A decomposition of homomorphic images of nearlattices. (English) Zbl 1123.06002

Let \(L\) be a bounded distributive lattice and \(c\in L\). It is well known that if \(c\) has a complement in \(L\) then \(L\) is isomorphic to the direct product \([c)\times (c]\), and if \(c\) is not complemented then the mapping \(\varphi_{c}:L\rightarrow [c)\times (c], \varphi_{c}(x)=(x\vee c,x\wedge c)\), is still an injective homomorphism and one can discuss whether the homomorphic image \(\varphi_{c}(L)\) is a subdirect product of \([c)\times (c]\). A nearlattice is a semilattice \(\mathcal{S}=(S;\vee)\) where for each \(a\in S\) the principal filter \([a)=\{x\in S:a\leq x\}\) is a lattice with respect to the induced order \(\leq \) of \(\mathcal{S}\). In this paper, the authors investigate which of the above results remain true in the case of nearlattices; it turns out that only for the class of so-called nested nearlattices all of these results are valid.


06A12 Semilattices
06B99 Lattices
06D99 Distributive lattices
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