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I was reading Dummit and Foote to be ready for my group theory text, but my teacher seems to be paying special attention to things with less structure than groups, for example monoids, semigroups, and other things I don't know the name for in english. We have proves problems similar to the 2001 A1 problem or the 2012 A2 problem. We also saw if $S$ is a semigroup with left and right cancelation then $S$ is a monoid, I later realized on my own $S$ is also a group.

I would like to obtain certain mastery and intuition on these group like structures, I am looking for texts, or references of any sort that will help me become better at dealing with these things.

Thank you very much in advance.

Regards.

vociferous_rutabaga
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Asinomás
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  • "We also saw if S is a semigroup with left and right cancelation then S is a monoid, I later realized on my own S is also a group." - Hm? – Martin Brandenburg Aug 08 '14 at 08:52
  • is that false? I thought since there is an identity and cancelation then for each a the equation xa=e must have a solution and the equation ax=e also, so it has a left and a right inverse, and if it has a left and right inverse they must be equal. Is this wrong? – Asinomás Aug 08 '14 at 11:58
  • I think I read somewehere that you don't even need the unicity of the cancelation i.e. if $S$ is a semigroup under composition such that for all $a,b,c,d$ we have $ax=b$ and $yc=d$ have solutions then $S$ is a group. – Asinomás Aug 08 '14 at 12:04
  • This appeared here quite recently. – James Aug 08 '14 at 18:35

1 Answers1

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I don't know of a really good general book at the level of Dummit and Foote. One fairly old reference is:

R.H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1971. (Google Books)(Amazon)

I'm not sure I'd recommend that as an undergraduate text, though. A good (but fairly heavy, and also harder to find) source for the theory of semigroups is:

A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Amer. Math. Soc., 1961.(Google Books)

The following book on quasigroups and loops is quite nice and readable.

H. O. Pflugfelder, Quasigroups and Loops: An Introduction, Heldermann-Verlag, 1990. (Amazon)

Another direction you might look at is texts on universal algebra. The best reference I know is

S.N. Burris and H.P. Sankappanavar, Universal Algebra, Springer-Verlag, 1981. (More)

Again, though, it's not really an undergraduate level text. Possibly more helpful is:

G. Gratzer, Universal Algebra, Second Edition, Springer-Verlag, 2008. (Amazon)

James
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