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A semigroup is a set $S$ together with a binary operation that is associative. Examples of semigroups are the set of finite strings over a fixed alphabet (under concatenation) and the positive integers (under addition, maximum, or minimum). A monoid is a semigroup with a neutral element. Of course, any group is also a monoid/semigroup.

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Examples of left reversible semigroups

I suppose that the non-zero elements of a left Ore domain would work --- presumably this is why they are sometimes called Ore semigroups. To expand: a ring is a left Ore domain if it has no non-trivi …
Simon Wadsley's user avatar
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Motivation and reference for Brauer algebras

For motivation I would advise starting with Brauer's original paper. You'll need a JSTOR login though: https://www.jstor.org/stable/1968843?origin=crossref&seq=1#metadata_info_tab_contents
Simon Wadsley's user avatar