Stable varieties of semigroups and groupoids

The paper deals with $\Sigma-$composition and $\Sigma$-essential composition of terms, which lead to stable and s-stable varieties of algebras. A full description of all stable varieties of semigroups, commutative and idempotent groupoids is obtained. We use an abstract reduction system which simplifies the presentations of terms of type $\tau=(2)$ to study the varietiy of idempotent groupoids and s-stable varieties of groupoids. They are used as an alternating of the stable varieties, aiming to highlight replacing the subterms of a term in a deductive systems instead of the usual replacing the variables with terms.