## 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.

### MSC:

 06A12 Semilattices 06B99 Lattices 06D99 Distributive lattices
Full Text:

### References:

 [1] Chajda I., Kolařík M.: Nearlattices. Discrete Math., submitted. · Zbl 1151.06004 [2] Cornish W. H.: The free implicative BCK-extension of a distributive nearlattice. Math. Japonica 27, 3 (1982), 279-286. · Zbl 0496.03046 [3] Cornish W. H., Noor A. S. A.: Standard elements in a nearlattice. Bull. Austral. Math. Soc. 26, 2 (1982), 185-213. · Zbl 0523.06006 [4] Grätzer G.: General Lattice Theory. : Birkhäuser Verlag, Basel. 1978. · Zbl 0436.06001 [5] Noor A. S. A., Cornish W. H.: Multipliers on a nearlattices. Comment. Math. Univ. Carol. (1986), 815-827. · Zbl 0605.06005 [6] Scholander M.: Trees, lattices, order and betweenness. Proc. Amer. Math. Soc. 3 (1952), 369-381. [7] Scholander M.: Medians and betweenness. Proc. Amer. Math. Soc. 5 (1954), 801-807. · Zbl 0056.26101 [8] Scholander M.: Medians, lattices and trees. Proc. Amer. Math. Soc. 5 (1954), 808-812. · Zbl 0056.26201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.