Let $A_{2}$ be a free associative algebra or polynomial algebra of rank two over a field of characteristic zero. The main results of this paper are the classification of noninjective endomorphisms of $A_{2}$ and an algorithm to determine whether a given noninjective endomorphism of $A_{2}$ has a nontrivial fixed element for a polynomial algebra. The algorithm for a free associative algebra of rank two is valid whenever an element is given and the subalgebra generated by this element contains the image of the given noninjective endomorphism.

An endomorphism φ of a polynomial ring is said to preserve outer rank if φ sends each polynomial to one with the same outer rank. For the polynomial ring in two variables over a field of characteristic 0 we prove that an endomorphism φ preserving outer rank is an automorphism if one of the following conditions holds: (1) the Jacobian of φ is a nonzero constant; (2) the image of φ contains a coordinate; (3) φ has a ‘fixed point’.

Recent research of the author has studied edge-labelled directed trees under a natural multiplication operation. The class of all such trees (with a fixed labelling alphabet) has an algebraic interpretation, as a free object in the class of adequate semigroups. We consider here a natural subclass of these trees, defined by placing a restriction on edge orientations, and show that the resulting algebraic structure is a free object in the class of left adequate semigroups. Through this correspondence we establish some structural and algorithmic properties of free left adequate semigroups and monoids, and consequently of the category of all left adequate semigroups.

Let P be a finite, connected partially ordered set containing no crowns and let Q be a subset of P. Then the following conditions are equivalent: (1) Q is a retract of P; (2) Q is the set of fixed points of an order-preserving mapping of P to P; (3) Q is obtained from P by dismantling by irreducibles.