Relative Algebraisierbarkeit von Untervoll-Verbänden eines Unteralgebrensystems

LetG be a closure system on a setA, and let ℌ ⊂G be a closure system consisting of allg-closed elements ofG, whereg is a family of unary operations onA. In this note we discuss whether ℌ is a closed sublattice ofG-this means, whether arbitrary suprema taken in ℌ, and taken inG, coincide. An example shows that ℌ is not a closed sublattice ofG in general. We give in theorem 2 a necessary and sufficient condition ofG that all unary-defined ℌ ⊂G are closed sublattices ofG. But of course very fewG fulfill this condition. For an arbitrary closure-systemG we get with the aid of the concept of ‘relative character’ and of theorem 1: IfG is of character ≤k then (theorem 3) a unary-defined ℌ is a closed sublattice ofG iff thek-small suprema of ℌ and ofG coincide. The fact that a closure-system is of character ≤k iff it is obtainable as the subalgebra-system of an at mostk-ary universal algebra now implies: IfG is the subalgebra-system Sub (A, f) of a universal algebra (A, f) then (theorem, 4) ℌ is a closed sublattice ofG iff ℌ is definable by endomorphisms of (A, f). If one defines automorphisms of a closure system in an obvious way, theorem 5 gives another sufficient condition for ℌ to be a closed sublattice ofG.