The lattice of closure endomorphisms of a Hilbert algebra
by
Jānis Cīrulis
Department of Computer Science, University of Latvia, Riga, Latvia

A closure endomorfism of a partially ordered algebra is its endomorphism that happens to be also a closure operator w.r.t. the selected ordering. It is known that closure operators of an implicative semilattice are just its Glivenko operators and form a distributive lattice. We present here some results concerning closure endomorphisms on a Hilbert algebra.

A selfmap \phi of a Hilbert algebra (A, --> , 1) turns out to be a closure endomorphism if and only if it an isotonic and right regular, i.e. satisfies the identity \phi(x --> y) = x --> \phi(y). Right regular maps on A are also known as right multipliers. We show that

(1) the set M of all right multipliers is closed under pointwise defined operation --> an forms an implication algebra (M, --> , \iota), where \iota is the unit map x --> 1;

(2) the identity map \epsilon is in M, and M is closed also under composition o , which serves as join in M;

(3) for any two right multipliers \phi, \psi in M and every a in A, (\phi /\ \psi)(a) = \phi(a) /\ \phi(b), where /\ at the left is the meet operation on M induced by --> , and at the right, the partial operation /\ on A defined by x /\ y : = min{z \colon x <= y --> z} (if x /\ y exists, then it is the g.l.b. of {x, y} in A) ;

(4) the algebra (M, o , /\ , \epsilon, \iota) is a Boolean lattice with complementation - defined by (-\phi)(x) = \phi(x) --> x;

(5) closure endomorphisms of A form a sublattice CE of M,

(6) each \phi in CE is a Glivenko operator w.r.t. the operation /\ on A.

Date received: May 14, 2004