## 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)

### MSC:

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

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