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Independent axiom systems for nearlattices. (English) Zbl 1249.06003

There exist several axiom systems for nearlattices. The authors checked two of them, namely the axiom system by R. Hickman [Commun. Algebra 8, 1653–1685 (1980; Zbl 0436.06003)] and the other by Chajda et al. (see, e.g., [I. Chajda, R. Halaš and J. Kühr, Semilattice structures. Lemgo: Heldermann Verlag (2007; Zbl 1117.06001)]). They show that both systems are redundant and, using of the automated deduction tool Prover9, they found which of the axioms can be omitted. Moreover, they set up a new axiom system for nearlattices consisting of two axioms in five variables.

MSC:

06A12 Semilattices
06B75 Generalizations of lattices
68T15 Theorem proving (deduction, resolution, etc.) (MSC2010)

Software:

Prover9; Mace4
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References:

[1] I. Chajda, R. Halaš: An example of a congruence distributive variety having no near-unanimity term. Acta Univ. M. Belii Ser. Math. 13 (2006), 29–31. · Zbl 1132.08002
[2] I. Chajda, R. Halaš, J. Kühr: Semilattice structures. Research and Exposition in Mathematics 30, Heldermann Verlag, Lemgo, 2007.
[3] I. Chajda, M. Kolařík: Nearlattices. Discrete Math. 308 (2008), 4906–4913. · Zbl 1151.06004
[4] R. Hickman: Join algebras. Commun. Algebra 8 (1980), 1653–1685. · Zbl 0436.06003
[5] W. McCune: Prover9 and Mace4, version 2009-11A. ( http://www.cs.unm.edu/mccune/prover9/ ).
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