6533b826fe1ef96bd1284577

RESEARCH PRODUCT

Lattice of closure endomorphisms of a Hilbert algebra

Jānis Cı̄rulis

subject

Pure mathematicsEndomorphismHilbert algebraGeneral Mathematics010102 general mathematicsAstrophysics::Instrumentation and Methods for AstrophysicsClosure (topology)Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing)010103 numerical & computational mathematics01 natural sciencesSet (abstract data type)Lattice (module)Computer Science::General LiteratureClosure operator0101 mathematicsMathematics

description

A closure endomorphism of a Hilbert algebra [Formula: see text] is a mapping that is simultaneously an endomorphism of and a closure operator on [Formula: see text]. It is known that the set [Formula: see text] of all closure endomorphisms of [Formula: see text] is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of [Formula: see text], anti-isomorphic to the lattice of certain closure retracts of [Formula: see text], and compactly generated. The set of compact elements of [Formula: see text] coincides with the adjoint semilattice of [Formula: see text]; conditions under which two Hilbert algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian extensions) are discussed. Several consequences are drawn also for implication algebras.

yearjournalcountryeditionlanguage
2019-09-03Asian-European Journal of Mathematics
https://doi.org/10.1142/s1793557119500827