## Ideals, congruences and annihilators on nearlattices.(English)Zbl 1147.06002

A semilattice $$\mathcal{N}=(N,\vee ),$$ where for each $$a\in N$$ the principal filter $$[a)=\{x\in N:a\leq x\}$$ is a lattice with respect to the induced order $$\leq$$ of $$\mathcal{N},$$ is called a nearlattice. For $$a,b,x\in N,$$ a relative annihilator of $$a$$ with respect to $$b$$ is defined to be the set $$\langle a,b\rangle=\{z\in N:z\leq x$$ where $$a\wedge x$$ exists and $$a\wedge x\leq b\}.$$ The purpose of this paper is to characterize some properties like distributivity, modularity or $$0$$-distributivity of nearlattices by means of certain properties of annihilators.

### MSC:

 06A12 Semilattices 06D99 Distributive lattices 06C99 Modular lattices, complemented lattices
Full Text:

### References:

 [1] Abbott J. C.: Semi-boolean algebra. Mat. Vestnik 4 (1967), 177-198. · Zbl 0153.02704 [2] Beazer R.: Hierarchies of distributive lattices satisfying annihilator conditions. J. London Math. Soc. 11 (1975), 216-222. · Zbl 0335.06008 [3] Chajda I., Kolařík M.: A decomposition of homomorphic images of nearlattices. Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 45 (2006), 43-52. · Zbl 1123.06002 [4] Chajda I., Kolařík M.: Nearlattices. Discrete Math., to appear. · Zbl 1151.06004 [5] Cornish W. H.: The free implicative BCK-extension of a distributive nearlattice. Math. Japonica 27, 3 (1982), 279-286. · Zbl 0496.03046 [6] Cornish W. H., Noor A. S. A.: Standard elements in a nearlattice. Bull. Austral. Math. Soc. 26, 2 (1982), 185-213. · Zbl 0523.06006 [7] Davey B.: Some annihilator conditions on distributive lattices. Algebra Universalis 4 (1974), 316-322. · Zbl 0299.06007 [8] Davey B., Nieminen J.: Annihilators in modular lattices. Algebra Universalis 22 (1986), 154-158. · Zbl 0613.06004 [9] Grätzer G.: General Lattice Theory. : Birkhäuser Verlag. Basel, 1978. [10] Halaš R.: Subdirectly irreducible distributive nearlattices. Math. Notes 7, 2 (2006), 141-146. · Zbl 1120.06003 [11] Hickman R.: Join algebras. Communications in Algebra 8 (1980), 1653-1685. · Zbl 0436.06003 [12] Mandelker M.: Relative annihilators in lattices. Duke Math. J. 40 (1970), 377-386. · Zbl 0206.29701 [13] Nieminen J.: The Jordan-Hölder chain condition and annihilators in finite lattices. Tsukuba J. Math. 14 (1990), 405-411. · Zbl 0721.06008 [14] Noor A. S. A., Cornish W. H.: Multipliers on a nearlattices. Commentationes Mathematicae Universitatis Carolinae (1986), 815-827. · Zbl 0605.06005 [15] Scholander M.: Trees, lattices, order and betweenness. Proc. Amer. Math. Soc. 3 (1952), 369-381. [16] Scholander M.: Medians and betweenness. Proc. Amer. Math. Soc. 5 (1954), 801-807. · Zbl 0056.26101 [17] 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.