Retract lattices

Abstract

In this article we study retract and weakly retract lattices — lattices whose congruences are generated by retractions or weakly retractions, respectively. A retraction (weak retraction) of a lattice is any idempotent lattice (semilattice) endomorphism.
We have obtained the structural properties of retract and weakly retract lattices (section 2).
It is proved that the class of all retract lattices is closed under homomorphic images (Theorem 1), finite direct products (Theorem 2), direct sums (Theorem 4), and passage to dual lattices (Remark 13), but not under taking sublattices (Proposition 1) and ordinal sums (Example 12). Example 11 shows that the finite products of chains are retract lattices. A wider class of weakly retract lattices is closed under homomorphic images, finite direct products, direct sums, and ordinal sums (Theorem 3).
In section 3, preliminary results are presented on the retractions of the direct product of an 𝑚-element and an 𝑛-element chain (Proposition 2, Examples 13 and 14). The problem of finding the number of retractions of such a product is posed.
Section 4 contains the first author’s results on the structure of retract semilattices, which complement the results on retract and weakly retract lattices.
Explanatory notes are made.