Orthomodular Lattices

Elżbieta Mądra 1  and Adam Grabowski 1
  • 1 Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok Poland

Orthomodular Lattices

The main result of the article is the solution to the problem of short axiomatizations of orthomodular ortholattices. Based on EQP/Otter results [10], we gave a set of three equations which is equivalent to the classical, much longer equational basis of such a class. Also the basic example of the lattice which is not orthomodular, i.e. benzene (or B6) is defined in two settings - as a relational structure (poset) and as a lattice.

As a preliminary work, we present the proofs of the dependence of other axiomatizations of ortholattices. The formalization of the properties of orthomodular lattices follows [4].

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  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

  • [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

  • [3] Grzegorz Bancerek. Complete lattices. Formalized Mathematics, 2(5):719-725, 1991.

  • [4] Ladislav Beran. Orthomodular Lattices. Algebraic Approach. Academiai Kiado, 1984.

  • [5] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

  • [6] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

  • [7] Adam Grabowski. Robbins algebras vs. Boolean algebras. Formalized Mathematics, 9(4):681-690, 2001.

  • [8] Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Formalized Mathematics, 6(1):117-121, 1997.

  • [9] Adam Grabowski and Markus Moschner. Formalization of ortholattices via orthoposets. Formalized Mathematics, 13(1):189-197, 2005.

  • [10] W. McCune, R. Padmanabhan, M. A. Rose, and R. Veroff. Automated discovery of single axioms for ortholattices. Algebra Universalis, 52(4):541-549, 2005.

  • [11] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.

  • [12] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.

  • [13] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

  • [14] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215-222, 1990.


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