Abstract
In this paper a cubic lattice L(S) is endowed with a symmetric implication structure and it is proved that L(S) \ {0} is a power of the three-element simple symmetric implication algebra. The Metropolis–Rota’s symmetries are obtained as partial terms in the language of symmetric implication algebras.
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Abbott, J.C.: Semi-boolean algebras. Mat. Vesn. 4(19), 177–198 (1967)
Abbott, J.C.: Implicational algebras. Bull. Math. Soc. Sci. Math. R\(\acute{e}\)pub. Social. Roum. 11, 3–23 (1967)
Bailey, C.G., Oliveira, J.S.: An axiomatization for cubic algebras. In: Sagan, B.E., Stanley, R.P. (eds.) Mathematical Essays in Honor of Gian-Carlo Rota. Birkhauser, Cambridge, MA (1998)
Bennet, M.K.: The face lattice of an n-dimensional cube. Algebra Univers. 14, 82–86 (1982)
Chen, W.Y.C., Oliveira, J.S.: Implication algebras and the Metropolis–Rota axioms for cubic lattices. J. Algebra 171, 383–386 (1995)
Díaz Varela, J.P., Torrens, A.: Decomposability of free tarski algebras. Algebra Univers. 50, 1–5 (2003)
Metropolis, N., Rota, G.C.: Combinatorial structure of the faces of the n-cube. SIAM J. Appl. Math. 35, 689–694 (1978)
Monteiro, A.: Sur les algèbres de Heyting symmétriques. Port. Math. 39(Fasc 1–4), 1–237 (1980)
Oliveira, J.S.: The theory of cubic lattices. PhD thesis, MIT (1992)
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Abad, M., Varela, J.P.D. Representation of Cubic Lattices by Symmetric Implication Algebras. Order 23, 173–178 (2006). https://doi.org/10.1007/s11083-006-9041-1
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DOI: https://doi.org/10.1007/s11083-006-9041-1