Two Classes of Semilattices with Retractive Properties

Two possible properties of semilattices are studied: absolute retractivity (being a retract of all super-semilattices) and retractivity of all sub-semilattices. Semilattices with given retractivity properties are described. It is proved that the absolute retractivity of a semilattice is equivalent to its completeness. It is shown that all sub-semilattices of a semilattice are its retracts if and only if the upper cones of all elements of the semilattice are chains and each chain in the semilattice is discrete.