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In the definition of a monoid firstly we should have associativity. What I wonder about is the definition of the identity element;

$\exists x \forall y\;\; x.y=y.x=y $

Which structure do we get if we change the order of quantifiers in the definition? That is;

$\forall y \exists x \;\; x.y=y.x=y $

ahmetselcuk
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    Well, we get something weaker certainly, though still of interest. For example, if $R$ is a ring with unity, then the set of sequences of elements from $R$, which are $0$ from some point, becomes a ring where there is no unit for multiplication, but where the multiplication has the weaker version of a unit you describe here. – Tobias Kildetoft Feb 06 '14 at 12:44

2 Answers2

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In some sense, neither property is that important, because if a semigroup doesn't have an identity you can just add one in (call it $e$) and define $ex = xe = x$ for all $x$. Then the associative property still holds, and now both of your properties are satisfied.

  • Global identity: there exists $e$ such that $xe = ex = x$ for all $x$

  • Local identities: for all $x$ there exists $e_x$ such that $e_x x = x e_x$.

What about semigroups with local identities but no global identities? I'm not sure. Tobias Kildetoft gives a good example. Many of the identities will work for multiple elements since $e_x (xyx) = (xyx) e_x = xyx$. I couldn't find any references on this property being studied or named before.

Caleb Stanford
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  • Indeed, in the example I mentioned, any finite subset will have a "common identity" (but if the ring $R$ is finite, then no infinite subset will have such a "common identity"). – Tobias Kildetoft Sep 20 '14 at 19:14
  • @Goos Thanks for the answer. As far as I can see the existence of only local identities boils down to this: If all elements of the monoid can be written as $xyx$ then $e_x$ is the global identity. What happens if the premise is not true, I really don't know. – ahmetselcuk Sep 21 '14 at 19:45
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The structure I asked about seems very much like square matrices. They satisfy associativity and there may be more than one element acting as identity on an element.

ahmetselcuk
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