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Does anyone know, how is called a semigroup in which every equation $ax=b$ has only a finite set (maybe empty) of solutions?

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Please give more motiviation and background for the question. –  Martin Brandenburg Nov 17 '11 at 16:12
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@Martin: I now return to my old article in 1996 with K.Iordjev (you can see it in my homepage). I used such semigroups in it and now would know whether they are some own name. –  Boris Novikov Nov 17 '11 at 18:31
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Pedro Silva and I introduced what we called finite geometric type for a finitely generated semigroup. The definition was that the in-degree of each vertex of the right Cayley graph be finite. An easy induction on the length of a shows this is equivalent to xa=b has finitely many solutions for any fixed a,b.

This should really be called right finite geometric type and so you should have left finite geometric type. Actually we probably should have used proper since for finitely generated semigroups it means the Cayley graph is a proper metric space.

Update. It is proved by Ellis in http://www.ams.org/journals/tran/2001-353-04/S0002-9947-00-02704-5/S0002-9947-00-02704-5.pdf that an infinite semigroup T has finite geometric type if and ony if $\beta T\setminus T$ is a right ideal where $\beta T$ is the Stone-Czech compactification of T.

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Thank you, Benjamin. Could you give me the reference to to this article? –  Boris Novikov Nov 17 '11 at 17:50
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@Boris, the relevant papers are: P. V. Silva and B. Steinberg, Extensions and subsemigroups of automatic monoids, Theoret. Comput Sci. 289 (2002), 727-754 and P. V. Silva and B. Steinberg, A geometric characterization of automatic monoids, Q. J. Math. 55 (2004), 333-356. I don't remember if both these papers have the notion or just the second one. They were both written around 2000 but took a while to appear. Of course it is quite possible that a name for this notion occurred long ago. In 2000 there was no MO and Google was not so good for finding these things. –  Benjamin Steinberg Nov 17 '11 at 20:21
    
@Benjamin, once more thank you. I found this papers. –  Boris Novikov Nov 17 '11 at 20:45
    
Ben, why on earth finite geometric type is intersting? Never understood this. Can you find nice examples where FGT would help or can you explode any big problem with FGT stuff? –  Victor Apr 18 '12 at 7:45
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This condition is needed to make the Cayley graph locally finite. Without this there is no hope to try and work geometrically with monoid Cayley graphs. Also homological properties behave badly without it. The condition is a weak form of cancellativity and makes a monoid behave more like a group. But one can of course find ugly semigroups with FGT. –  Benjamin Steinberg Apr 18 '12 at 10:23
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