# Algebraical Structures classification

Someone have a good reference list for the classification of an algebrical structures? An example? Be: "a" for associative, "c" for commutative, "d" for distributive, "n" for neutro element, "o" for opposite/inverse, so, R(+,*) is d+acno*acno!0 and it's called "field". Tricky? "d" it's the last with parentesis for explain distributive related who, and if it's left or right, the simbols have the related property to their right! ("!0" mean not 0, it has not inverse about product in real)

Could someone prove a list of this type with relative name? Like before:

R(+,*): acno*acno!0 d(+, l, r) -> field

It will help me, because I remember the operation's property, but not the correct name.

Thank's previously and sorry for the bad Eglish(I'm Italian)

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Why would anyone want to complicate matters by adding an obscure abbreviation? – lhf Jan 31 '12 at 16:54
Abbreviations of the type you suggest can be seen in some parts of mathematics, and even more frequently in theoretical computer science. So they are obviously viewed as useful by many pople. – André Nicolas Jan 31 '12 at 17:17
Not that it is directly relevant, but a structure with two binary operations, $+$, and $*$, where each is associative, commutative, and has a neutral element, and in which $*$ distributes over $+$ (your notation does not seem to distinguish who distributes over who; this is important!) is not necessarily a field, it's not even necessarily a semiring! You are missing additive inverses to get a commutative ring, and multiplicative inverses of nonzero elements to get a field. – Arturo Magidin Jan 31 '12 at 17:31
I have trouble understanding your request; you say you want a list in which the properties of the operations are described by a single word (which, so far, is related to its standard name); yet you say you want the list because you have trouble remembering the standard name of the properties and remember only what they are. If that is the case, it would seem that such a list would be confusing! You would need to: (i) look up the abbreviation, and then (ii) look up the definition. How is this better than looking up the definition? (cont) – Arturo Magidin Feb 1 '12 at 5:02
For a slightly more in-depth discussion of the point @Arturo raised in his last comment, see this wikipedia page. If I understand correctly, Arturo is trying to draw your attention to the difference between the first section of that page and the second section. – Willie Wong Feb 1 '12 at 15:59

Not the symbolic solution you're looking for but a nice picture is available at http://mathoverflow.net/questions/72867/an-image-of-the-hierarchy-of-algebraic-structures:

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Neat! It would be fun to play with the source of this, to expand it to include some of the missing links. For example, the jump from the rationals to reals and complexes puts in one tiny arrow almost the entirety of number theory. (And it looks like fields get commutativity added twice, skipping skew fields...) – Cam McLeman Feb 1 '12 at 16:22

This page hosted by Peter Jipsen contains a list of 311 (as of today) named algebraic structures.

This wikipedia page also contains a lot of classification information on algebraic structures.

While neither of them has a simple table of the form you aspire from which to read off immediately the classification, you can presumably try to compile such a table yourself based on the information there. It may even be a worthwhile "cheatsheet" for future students of universal algebra.

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It's what I'm thinking to do! A simple ordinated cheatsheet ;) – Pizzirani Leonardo Feb 1 '12 at 17:53