Weak relative pseudo-complements of closure operators. (English) Zbl 0901.06003

Let \(\mathcal L=\langle L,\leq ,\wedge \rangle \) be a meet semilattice, \(x\), \(y\in L\), \(y\leq x.\) A weak relative pseudocomplement of \(x\) w.r.t. \(y,\) if it exists, is the unique element \(\operatorname {wr}(x,y)\in L\) s.t.
{(i)} \(x\wedge \big (\operatorname {wr}(x,y)\big)=y\)
{(ii)} for each \(z\in L,\) if \(x\wedge z=y,\) then \(z\leq \operatorname {wr}(x,y).\)
This notion is strictly weaker than relative pseudocomplement and stronger than pseudocomplement. For a complete lattice \(\mathcal L\) an upper closure operator on \(\mathcal L\) is a mapping \(\rho \: L\to L\) s.t. \(x\leq \rho (x)\), \(\rho \big (\rho (x)\big)=\rho (x)\) and \(x\leq y\Rightarrow \rho (x) \leq \rho (y).\) Now \(\rho \) is called continuous if \(\rho (\vee C)=\vee \rho (c)\) for any chain \(C\) in \(L.\) The set \(\operatorname {uco} (\mathcal L)\) of all closure operators form a complete lattice with respect to the natural pointwise ordering \(\sqsubseteq .\) The main result is the following Theorem: If \(\mathcal L\) is a complete, meet-continuous lattice, then for all \(\rho \), \(\eta \in \operatorname {uco} (\mathcal L)\) such that \(\eta \sqsubseteq \rho \) and \(\eta \) is continuous, there exists \(\operatorname {wr}(\rho ,\eta).\)
Reviewer: R.Halaš (Olomouc)


06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B23 Complete lattices, completions
06A12 Semilattices
06D15 Pseudocomplemented lattices
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