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I have a set $M$ and an associative binary relation $+ : M \times M \to M$.

There exists an inversion operator $-$ , such that if $ m \in M$, then $m+(-m) \in Z$ where $Z$ is the set of all zeros ($Z \subset M$).

Additionally, if $z \in Z$ and $m \in M$, then $m+z=m$

If $|Z| = 1,$ then this would form a group, right? But since $|Z|>1$, what would I call it? Is there any theory about categories like this?


Also, what if there was no identity at all? That is, $m_1 + m_2 + (-m_2) = m_1$, but $m_2 + (-m_2)$ is undefined. In this case, associativity would not hold.

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    $\begingroup$ if it's a two-sided identity, then there can only be one. Further, even though you didn't explicitly state it, the use of $+$ is generally reserved for commutative operations, which would make it two-sided. $\endgroup$ Aug 2, 2012 at 19:25
  • $\begingroup$ @JackSchmidt Yep. Fixed it. $\endgroup$ Aug 2, 2012 at 19:28
  • $\begingroup$ @DustanLevenstein I have a specific case in computer science in mind, and I'm pretty sure it has multiple two sided identities. (The operation I have in mind is commutative BTW.) $\endgroup$ Aug 2, 2012 at 19:30
  • $\begingroup$ @MikeIzbicki: if the inverse is "unique", then this is a very specific mathematical object. If the inverses aren't unique, then I think you still have some sort of thing, but I'd have to go look its name and axioms up. $\endgroup$ Aug 2, 2012 at 19:32
  • $\begingroup$ @MikeIzbicki, no it doesn't, if it's associative. Suppose $eg=ge=fg=gf=g$ for all $g \in G$. Then $ef=fe=f$ but $fe=ef=e.$ Ah-maybe associativity's only needed to get a unique two-sided inverse out of a one-sided one... $\endgroup$ Aug 2, 2012 at 19:33

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Suppose $M$ is a set with an associative binary operator $+$ such that for all $m \in M$ there is a unique $n$ such that $m+n+m = m$ and $n+m+n = n$ (thinking of $z_r=m+n$ we get $z_r + m = m$ and $n+z_r = n$; thinking of $z_l =n+m$ we get $m+z_l = m$ and $z_l+n=n$), then you have a well studied object called an “inverse semigroup”.

The set $Z$ you want is called the set of idempotents, $Z = \{ m \in M : m + m = m \}$. It also consists exactly of all $m + (-m)$ for $m \in M$. I also call them "partial identities", because:

There is a partial order $\leq$ on every inverse semigroup: $x \leq y$ iff $x = x + (-x) + y$. This let's you express how close to an identity each $z \in Z$ is: $z + x = x$ whenever $x \leq z$.

For every partial identity $z=g+(-g)$ there is an honest to goodness group called the $H$-class (or more accurately its Schutzenberger group) associated to $z$ consisting of all invertible $z+m+z$ for $m \in M$.

Inverse semigroups as semigroups of bijections

Every inverse semigroup is isomorphic to a semigroup consisting of bijections between subsets of a set $X$, where multiplication is the best version of composition you can manage: if $f:A\to B$ and $g:C \to D$ then $f\cdot g$ takes $f^{-1}(x)$ to $g(x)$ whenever $x \in B \cap C$.

In terms of bijections, the set $Z$ consists of all identity bijections $f:A \to A :a \mapsto a$ where $A$ is the domain of some $m \in M$. In terms of bijections, $-g$ is the inverse function $g^{-1} : D \to C : g(c) \mapsto c$. The partial identities $g+( -g)$ and $-g+g$ are the identity bijections on the domain and range of $g$.

The natural partial order is actually just “subset” if you consider a bijection to be a set of ordered pairs $(a,f(a))$, so that $f \leq g$ if and only if $f$ is the restriction of $g$ to the domain of $f$.

The H-class of an idempotent $f:A\to A:a\mapsto a$ is all the permutations of $A$, that is, bijections from $A$ to $A$.

Bibliography

I recommend both of these books as very clear and motivated introductions to these structures. Lawson's book is particularly readable and draws connections to topology, symmetry, groupoids, category theory, and computer science.

  • Lawson, Mark V. Inverse semigroups: The theory of partial symmetries. World Scientific Publishing Co., Inc., River Edge, NJ, 1998. xiv+411 pp. ISBN: 981-02-3316-7 MR1694900 DOI10.1142/9789812816689
  • Petrich, Mario. Inverse semigroups. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1984. x+674 pp. ISBN: 0-471-87545-7 MR752899
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  • $\begingroup$ Thanks. This is what I was looking for. The inverses are unique for each m, but not in general the same. $\endgroup$ Aug 2, 2012 at 19:38
  • $\begingroup$ Wikipedia seems to have a pretty good page on inverse semigroups. Is there anything else I should take a look at? I'd like to get a pretty in depth feel for what's been studied about these things. $\endgroup$ Aug 2, 2012 at 19:44

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