## Remarks on endomorphisms of the closure of implicative semilattices.(Russian)Zbl 0621.06002

An algebra $$(L;\wedge,\to,l)$$ of type $$(2,2,0)$$ is called an implicative (Brouwerian) semilattice if $$(L;\wedge,l)$$ is a meet-semilattice with the greatest element $$l$$ and $$\to$$ is the operation of a relative pseudocomplement, i.e. $$a\wedge x\leq b$$ if and only $$x\leq a\to b$$. An endomorphism $$\phi$$ of an implicative semilattice $$L$$ is said to be closed if $$x\leq x\phi$$ and $$x\phi =(x\phi)\phi$$ for every $$x\in L$$.
The main results:
(1) The closed endomorphisms are characterized, for example, $$\phi$$ is a closed endomorphism if and only if $$\phi$$ is isotone and $$x\to (y\phi)=(x\to y)\phi$$ for every $$x, y$$.
(2) The set of all closed endomorphisms forms a distributive lattice (see also C. Tsinakis [Houston J. Math. 5, 427–436 (1979; Zbl 0431.06003)]).
(3) A filter $$F$$ is the kernel of some closed endomorphism if and only if $$F$$ is comonomial, i.e. every congruence class $$\theta [F]$$ possesses a greatest element.

### MSC:

 06A12 Semilattices 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06B10 Lattice ideals, congruence relations 08A35 Automorphisms and endomorphisms of algebraic structures 06D20 Heyting algebras (lattice-theoretic aspects)

Zbl 0431.06003