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JP-semilattice; meet semilattice; nearlattice; partial lattice
In recent papers, S. N. Begum and A. S. A. Noor have studied join partial semilattices (JP-semilattices) defined as meet semilattices with an additional partial operation (join) satisfying certain axioms. We show why their axiom system is too weak to be a satisfactory basis for the authors' constructions and proofs, and suggest an additional axiom for these algebras. We also briefly compare axioms of JP-semilattices with those of nearlattices, another kind of meet semilattices with a partial join operation.
[1] Begum, S. N., Noor, A. S. A.: Some characterizations of modular and distributive JP-semilattices. Atl. Electron. J. Math. 4 (2011), 56-69. MR 2900988
[2] Begum, S. N., Noor, A. S. A.: Congruence kernels of distributive PJP-semilattices. Math. Bohem. 136 (2011), 225-239. MR 2893973 | Zbl 1249.06004
[3] Chajda, I., Seidl, Z.: An algebraic approach to partial lattices. Demonstr. Math. 30 (1997), 485-494. MR 1601809 | Zbl 0910.06006
[4] Cīrulis, J.: Subtractive nearsemilattices. Proc. Latv. Acad. Sci., Sect. B, Nat. Exact Appl. Sci. 52 (1998), 228-233. MR 1788173 | Zbl 1027.06007
[5] Cīrulis, J.: Hilbert algebras as implicative partial semilattices. Centr. Eur. J. Math. 5 (2007), 264-279. DOI 10.2478/s11533-007-0008-2 | MR 2300273 | Zbl 1125.03047
[6] al., G. Grätzer et: General Lattice Theory. Second edition. Birkhäuser, Basel (1998). MR 1670580
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