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We prove that, if A is an algebra (over a field of characteristic zero) with no nonzero joint divisor of zero and containing a nonzero central idempotent, and if A satisfies the identities (x p , x q , x r) = 0 and (x p ′ , x q ′ , x r ′) = 0 with exponents p , q , r , p ′ , q ′ , r ′ belonging to {1, 2} such that p ≠ r , (p ′ , q ′ , r ′) ≠ (p , q , r) and (p ′ , q ′ , r ′) ≠ (3 − r , 3 − q , 3 − p) , then A is a unital power-associative algebra. [ABSTRACT FROM AUTHOR]

We study the ideal of polynomial identities of a single indeterminate satisfied by all backcrossing algebras. For this we distinguish two categories according to whether or not these algebras satisfy an identity for the plenary powers. For each category, we give the generators for the vector space of identities, a condition for any object belonging to one of these two categories verify a given identity, a necessary and sufficient condition that a polynomial is an identity and we study the existence of an idempotent element. We give a method which brings the search of identities satified by the backcrossing algebras to the solution of linear systems and we illustrate this method by constructing generators of homogeneous and non homogeneous identities of degrees less than 8. [ABSTRACT FROM AUTHOR]

We describe all endomorphisms of a free trioid of rank 1 and construct a semigroup which is isomorphic to the endomorphism monoid of such free trioid. Also, we give an abstract characteristic for the endomorphism monoid of a free trioid of rank 1 and prove that free trioids are determined by their endomorphism monoids. [ABSTRACT FROM AUTHOR]