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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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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.