 757:), the 684:librsvg 646:covered 609:Lambiam 590:Lambiam 332:Algebra 262:Lambiam 242:Lambiam 39:archive 3216:gundam 3006:clear? 2819:109:L3 2802:109:L3 2781:12:L5 2757:17:L9 2651:17:L8 2634:17:L8 2607:109:L4 2590:109:L4 2525:109:L1 2508:109:L1 2484:109:L2 2467:109:L2 2384:is not 2315:values 2044:, and 1623:boldly 959:. The 790:claims 766:. The 523:StuRat 2913:(PDF) 2822:8:L1 2805:8:L1 2610:8:L4 2593:8:L4 2528:8:L3 2511:8:L3 2487:8:L2 2470:8:L2 2128:WP:OR 1998:Adler 1822:, or 1563:still 1458:LaTeX 1277:: --> 909:. -- 839:, or 780:KSmrq 776:Opera 759:Batik 724:KSmrq 719:valid 696:KSmrq 240:. -- 16:< 3211:wing 3196:and 3183:talk 3105:talk 3043:talk 2995:talk 2977:talk 2958:talk 2931:talk 2898:talk 2859:talk 2842:Bir: 2402:and 2367:talk 2351:talk 2331:talk 2280:talk 2266:talk 2258:that 2256:not 2247:talk 2225:talk 2205:talk 2169:talk 2163:??? 2141:talk 2132:this 2118:talk 2099:talk 2088:ISBN 2057:talk 2023:talk 1995:Hans 1973:talk 1954:talk 1942:self 1933:see 1911:talk 1874:talk 1856:talk 1836:talk 1780:talk 1766:and 1744:talk 1713:talk 1693:talk 1670:talk 1611:talk 1571:talk 1539:talk 1524:talk 1463:HTML 1402:HTML 1386:talk 1311:< 1190:< 1013:talk 971:talk 905:aka 889:The 799:talk 774:and 755:beta 715:here 690:and 545:. -- 436:and 346:and 338:and 325:and 317:and 147:and 2830:DP: 2689:144 2672:144 2648:144 2631:144 2549:131 2445:Bir 2404:Bir 2396:SEP 2363:JBL 2276:JBL 2243:JBL 2201:CBM 1925:or 1757:or 1636:or 1607:CBM 1513:PNG 1497:PNG 1451:PNG 1009:CBM 967:CBM 843:). 633:. — 541:or 440:to 432:to 260:-- 3185:) 3163:= 3159:∨ 3151:∧ 3147:= 3107:) 3045:) 2997:) 2979:) 2960:) 2933:) 2919:. 2915:. 2900:) 2872:. 2861:) 2692:. 2675:. 2645:10 2569:. 2440:DP 2430:WP 2400:DP 2394:, 2392:WP 2388:is 2369:) 2361:-- 2353:) 2333:) 2282:) 2268:) 2249:) 2227:) 2203:· 2171:) 2143:) 2120:) 2101:) 2059:) 2040:, 2025:) 1975:) 1956:) 1913:) 1876:) 1858:) 1838:) 1818:, 1782:) 1746:) 1715:) 1695:) 1672:) 1664:-- 1659:.) 1609:· 1573:) 1541:) 1526:) 1507:∩( 1483:∅ 1473:∅ 1437:∅ 1415:∩ 1381:H. 1343:↔ 1337:⊕ 1326:, 1305:≤ 1294:, 1286:)} 1271:≥ 1259:, 1244:∧ 1238:∨ 1224:. 1209:∧ 1204:¯ 1196:⇔ 1161:∧ 1156:¯ 1145:↔ 1136:≥ 1107:∨ 1102:¯ 1094:⇔ 1088:≤ 1062:≥ 1056:≤ 1011:· 969:· 835:, 801:) 722:-- 96:) 3181:( 3167:. 3165:b 3161:b 3157:a 3153:a 3149:b 3145:a 3103:( 3041:( 3020:1 3016:2 3014:E 2993:( 2975:( 2956:( 2945:☎ 2929:( 2921:5 2896:( 2877:☎ 2857:( 2816:. 2813:. 2799:. 2796:. 2778:. 2775:. 2772:. 2754:. 2751:. 2748:. 2730:. 2727:. 2724:. 2713:. 2710:. 2707:. 2686:. 2683:3 2669:. 2666:3 2642:5 2628:9 2625:5 2604:8 2601:. 2587:7 2584:. 2566:. 2563:6 2560:4 2546:5 2543:4 2522:4 2519:1 2505:3 2502:1 2481:2 2478:2 2464:1 2461:2 2365:( 2349:( 2329:( 2293:☎ 2278:( 2264:( 2245:( 2223:( 2207:) 2199:( 2187:☎ 2167:( 2139:( 2116:( 2097:( 2081:s 2055:( 2048:. 2021:( 1971:( 1952:( 1929:? 1909:( 1872:( 1854:( 1834:( 1778:( 1742:( 1726:☎ 1711:( 1691:( 1668:( 1651:. 1613:) 1605:( 1569:( 1537:( 1522:( 1509:A 1505:A 1434:= 1431:) 1426:C 1422:A 1418:( 1412:A 1388:) 1384:( 1346:) 1340:, 1334:( 1314:) 1308:, 1302:( 1280:) 1274:, 1268:( 1247:) 1241:, 1235:( 1212:B 1201:A 1193:B 1187:A 1167:) 1164:B 1153:A 1148:( 1142:) 1139:B 1133:A 1130:( 1110:B 1099:A 1091:B 1085:A 1065:) 1059:, 1053:( 1027:☎ 1015:) 1007:( 996:☎ 973:) 965:( 797:( 704:— 444:. 189:. 50:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index