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Sorry if this is a silly question.

Define that a right-cancellative semigroup is a set $G$ together with an associative operation $*$ such that for all $a,b,x \in G$ it holds that $ax=bx \Rightarrow a=b$. Do right-cancellative semigroups form a variety of algebras?

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A variety of algebras is closed under quotients (and under products and substructures; these are the three conditions in Birkhoff's Theorem). But the cancellation property breaks down for quotients. For example, consider the semigroup defined by generators and relations $\langle a,b,x : ax=bx \rangle$. It is not cancellative, but it is a quotient of the free semigroup in three generators, which is cancellative.

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