RETRACTS OF TREES AND FREE LEFT ADEQUATE SEMIGROUPS

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.