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Conditional associativity in orthomodular lattices

Jeannine J. M. Gabriëls
From the journal Mathematica Slovaca

Abstract

For orthomodular lattices the word problem is still not decided, an important obstacle is the lack of distributivity. But also the absence of the associativity law for operations other than the lattice operations meet and join causes difficulties. In this paper we treat some aspects of associativity in orthomodular lattices.

There are six out of 96 orthomodular operations which are associative. We search for orthomodular lattice operations which fulfil the associative identity under some conditions. Specifically we assume that some arguments commute and identify, for each operation, sufficient conditions under which a triple of elements fulfils the respective associativity identity.

MSC 2010: Primary 06C15

(Communicated by Mirko Navara)


Acknowledgement

The author thanks the anonymous reviewers for careful proofreading.

References

[1] Beran, L.: Orthomodular Lattices. Algebraic Approach, Academia, Praha, 1984. Search in Google Scholar

[2] Boone, W. W.: The word problem, Proc. Natl. Acad. Sci. USA 44(10) (1958), 1061–1065. Search in Google Scholar

[3] Bruns, G.: Free ortholattices, Canad. J. Math. 28 (1976), 977–985. Search in Google Scholar

[4] Dedekind, R.: Über die drei Moduln erzeugte Dualgruppe, Math. Ann. 53 (1900), 371–403. Search in Google Scholar

[5] D’Hooghe, B.—Pykacz, J.: On some new operations on orthomodular lattices, Internat. J. Theor. Phys. 39 (2000), 641–652. Search in Google Scholar

[6] Freese, R.: Free modular lattices, Trans. Amer. Math. Soc. 261 (1980), 81–90. Search in Google Scholar

[7] Gabriëls, J. J. M.—Navara, M.: Associativity of operations on orthomodular lattices, Math. Slovaca 62 (2012), 1069–1078. Search in Google Scholar

[8] Gagola III, S. M.—Gabriëls, J. J. M.—Navara, M.: Weaker forms of associativity in orthomodular lattices, Algebra Universalis 73 (2015), 249–266. Search in Google Scholar

[9] Herrmann, C.: On the word problem for the modular lattice with four free generators, Math. Ann. 265 (1983), 513–527. Search in Google Scholar

[10] Hyčko, M.: Implications and equivalences in orthomodular lattices, Demonstratio Math. 38 (2005), 777–792. Search in Google Scholar

[11] Hyčko, M.: Computations in OML, http://www.mat.savba.sk/~hycko/oml (7 June 2011). Search in Google Scholar

[12] Kalmbach, G.: Orthomodular Lattices, Academic Press, London, 1983. Search in Google Scholar

[13] Kröger, H.: Zwerch-Assoziativität und verbandsähnliche Algebren, In: Sonderdruck 3 aus den Sitzungsberichten 1973, Mathematisch-Naturwissenschaftliche Klasse, Bayer. Akad. Wiss. Philos., pp. 23–48. Search in Google Scholar

[14] Megill, N. D.—Pavičić, M.: Orthomodular lattices and a quantum algebra, Internat. J. Theoret. Phys. 40 (2001), 1387–1410. Search in Google Scholar

[15] Megill, N. D.—Pavičić, M.: Equivalencies, identities, symmetric differences, and congruences in orthomodular lattices, Internat. J. Theoret. Phys. 42 (2003), 2797–2805. Search in Google Scholar

[16] Megill, N. D.—Pavičić, M.: Quantum implication algebras, Internat. J. Theoret. Phys. 42 (2003), 2807–2822. Search in Google Scholar

[17] Navara, M.: On generating finite orthomodular sublattices, Tatra Mt. Math. Publ. 10 (1997), 109–117. Search in Google Scholar

[18] Novikov, P. S.: On the algorithmic unsolvability of the word problem in group theory, Proc. Steklov Inst. Math. 44 (1955), 1–143. (Russian) Search in Google Scholar

[19] O’Connor, J. J.—Robertson, E. F.: http://www-history.mcs.st-andrews.ac.uk/PrintHT/Word_problems.html (14 September 2014). Search in Google Scholar

[20] Whitman, P. M.: Free lattices, Ann. of Math. 42 (1941), 325–330. Search in Google Scholar

[21] Whitman, P. M.: Free lattices II, Ann. of Math. 43 (1942), 104–115. Search in Google Scholar

Received: 2015-4-10
Accepted: 2015-9-17
Published Online: 2017-6-5
Published in Print: 2017-6-27

© 2017 Mathematical Institute Slovak Academy of Sciences